# Question about total absolute magnitudes of galaxies - negative or not?

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I have seen this in a couple of places, but when the authors in the paper - http://chandra.as.utexas.edu/~kormendy/kfcb-accepted.pdf - wrote:

"All 10 of our ellipticals with total absolute magnitudes M_VT ≤ −21.66 have cuspy cores - “missing light” - at small radii. Cores are well known and naturally scoured by binary black holes formed in dissipationless (“dry”) mergers. All 17 ellipticals with −21.54 ≤ M_VT ≤ −15.53 do not have cores."

Since absolute magnitude is the magnitude of that object at 10 pc away from Earth, then this is why the galaxies have such negative (meaning, very bright) values for total absolute magnitude, correct?

However, in this paper - http://arxiv.org/pdf/astro-ph/9602044v1.pdf - the author writes:

"The morphological properties of galaxies between 21 mag < I < 25 mag in the Hubble Deep Field are investigated using a quantitative classification system based on mea- surements of the central concentration and asymmetry of galaxian light."

The galaxies here are between 21 to 25 absolute magnitudes (?) in the infrared (I?) band, but why are these magnitudes not negative?

The morphological properties of galaxies between 21 mag < I < 25 mag in the Hubble Deep Field are investigated using a quantitative classification system based on mea- surements of the central concentration and asymmetry of galaxian light.

Those are apparent magnitudes, not absolute.

The convention is to use $M_X$ for absolute magnitude and just $X$ for apparent mag (or m$_X$), where X is the bandpass symbol (U, B, V, R, I, J, L, H, K, etc).

And, the $V_T$ and $M_{V_T}$ (note the correct subscripting, which the authors failed to do) means the V band radial profile is extrapolated to infinite radius to give a total for the whole galaxy.

## Question about total absolute magnitudes of galaxies - negative or not? - Astronomy

If we have fluxes or magnitudes at various wavelengths, and redshifts or other distance measures, one may examine group properties of large numbers of galaxies to look for features that can tell about galaxy formation or evolution, as well as being essential to understanding the present-epoch population of galaxies. One might look at

Observational selection effects must first be understood when they cannot be eradicated. The foremost example is Malmquist bias, the fact that in a typical flux-limited sample we see only atypically bright objects at larger distances. This is a major fact of life in extragalactic astronomy as Trimble (1996, PASP 108, 1073) points out, "Any large gathering of observational cosmologists today will include at least one person who thinks that someone else in the room does not understand the Malmquist effect". To allow for this effect, we need to know or guess the luminosity function (LF). As a simple example, take a class of objects with a uniform spatial distribution and a Gaussian LF extreme values only occur within large volumes and thus at large distances, and the detection threshold (slanting line in the picture, where we detect only objects above it) means that the mean luminosity of the sample grows with distance even if the population does not change at all. The situation will look like this:

For real galaxies the situation is even worse, because the LF is very steep and very deep this is why we don't know much about the population of dwarf galaxies. It is always safe (but seldom possible) to search much deeper than you might need to - otherwise elaborate statistical manipulations, such as survival analysis, will be needed to reconstruct the true properties of the sample distribution. Clusters of galaxies are popular for studies of the luminosity function for the same reason that star clusters are - distance-dependent effects are usually insignificant within a single cluster, so that the galaxy population in the cluster may be evaluated free of Malmquist bias. However, using clusters at a range of distances suffers not only from the Malmquist bias, but from the Scott effect - the fact that to be recognized as such at great distances, clusters become less and less typical in population. Note also all the selection effects mentioned at the outset of the course - surface brightness and compactness limits - mean that some kinds of galaxies are barely represented in existing catalogs. These selection effects are especially damaging for distance-scale problems.

Statistics with Hubble type

The number of galaxies of various Hubble types in a magnitude-limited sample is typified by these numbers from the RSA catalog:

 Ordinary Barred E+E/S0 173 SB0+SB0/Sba 48 S0+S0/a 142 SBa+SBab 42 Sa+Sab 123 SBb+SBbc 96 Sb+Sbc 187 SBc 77 Sc 293 SBcd+SBd 8 Scd+Sd 26 SBm+IBm 9 Sm+Im 13 . . S 16 . . Special 18 . . Totals 991 . 285

Types Sd,Sm are underrepresented in this flux-limited compilation because they are intrinsically fainter than the earlier spiral classes Sa-Sc. Some ellipticals (the sequence continuing into dwarfs) have similar problems. Only the types S0-Sc are probably fairly represented - these are giant galaxies and can be seen at large distances. If we regard Hubble type as mapping a continuous structural variable, the number of galaxies tells us about the bin widths of the Hubble classes in this variable.

Correlations with Hubble type may be examined in detail by using de Vaucouleurs' type index T, assigned as follows:

 Type E E/S0 S0 Sa Sb Sc Sd Im T -4 -2 0 1 3 5 7 10

A further luminosity-class index L (ranging 1-5) is defined for spirals and irregulars. The joint distribution of these for galaxies in the RC2 is given by de Vaucouleurs 1977 (Evolution of Galaxies and Stellar Populations, p. 43). Many useful quantities correlate with T, as shown in his Fig. 2.

Later-type galaxies are fainter in the mean - the scatter is quite large. Note that corrections for internal extinction needed to be made. As well as total optical luminosity and H I content, optical spectra and therefore color indices that relate to SFR history change with Hubble type. Some well-known examples are the UBV system indices U-B,B-V. These three passbands are centered near 3500, 4300, and 5800 Angstroms with passband widths 600-1400 Angstroms. Fig. 6 from de Vaucouleurs 1977 shows their variation (integrated across the whole galaxy) with T.

Early types E/S0 have red colors, as expected for systems with very low SFR. Later types have bluer colors, indicating a larger relative rate of recent star formation. This test alone does not tell whether this is due only to bulge/disk variations, chemical abundance effects or a real difference in the disk histories (I recall a very probing conversation with Sandy Faber about this, as a lowly grad student). We may regard this as a very low-spectral-resolution kind of spectral synthesis. The color-type relation is shown in this figure from Roberts and Haynes (1994 ARA&A 32, 115, reproduced from the ADS). That review also summarizes the evidence for changes in chemical abundance, dynamical mass, and H I content along the Hubble sequence.

It has long been known that the colors of E/S0 galaxies form a very well-defined sequence (the red sequence), reddening for brighter galaxies due to metallicity. It took the large uniform data set from the SDSS, augmented by GALEX, to show that star-forming galaxies have a set of colors which is surprisingly well defined in its own right, the blue sequence or blue cloud there is a genuine minimum between the two populations (the green valley). This is shown in a color-derived mass plot (courtesy of Kevin Schawinski) from SDSS data:

The color bimodality is similar to the morphologcal dichotomy (E/S0 versus spiral/irregular), but not identical. There exist populations of blue early-type galaxies and red spirals, with blue ellipticals most numerous in low-density environments and red spirals just outside the densest regions (Bamford et al. 2009 MNRAS 339, 1324). The "green valley" is too sparse for many red galaxies to become blue by adding starbursts it is potentially very important that these galaxies in transition have the highest probability of hosting AGN.

An important description of the distribution and occurrence of galaxies is the luminosity function &Phi : &Phi(L) dL is the number of galaxies in the interval L +/- dL/2 per unit volume. This may be defined for optical luminosity, radio power, far-IR power, . One always has the hope that this function is fundamental in telling how galaxy masses are distributed that is, that all kinds of galaxies have about the same visible to invisible mass ratio. The determination of &Phi over a wide luminosity range is complicated by Malmquist bias, and the need to reduce all measurements to a common emitted-wavelength frame - this is a special problem for QSOs and very luminous galaxies, for which we must look to significant redshifts to see any of the brightest examples. The luminosity function may be determined, in principle, very simply for objects in luminosity bin i, the luminosity function is simply

where Vm is the volume within which each object could have been detected, and the sum runs over all objects in bin i. All the selection effects fall into determining Vm for a given threshold condition, which may be nontrivial. Actually, it always seems to be nontrivial. An important application of Vmis the Schmidt V/Vm test, which can show whether the sample is complete or at least uniform, and can show the presence of some kinds of evolution with cosmic time when applied over a large redshift range. If the objects are uniformly distributed within the survey volume, the sample mean of the statistic V/Vm, where V is the volume of a sphere centered here and with the object at its surface, will be 1/2. [For cosmological applications one must worry about whether this is the right prescription for the volume of the sphere, integrating volume elements to the distance in question.] A value smaller than this implies that there are more objects close to us, which for galaxies normally means that the sample is more incomplete at large distances than we initially assumed. A mean value greater than 1/2 almost always implies cosmological evolution, as for QSOs. The fact that gamma-ray bursts show a value significantly smaller than 1/2 even for the most complete flux samples is one of their major puzzles.

From the magnitude-limited RSA catalog, the redshift distribution of catalog entries is shown here (less a single object at 9875 km/s).

From the wide range of cz we see that the volumes sampled at various luminosities differ by factors of order 10 6 . Thus careful allowances for sample properties is crucial to measuring the LF. This is clear when comparing the distribution of apparent and absolute magnitude shown in the figures below (again from the RSA, skipping three naked-eye members of the Local Group):

To derive the proper relative numbers, one must correct for the different volumes within which each object would appear in the catalog. This apparent distribution declines fainter than absolute blue magnitude -21.5, while the space density continues to increase greatly to fainter absolute magnitudes.

An important analytic approximation to the overall galaxy luminosity function is the Schechter (1976, ApJ 203, 297) form &Phi (L) dL = &phi* (L/L*) &alpha e -(L/L*) (dL/L*) where &phi * (L/L*) is the normalizing factor, set by the number of galaxies per Mpc 3 , L* is a characteristic luminosity, and &alpha is an asymptotic slope to be fit a value around -5/4 usually agrees with the data. The plot (from Schechter's paper, reproduced by permission of the AAS) shows the fit to the mean of galaxy counts in 13 clusters.

L* appears to be constant among various clusters, and maybe even for non-cluster galaxies, at a given cosmic epoch, so that one may read references to "an L* galaxy". This is sometimes taken as a characteristic scale for galaxy formation. The brightest cD galaxies may require some additional process they may violate the LF shape in that there should be virtually no galaxies so luminous in the observable Universe if the Schechter function held absolutely.

Different kinds of galaxies have different LF shapes and normlizations this explains why Hubble thought of the LF as approximately Gaussian, from studies of (giant) spirals, while Zwicky counted everything, dissented vigorously and as usual correctly, and found a divergence at faint magnitudes. Zwicky distinguished dwarf, pygmy, and gnome galaxies (see his idiosyncratic book Morphological Astronomy). The LF must converge somewhere to avoid Olbers' paradox. The LF is simple only for dwarfs the various types are distributed in Virgo as follows, from Fig. 1 of Binggeli, Sandage, and Tammann 1988 (Ann. Rev. 26, 509 - an excellent reference for the whole topic, figure reproduced from the ADS).

The differences may be clues to how different galaxy types form - in some biassing schemes, for example, ellipticals need stronger peaks than disks. On the other hand, if merging is important, this may tell us about the history of mergers rather than galaxy formation. It does seem to be quite consistent in shape among clusters of galaxies, so that it tells something basic and general about how galaxies have developed.

Similar clues are hidden in some of the basic correlations among global galaxy properties involving dynamics - the Tully-Fisher and Faber-Jackson relations. The Tully-Fisher relation, often employed as a distance indicator for spirals, is a tight relation between galaxy absolute magnitude and velocity scale of the disk (for example, at the 20% - of - peak level in an integrated H I profile, with appropriate inclination corrections). There are broad theoretical reasons why such a relation might hold, but no deep understanding at this point. The Faber-Jackson relation was also found empirically, from the fact that elliptical-galaxy luminosity and central velocity dispersion &sigma are related approximately as L

&sigma 4 . A generalization, the fundamental plane, was found by noting that scatter about the F-J relation is correlated with metallicity (usually through a simple index of Mg absorption), although it turned out that this may not be the most basic parametrization. Not only is the fundamental plane a useful distance and environment probe, but it sets strong constraints on dynamical evolution any transformations of galaxies must leave them very close to this plane. Since (in log space) the fundamental plane is, as far as we can tell, a plane, there are transformations of variables which correspond to orthogonal variables imbedded in it Burstein and coworkers have explored the interpretation of these so-called &kappa parameters.

Basics of the "fundamental plane" may be found in the review by Kormendy and Djorgovski (1989, ARA&A 27, 235). Their Figure 2 (reproduced from the ADS) compares the observed Faber-Jackson relation (upper left) to more exact projections of the galaxy distribution in the volume of radius, surface brightness, and velocity dispersion.

In the observable parameters, R

&sigma 1.4 +/- 0.15 I -0.9 +/- 0.1 where R is an effective or core (but not isophotal) radius, &sigma is the velocity dispersion (one-dimensional, in the line of sight), and I is an averaged intensity (commonly the mean within the effective radius). Some of the earlier relations, such as L

&sigma 4/3 for diameter to reach a particular mean surface brightness, are projections of this plane along different observable axes. One mapping of particular theoretical interest (in which galaxies are widely spread) is the &sigma - &mu plane, corresponding roughly to the density - cooling rate prescription needed to describe a galaxy's initial collapse. The virial theorem suggests a relation of about the FP form, with departures of the scaling exponents from 2 and 1 coming about if there is systematic variation in the (M/L) ratio with luminosity or other global parameters. The narrowness of the fundamental plane tells us that evolution by merging, if it is significant for ellipticals, must carry galaxies along but not across the plane. There are simulations suggesting that the FP parameters are indeed preserved during (some kinds of) merging. Recent work indicates the the fundamental plane shifts at least in luminosity with redshift, as expected for a broad class of galaxy-evolution schemes.

## Magnitude

The apparent magnitude, symbol: m , is a measure of the brightness of a star, etc., as observed from Earth. Its value depends on the star's luminosity (i.e. its intrinsic brightness), its distance, and the amount of light absorption by interstellar matter between the star and Earth. In ancient times the visible stars were ranked in six classes of apparent magnitude: the brightest stars were of first magnitude and those just visible to the naked eye were of sixth magnitude. This system became inadequate as fainter stars were discovered with the telescope and instruments became available for measuring apparent brightness. In the 1850s it was proposed that the physiological response of the eye to a physical stimulus was proportional to the logarithm of that stimulus (Weber-Fechner law). A difference in apparent magnitude of two stars is thus proportional to the difference in the logarithms of their brightness, i.e. to the logarithm of the ratio of their brightness.

In order to make the magnitude scale precise the English astronomer N.R. Pogson proposed, in 1856, that a difference of five magnitudes should correspond exactly to a brightness ratio of 100 to 1. (W. Herschel had shown this to be approximately true.) Hence two stars that differ by one magnitude have a brightness ratio of 100 1/5 :1, i.e. a ratio – known as the Pogson ratio – of 2.512. A star two magnitudes less than another is (2.512) 2 , i.e. 6.3 times, brighter, and so on. In general the apparent magnitudes m 1 and m 2 of two stars with apparent brightness I 1 and I 2 are related by:

The Pogson scale, based on the Pogson ratio, is now the universally adopted scale of magnitude (see table). The zero of the scale was established by assigning magnitudes to a group of standard stars near the north celestial pole, known as the North Polar Sequence, or more recently by photoelectric measurements. The class of magnitude-one stars was found to contain too great a range of brightness and zero and negative magnitudes were consequently introduced: the higher the negative number, the greater the brightness. The scale also had to be extended in the positive direction as fainter objects were discovered. Values can be recorded in tenths, hundredths, even thousandths of a magnitude.

Originally apparent magnitude was measured by eye – visual magnitude, m vis – usually in conjunction with an instrument by which brightnesses could be compared. It is now measured much more accurately by photometric techniques but previous to that photographic methods were used. Photographic magnitudes, m pg, are determined from the optical density of images on ordinary film, i.e. film that has a maximum response to blue light. Photo?-visual magnitudes, m pv, are measured using film that has been sensitized to light – yellowish green in color – to which the human eye is most sensitive. In the early International Color System these two magnitudes were measured with films having a maximum response to a wavelength of 425 nanometers (m pg) and 570 nm (m pv ), the magnitudes being equal for A0 stars.

A suitable combination of photometer and filter can select light or other radiation of a desired wavelength band and measure its intensity. These photoelectric magnitudes can be measured over either narrow or broad bands. The UBV system of stellar magnitudes is based on photoelectric photometry and has been widely adopted as the successor of the International System. The photoelectric magnitudes, denoted U , B , and V are measured at three broad bands: U (ultraviolet radiation) centered on a wavelength of 365 nm, B (blue) centered on 440 nm, and V (visual, i.e. yellowish green) centered on 550 nm. These magnitudes can also be written m U, m B, and m V. The zero point of the UBV system is defined in terms of standard stars having a carefully studied and agreed magnitude. Another important photometric system, the uvby system, uses filters passing narrower wavelength bands than in the UVB system.

The UBV system has been extended by the use of magnitudes at red and infrared wavelengths. The photometric designations are R (700 nm, i.e. 0.7 μm), I (0.9 μm), J (1.25 μm), H (1.6 μm), K (2.20 μm), L (3.40 μm), M (5.0 μm), N (10.2 μm), Q (21 μm). The designations JQ relate to the infrared atmospheric windows two alternative values for R (640 nm) and I (800 nm) have recently gained acceptance.

Apparent magnitude is a measure of the radiation in a particular wavelength band, say of blue light, received from the celestial body. Apparent bolometric magnitude, m bol, is a measure of the total radiation received from the body. The bolometric correction (BC) is the difference m Vm bol between the apparent visual (V) and bolometric magnitudes it is generally defined to be zero for stars with surface temperatures similar to the Sun. Other wavebands apart from V are sometimes used in calculating BC.

Apparent magnitude gives no indication of a body's luminosity: a very distant very luminous star may have a similar apparent magnitude as a closer but fainter star. Luminosity is defined in terms of absolute magnitude, M , which is the apparent magnitude of a body if it were at a standard distance of 10 parsecs. It can be shown that the two magnitudes of a body are related to distance d (in parsecs) or its annual parallax, π (in arc seconds):

A is the interstellar extinction. As with apparent magnitude there are values of absolute photoelectric magnitudes: M U , M B, M V, etc. of bolometric magnitude: M bol and of photographic magnitudes: M pg and M pv. Knowledge of a body's absolute bolometric magnitude enables its luminosity to be found. The flux density in jansky of a body can be determined from a value of absolute magnitude at one of the photometric designations J–Q.

## Measures of flux and magnitudes

This page provides detailed descriptions of various measures of magnitude and related outputs of the photometry pipelines. We also provide discussion of some methodology. For details of the Photo pipeline processing please read the corresponding EDR paper section. There are also separate pages describing the creation of flat-fields and the photometric flux calibration.

### The SDSS asinh magnitude system

Magnitudes within the SDSS are expressed as inverse hyperbolic sine (or asinh'') magnitudes, described in detail by Lupton, Gunn, & Szalay (1999). They are sometimes referred to informally as luptitudes . The transformation from linear flux measurements to asinh magnitudes is designed to be virtually identical to the standard astronomical magnitude at high signal-to-noise ratio, but to behave reasonably at low signal-to-noise ratio and even at negative values of flux, where the logarithm in the Pogson magnitude fails. This allows us to measure a flux even in the absence of a formal detection we quote no upper limits in our photometry.

The asinh magnitudes are characterized by a softening parameter b, the typical 1-sigma noise of the sky in a PSF aperture in 1'' seeing. The relation between detected flux f and asinh magnitude m is:

Here, f0 is given by the classical zero point of the magnitude scale, i.e., f0 is the flux of an object with conventional magnitude of zero. The quantity b is measured relative to f0, and thus is dimensionless it is given in the table of asinh softening parameters (Table 21 in the EDR paper), along with the asinh magnitude associated with a zero flux object. The table also lists the flux corresponding to 10f0, above which the asinh magnitude and the traditional logarithmic magnitude differ by less than 1% in flux. For details on converting asinh magnitudes to other flux measures, see converting counts to magnitudes.

### Fiber magnitudes

The flux contained within the aperture of a spectroscopic fiber (3" in diameter) is calculated in each band.

#### Notes

• For children of deblended galaxies, some of the pixels within a 1.5" radius may belong to other children we now measure the flux of the parent at the position of the child this properly reflects the amount of light which the spectrograph will see.
• Images are convolved to 2" seeing before fiberMags are measured. This also makes the fiber magnitudes closer to what is seen by the spectrograph.

### Model magnitudes

The computation of model magnitudes in the DR1 and EDR processing had a serious bug which implied that model magnitudes from the EDR and DR1 should not be used for scientific analysis. The imaging data have all been processed through a new version of the SDSS imaging pipeline, that most importantly fixes an error in the model fits to each object. The result is that the model magnitude is now a good proxy for point spread function (PSF) magnitude for point sources, and Petrosian magnitude (which have larger errors than model magnitude) for extended sources.

Just as the PSF magnitudes are optimal measures of the fluxes of stars, the optimal measure of the flux of a galaxy would use a matched galaxy model. With this in mind, the code fits two models to the two-dimensional image of each object in each band:

1. A pure deVaucouleurs profile
I(r) = I0 exp<-7.67 [(r/re) 1/4 ]>
(truncated beyond 7re to smoothly go to zero at 8re, and with some softening within r=re/50).
2. A pure exponential profile
I(r) = I0 exp(-1.68r/re)
(truncated beyond 3re to smoothly go to zero at 4re.

Each model has an arbitrary axis ratio and position angle. Although for large objects it is possible and even desirable to fit more complicated models (e.g., bulge plus disk), the computational expense to compute them is not justified for the majority of the detected objects. The models are convolved with a double-Gaussian fit to the PSF, which is provided by psp . Residuals between the double-Gaussian and the full KL PSF model are added on for just the central PSF component of the image. These fitting procedures yield the quantities

• r_deV and r_exp , the effective radii of the models
• ab_deV and ab_exp , the axis ratio of the best fit models
• phi_deV and phi_exp , the position angles of the ellipticity (in degrees East of North).
• deV_L and exp_L , the likelihoods associated with each model from the chi-squared fit
• deVMag and expMag , the total magnitudes associated with each fit.

Note that these quantities correctly model the effects of the PSF. Errors for each of the last two quantities (which are based only on photon statistics) are also reported. We apply aperture corrections to make these model magnitudes equal the PSF magnitudes in the case of an unresolved object.

#### Cmodel magnitudes

The code now also takes the best fit exponential and de Vaucouleurs fits in each band and asks for the linear combination of the two that best fits the image. The coefficient (clipped between zero and one) of the de Vaucouleurs term is stored in the quantity fracDeV in the CAS. (In the flat files of the DAS, this parameter is misleadingly termed fracPSF .) This allows us to define a composite flux:

where FdeV and Fexp are the de Vaucouleurs and exponential fluxes (not magnitudes) of the object in question. The magnitude derived from Fcomposite is referred to below as the cmodel magnitude (as distinct from the model magnitude, which is based on the better-fitting of the exponential and de Vaucouleurs models in the r band).

In order to measure unbiased colors of galaxies, we measure their flux through equivalent apertures in all bands. We choose the model (exponential or deVaucouleurs) of higher likelihood in the r filter, and apply that model (i.e., allowing only the amplitude to vary) in the other bands after convolving with the appropriate PSF in each band. The resulting magnitudes are termed modelMag . The resulting estimate of galaxy color will be unbiased in the absence of color gradients. Systematic differences from Petrosian colors are in fact often seen due to color gradients, in which case the concept of a global galaxy color is somewhat ambiguous. For faint galaxies, the model colors have appreciably higher signal-to-noise ratio than do the Petrosian colors.

There is now excellent agreement between cmodel and Petrosian magnitudes of galaxies, and cmodel and PSF magnitudes of stars. Cmodel and Petrosian magnitudes are not expected to be identical, of course as Strauss et al. (2002) describe, the Petrosian aperture excludes the outer parts of galaxy profiles, especially for elliptical galaxies. As a consequence, there is an offset of 0.05-0.1 mag between cmodel and Petrosian magnitudes of bright galaxies, depending on the photometric bandpass and the type of galaxy. The rms scatter between model and Petrosian magnitudes at the bright end is now between 0.05 and 0.08 magnitudes, depending on bandpass the scatter between cmodel and Petrosian magnitudes for galaxies is smaller, 0.03 to 0.05 magnitudes. For comparison, the code that was used in the EDR and DR1 had scatters of 0.1 mag and greater, with much more significant offsets.

The cmodel and PSF magnitudes of stars are in good agreement (they are forced to be identical in the mean by aperture corrections, as was true in older versions of the pipeline). The rms scatter between model and PSF magnitudes for stars is much reduced, going from 0.03 mag to 0.02 magnitudes, the exact values depending on bandpass. In the EDR and DR1, star-galaxy separation was based on the difference between model and PSF magnitudes. We now do star-galaxy separation using the difference between cmodel and PSF magnitudes, with the threshold at the same value (0.145 magnitudes).

Due to the way in which model fits are carried out, there is some weak discretization of model parameters, especially r_exp and r_deV . This is yet to be fixed. Two other issues (negative axis ratios, and bad model mags for bright objects) have been fixed since the EDR.

### Petrosian magnitudes

Stored as petroMag . For galaxy photometry, measuring flux is more difficult than for stars, because galaxies do not all have the same radial surface brightness profile, and have no sharp edges. In order to avoid biases, we wish to measure a constant fraction of the total light, independent of the position and distance of the object. To satisfy these requirements, the SDSS has adopted a modified form of the Petrosian (1976) system, measuring galaxy fluxes within a circular aperture whose radius is defined by the shape of the azimuthally averaged light profile.

We define the Petrosian ratio'' RP at a radius r from the center of an object to be the ratio of the local surface brightness in an annulus at r to the mean surface brightness within r, as described by Blanton et al. 2001a, Yasuda et al. 2001:

where I(r) is the azimuthally averaged surface brightness profile.

The Petrosian radius rP is defined as the radius at which RP(rP) equals some specified value RP,lim, set to 0.2 in our case. The Petrosian flux in any band is then defined as the flux within a certain number NP (equal to 2.0 in our case) of r Petrosian radii:

In the SDSS five-band photometry, the aperture in all bands is set by the profile of the galaxy in the r band alone. This procedure ensures that the color measured by comparing the Petrosian flux FP in different bands is measured through a consistent aperture.

The aperture 2rP is large enough to contain nearly all of the flux for typical galaxy profiles, but small enough that the sky noise in FP is small. Thus, even substantial errors in rP cause only small errors in the Petrosian flux (typical statistical errors near the spectroscopic flux limit of r

17.7 are < 5%), although these errors are correlated.

The Petrosian radius in each band is the parameter petroRad , and the Petrosian magnitude in each band (calculated, remember, using only petroRad for the r band) is the parameter petroMag .

In practice, there are a number of complications associated with this definition, because noise, substructure, and the finite size of objects can cause objects to have no Petrosian radius, or more than one. Those with more than one are flagged as MANYPETRO the largest one is used. Those with none have NOPETRO set. Most commonly, these objects are faint (r > 20.5 or so) the Petrosian ratio becomes unmeasurable before dropping to the limiting value of 0.2 these have PETROFAINT set and have their Petrosian radii'' set to the default value of the larger of 3" or the outermost measured point in the radial profile. Finally, a galaxy with a bright stellar nucleus, such as a Seyfert galaxy, can have a Petrosian radius set by the nucleus alone in this case, the Petrosian flux misses most of the extended light of the object. This happens quite rarely, but one dramatic example in the EDR data is the Seyfert galaxy NGC 7603 = Arp 092, at RA(2000) = 23:18:56.6, Dec(2000) = +00:14:38.

How well does the Petrosian magnitude perform as a reliable and complete measure of galaxy flux? Theoretically, the Petrosian magnitudes defined here should recover essentially all of the flux of an exponential galaxy profile and about 80% of the flux for a de Vaucouleurs profile. As shown by Blanton et al. (2001a), this fraction is fairly constant with axis ratio, while as galaxies become smaller (due to worse seeing or greater distance) the fraction of light recovered becomes closer to that fraction measured for a typical PSF, about 95% in the case of the SDSS. This implies that the fraction of flux measured for exponential profiles decreases while the fraction of flux measured for deVaucouleurs profiles increases as a function of distance. However, for galaxies in the spectroscopic sample (r<17.7), these effects are small the Petrosian radius measured by frames is extraordinarily constant in physical size as a function of redshift.

### PSF magnitudes

Stored as psfMag . For isolated stars, which are well-described by the point spread function (PSF), the optimal measure of the total flux is determined by fitting a PSF model to the object. In practice, we do this by sync-shifting the image of a star so that it is exactly centered on a pixel, and then fitting a Gaussian model of the PSF to it. This fit is carried out on the local PSF KL model at each position as well the difference between the two is then a local aperture correction, which gives a corrected PSF magnitude. Finally, we use bright stars to determine a further aperture correction to a radius of 7.4'' as a function of seeing, and apply this to each frame based on its seeing. This involved procedure is necessary to take into account the full variation of the PSF across the field, including the low signal-to-noise ratio wings. Empirically, this reduces the seeing-dependence of the photometry to below 0.02 mag for seeing as poor as 2''. The resulting magnitude is stored in the quantity psfMag . The flag PSF_FLUX_INTERP warns that the PSF photometry might be suspect. The flag BAD_COUNTS_ERROR warns that because of interpolated pixels, the error may be under-estimated.

### The Reddening Correction

Reddening corrections in magnitudes at the position of each object, reddening , are computed following Schlegel, Finkbeiner & Davis (1998). These corrections are not applied to the magnitudes in the databases. Conversions from E(B-V) to total extinction Alambda, assuming a z=0 elliptical galaxy spectral energy distribution, are tabulated in Table 22 of the EDR Paper.

### Which Magnitude should I use?

Faced with this array of different magnitude measurements to choose from, which one is appropriate in which circumstances? We cannot give any guarantees of what is appropriate for the science you want to do, but here we present some examples, where we use the general guideline that one usually wants to maximize some combination of signal-to-noise ratio, fraction of the total flux included, and freedom from systematic variations with observing conditions and distance.

Given the excellent agreement between cmodel magnitudes (see cmodel magnitudes above) and PSF magnitudes for point sources, and between cmodel magnitudes and Petrosian magnitudes (albeit with intrinsic offsets due to aperture corrections) for galaxies, the cmodel magnitude is now an adequate proxy to use as a universal magnitude for all types of objects. As it is approximately a matched aperture to a galaxy, it has the great advantage over Petrosian magnitudes, in particular, of having close to optimal noise properties.

#### Example magnitude usage

• Photometry of Bright Stars: If the objects are bright enough, add up all the flux from the profile profMean and generate a large aperture magnitude. We recommend using the first 7 annuli.
• Photometry of Distant Quasars: These will be unresolved, and therefore have images consistent with the PSF. For this reason, psfMag is unbiased and optimal.
• Colors of Stars: Again, these objects are unresolved, and psfMag is the optimal measure of their brightness.
• Photometry of Nearby Galaxies: Galaxies bright enough to be included in our spectroscopic sample have relatively high signal-to-noise ratio measurements of their Petrosian magnitudes. Since these magnitudes are model-independent and yield a large fraction of the total flux, roughly constant with redshift, petroMag is the measurement of choice for such objects. In fact, the noise properties of Petrosian magnitudes remain good to r=20 or so.
• Photometry of Galaxies: Under most conditions, the cmodel magnitude is now a reliable estimate of the galaxy flux. In addition, this magnitude account for the effects of local seeing and thus are less dependent on local seeing variations.
• Colors of Galaxies: For measuring colors of extended objects, we continue to recommend using the model (not the cmodel ) magnitudes the colors of galaxies were almost completely unaffected by the DR1 software error. The model magnitude is calculated using the best-fit parameters in the r band, and applies it to all other bands the light is therefore measured consistently through the same aperture in all bands.

Of course, it would not be appropriate to study the evolution of galaxies and their colors by using Petrosian magnitudes for bright galaxies, and model magnitudes for faint galaxies.

Finally, we note that azimuthally-averaged radial profiles are also provided, as described below, and can easily be used to create circular aperture magnitudes of any desired type. For instance, to study a large dynamic range of galaxy fluxes, one could measure alternative Petrosian magnitudes with parameters tuned such that the Petrosian flux includes a small fraction of the total flux, but yields higher signal-to-noise ratio measurements at faint magnitudes.

### Radial Profiles

The frames pipeline extracts an azimuthally-averaged radial surface brightness profile. In the catalogs, it is given as the average surface brightness in a series of annuli. This quantity is in units of maggies'' per square arcsec, where a maggie is a linear measure of flux one maggie has an AB magnitude of 0 (thus a surface brightness of 20 mag/square arcsec corresponds to 10 -8 maggies per square arcsec). The number of annuli for which there is a measurable signal is listed as nprof , the mean surface brightness is listed as profMean , and the error is listed as profErr . This error includes both photon noise, and the small-scale bumpiness'' in the counts as a function of azimuthal angle.

## Question about total absolute magnitudes of galaxies - negative or not? - Astronomy

The revised 3-dimensional Hubble classification system (de Vaucouleurs 1959, 1963, 1974 Sandage 1961, 1975) is still basic. Other systems (Yerkes: Morgan 1958, 1959 DDO: van den Bergh 1960 RDDO: van den Bergh 1976 Byurakan: Tovmassian 1964) provide important additional information. Revised T types are now available for over 4000 galaxies, but for less than 1000 in other systems (Table 1). Even though all such systems are still entirely subjective, independent classifications by trained observers agree fairly closely (Corwin 1968, Nilson 1973, Buta 1977). In general the reliability of a classification depends on telescope aperture, plate scale and image area as well as inclination. In good cases of nearly face-on objects classified on original plates with N > 10 4 pixels in image area the m.e. of T types can be as small as 5 percent of the range classifications on small-scale, high contrast prints such as the POSS copies have much lower weight (at the limit all galaxies appear "elliptical").

Because most systems are highly correlated (de Vaucouleurs 1963) and the revised Hubble system gives the best "resolution" (range / ), the closest correlation with physical parameters as well as the most extensive overlap with other systems, it is expedient to consider T as the basic qualitative parameter and to correlate it with other qualitative systems and with quantitative parameters. The following report updates or supersedes results reviewed at the Canberra (IAU 58) 1973 symposium (de Vaucouleurs 1974) as far as possible statistics are based on the Second Reference Catalogue of Bright Galaxies (RC2) (de Vaucouleurs et al. 1976).

The DDO luminosity class L is highly correlated with type T (Fig. 1) this is because the DDO system used only the original Sa, Sb, Sc Hubble stages and essentially bridges the gap to magellanic irregulars Im by means of luminosity classes IV, V attached to Sc conversely, the revised Hubble system introduces late-spiral stages Sd, Sm across this gap, but has no luminosity classes. Each system provides some information not given by the other because of this and the fact that each is subject to classification errors of

12 percent (over the restricted range 2 T 10, i.e., Sab to Im, over which the L classification is valid) it is convenient to combine them into a composite luminosity index = (T + L) / 10 which is more closely correlated with absolute magnitude than either T or L alone (de Vaucouleurs 1976).

Excluding a few early types (L, Sa) which were misclassified as later-type spirals in the DDO system because of inadequate resolution (POSS paper prints!), the mean relation is simply T = L + 1.0, with a s.d. (L - T) = 1.4, consistent with mean errors (L) (T) 1.0 step for each system. The total range of is from 0.3 (Sab I) to 1.9 (Im V) with a resolution = range / = 1.6 / 0.14 = 11. Extensive tests of the correlation between and the corrected absolute magnitude M°T (in the B°T system of RC2) lead (without arbitrary extra-polation) to the adopted regression line

The constant is derived from 9 nearby galaxies (0.5 1.5) having distances from primary and secondary indicators (de Vaucouleurs 1977b) the coefficient of 2 is derived from the apparent magnitudes of 22 galaxies in the Virgo cluster and from provisional absolute magnitudes of 351 galaxies calculated from their redshifts (assuming linearity) (Fig. 2). (1)

If B°T(1) = B°T - 1.4( 2 - 1), the geometric distance modulus of any spiral for which B°T, T and L are known is given by µ0 = 19.25 + B±T(1) with an estimated mean error (µ0) 0.5 mag if 6, L > 5) in particular, both nearby low-luminosity and distant high-luminosity systems are mixed among the DDO "dwarfs" (Fisher and Tully 1975). Magellanic irregulars span a very large range of luminosities from extreme dwarfs such as DDO 155 = GR 8 = A 1256 + 14 with M°T -11 (Hodge 1974, de Vaucouleurs 1977b) up to "clumpy" giant systems such as Mark 325 at M°T -20 (Casini and Heidmann 1976), but the average still agrees with equation (1).

The correlation between the two Yerkes lists Y1, Y2 and the revised Hubble typo T has been analyzed previously (de Vaucouleurs 1963) (Fig. 3). The main interest of the Y types is in its providing additional information on the bulge/disk ratio and in its (weak) correlation with color residuals at constant type T (Section 2.2).

Byurakan nuclear types N' were examined for systematic errors dependent on apparent diameter and inclination. No significant effect of apparent diameter was found, but an effect of axis ratio R = a/b is present for types t > -2, that is

where c(t) -1.5 ± 0.3 for t > -2 (0 for t -2). This is clearly an effect of internal obscuration of the nucleus in dusty galaxies. The inclination-corrected type N'c = N + 1.5(log R - 0.2) shows a loose, but significant correlation with morphological type T in the expected sense as shown in Fig. 4 that is, late types (t 5) have weaker nuclei than early types, but the scatter is large. Occasional confusion between nuclei and bright HII regions has been noted (e.g. in NGC 55), especially among late types.

The BGC nuclear classes N'' coded as in RC2 (Table 1) were likewise investigated for resolution and inclination effects using the weight (log W) and axis ratio (log R) as parameters. As for the Byurakan data, an effect of inclination (obscuration) was indicated with c(t) = 0 for t -2, c(t) = -1.5 ± 0.2 for t > -2. The effect of resolution measured by the weight log W of the morphological classification (function of image size and plate scale as explained in RC2) was expressed by

where t = 0 for t -4 (E), 0.058 ± 0.014 for -3 t 0 (L), 0.115 ± 0.013 for 1 t 4 (Sa-Sbc) and 0.049 ± 0.023 for 5 t 10 (Sc-Im). The corrected class N''c is correlated with morphological type in the same sense as the Byurakan types (Fig. 4).

There is a loose correlation between the revised DDO system (van den Bergh 1976) and the revised Hubble types T (Table 3). RDDO has more steps among lenticulars and early spirals (making up the S0 and A classes of RDDO) and fewer among the late-type spirals and irregulars (comprising the Sc, Sc/Irr, and Irr of RDDO). Essentially, the late L and early S stages (-2 t 2) have been depleted to form the "anemic spiral" A class. The rationale offered for this reshuffling is that it improves the correlation with color indices and bulge/disk ratio. I have been unable to substantiate this claim from a study of available quantitative data for the small set of 126 examples of RDDO types in the Hubble Atlas. In particular the claim that anemic spirals (A) are redder than normal spirals (5) of the same Hubble stage is not verified in the best documented case the mean color of RDDO Ab objects of types t = 1 to 3 (Sa - Sb) is = 0.73 compared with 0.74 for RDDO Sb in the same t range (see Section 2.2c).

(1) The constant term implies 90 (1 ± 0.1) km s -1 Mpc -1 for the local all-sky average of the Hubble ratio for 300 spirals having distance moduli µ0 *****

## Voyage to the Great Attractor

In the mid-1970's, astronomers studying the Cosmic Microwave Background radiation to study the large scale homogeneity of the Universe discovered a startling dipole pattern on the sky. Indeed more modern observations with the COBE (Cosmic Background Explorer) satellite have shown this to be essentially a perfect dipole. The amplitude of the dipole is only 3.358 +/- 0.001 milli Kelvins in the direction (l = 264.31 +/-0.16, b = +48.05 +/-0.09, Galactic coordinates, or RA = 11.199 h , Dec = -7.22, celestial coordinates). The dipole was almost immediately interpreted as a motion of the Milky Way or the whole Local Group. In raw form, since the observation is made from the solar system, its amplitude is only about 369.0 +/- 2.5 km/s and is in a direction almost opposite the rotation direction of the Sun around the center of the MW. After correction to the reference frame (barycenter) of the Local Group of galaxies, however, the velocity amplitude rises to 627 +/-22 km/s towards (l = 276 +/-3, b= +30 +/-2) where the increase in the uncertainty is due to the uncertainty in the coordinate transformation (c.f. Lineweaver 1996).

Even before the discovery of this dipole motion, astronomers had been "sniffing" around the idea that the expansion of the Universe might not be uniform, especially on small scales. For example, we know that the Local Group of galaxies is bound and therefore the galaxies in it are not participating in the global expansion at least relative to each other. We also know that galaxies in the cores of rich clusters of galaxies are bound to those much more massive objects and are in orbit around their centers with "peculiar velocities" that are substantial, thousands of km/s, such that in the nearest rich cluster, Virgo, the gravitationally induced velocity can dwarf the "Hubble" velocity (the velocity due to the expansion of the Universe). In the early 1950's, Vera Rubin (1951) first attempted to measure peculiar velocities in the nearby Universe, and in the mid-1950's Gerard deVaucouleurs (1956, 1958) hypothesized that the Milky Way should actually be falling into (i.e. expanding away from more slowly) the Local Supercluster (c.f. the seminar projects on the Virgo Cluster and on the Local Supercluster).

The fundamental idea here is that the non-cosmological motion of the Local Group would be caused by some large mass concentration pulling us towards it. In other words, Gauss's theorem says that if matter in the Universe were truly uniformly distributed there would be no net force on any galaxy, but since we know that the Universe is lumpy, galaxies will be accelerated by large masses such as othr galaxies, galaxy groups and galaxy clusters that are near them. Since the pull of gravity decreases as the square of the separation, nearby masses have a much greater effect than distant masses. Also, we still do think that on very large scales, in the words of Edwin Hubble, "the Universe is sensibly uniform."

Given that, the first place astronomers looked for the cause of our motion w.r.t. the CMB was the Local Supercluster following deVaucouleurs (c.f. Schechter 1980 Davis et al. 1980 Aaronson et al. 1982). The Local Supercluster dominates the galaxy distribution inside a distance of a few thousand km/s (40-50 Megaparsecs or 120-150 million light years). We found a definite "flow" into the LSC, the Local Group is expanding away from Virgo about 250 km/s slower than it would be if we were freely participating in the Hubble Flow, but that 250 km/s motion is much less than the

630 km/s and not quite in the right direction.

Since Virgo and the Local Supercluster can't do it, astronomers looked for bigger things further away.

The purpose of this project is to study the properties of the Great Attractor (GA) with a new and deep all-sky survey of galaxies, the 2MASS Redshift Survey (2MRS). We will measure several global properties of the GA, including its size and shape. We will examine maps of the distribution of galaxies in this supercluster and the distribution by morphological type.

## Realistic telescopic limiting magnitudes

Okay, time to open a can of worms. I have been looking through Schaefer and Crumey's reports on telescopic limiting magnitudes and while educational, I have beefs with both. The most obvious is that they are far too conservative in their results. I have to use

0.4 observer factor, F, to match my results in various scopes. That is less than half Crumey's most optimistic values and it isn't as good as some other observers reported. I run about 0.9 F naked eye, despite vision issues as I age. Part of the problem is that in some respects they lump observers with average or poor vision in with those of what I consider good acuity (reading the raw data) and try to curve fit from there using NELM and assumptions about experience, etc. The result is that they miss by over a magnitude compared to what my eye can show in the eyepiece (and nearly as much even today NELM wise despite my having lost about half a magnitude from my younger years), even in mediocre seeing.

Schaefer also tried to predict results for non-dark locations. which is so rife with problems as to not be a useful approach in my opinion. at least until a very good dark sky correlation has been achieved to work back from. Stick to what experienced observers with good acuity can do in dark skies, and then you have a measure of what the eye can do aided by a telescope. Nerfing everything for a mix of poor conditions or inexperience/compromised acuity is going to give unsatisfactory results because there is not good way to correlate NELM with telescopic since the telescope corrects various aspects of naked eye's vision flaws. The latter disconnect is a huge problem

Another problem is the assumption that

25 MPSAS = black. This is most noticeable in Crumey's otherwise excellent work and shows up as an exit pupil cut off of about 2.0 mm in very dark skies (maybe a little less.) Despite literature references supporting it, this is certainly wrong in my opinion, as analysis of Schaefer's individual data proves: the data consistently show smaller exit pupils taking experienced observers in dark/semi-dark skies much deeper (typically 0.7 magnitudes or more versus the assumed limit which is well past the significance of the model.) The impact is somewhat reduced by increasing aperture which flattens the curve, because seeing affects turn stars into extended objects at small exit pupils, and these are more difficult to see as we all know. So a seeing term is a critical consideration once aperture exceeds perhaps 4 inches (although I already use a pupil several times smaller at 4.3" to go well past the correlation predictions.) Past 10" seeing is one of the most important factors in my experience. Crumey addresses exit pupil for extended objects, but missed the opportunity for point sources versus diffraction/seeing effects for stars (typically point sources. except for small exit pupil and/or poor seeing.)

Back in black. where do others here see the field and the field stop become indistinguishable? I have done some testing in only moderately dark conditions (21.38 MPSAS) and found that in a relatively dim portion of the overhead sky, I become unable to detect the field stop except when a sufficiently bright star illuminates it at about 0.4mm exit pupil with the 20". At 0.5mm exit pupil and above, after a time I can detect the field stop from the field and trace it in averted vision. This means I can detect a point source from the field, if it is still a point source or nearly so. Below that I am blind to the transition and can't find much to focus on other than the blur of dim field stars as extended objects (not surprisingly this tends to coincide with poor seeing at over 1000x.) The eye lens stays black and I fumble to find something to focus on. The current models assume 25 MPSAS as the cut off based on Blackwell data, while in reality I see 27 MPSAS or more. although certainly at a reduced acuity that offsets some of the gain, particularly when seeing is considered. I consider my dark adaptation/threshold sensitivity good, but I don't find it to be particularly impressive, others have demonstrated considerably better (Skiff in particular, and of course O'Meara as well as several observers on CN.)

I have hunted down the Schaefer request for data in the March 1989 S&T. It uses M67 as the test field with a sequence of stars marked in roughly 0.35 mag increments on average. While systematic these are still rather coarse increments that will bias toward low/conservative values. The bias is likely half the delta assuming perfect accuracy, because one is unlikely to report confidently sighting past what was actually seen. Seeing is not specifically requested (it is only suggested as a supplemental note.) This becomes more problematic for dim stars near much brighter cluster stars. Multiple/more distant field stars might have been helpful past about 16th magnitude. 17+ mag stars are very hard to detect near bright field stars in my experience. A 17.64 mag star sandwiched between 12 and 13 mag stars is substantially tougher than one near 15/16 mag stars even in good seeing. The next step is 18.04. very near a 13 mag star. Less obvious is why the 17.38 star was not seen, since it is more isolated, however, there were few reports of large aperture in dark sky. Instead 17.05 was reported as uncertain for a 17.5" in 7.2 mag skies, a credible result.

Back when Schaefer's data request was made, SQM/SQM-L meters were not yet available and NELM estimates were about all that one could expect. Now that better instruments are commonly available, it seems a good time for someone to repeat the exercise using SQM data for actual sky brightness. vs. heavily error prone approximations. With the meters, one variable in the equations can be fixed with relative precision compared to the others. I hope to have the seeing this next month to attempt Schaefer's sequence in M67. I don't know if I will have time to experiment with more than one aperture or run through various magnification levels. Seeing will likely be a major factor. but at least I intend to attempt to rate it on a Pickering scale for any aperture used.

It is not that the prior work was poor and certainly not for the basis at the time, it is more that time and improved tech provide an opportunity to be more precise and address problematic aspects of prior approaches. It is a process of building on prior work.

## Magnitude

A means of measuring the brightness of a star or other body: the lower the body's magnitude value, the brighter it appears in the sky. A star of magnitude +5.0 is dim, and hardly visible to the naked eye, while a star with a magnitude close to 0.0 (such as Capella in Auriga) is very bright indeed.

The system of stellar magnitudes is more than two thousand years old. Its first known use was by the Greek astronomer Hipparchus in about 120 BCE, whose informal system classified stars as first magnitude for the brightest down to sixth magnitude for the faintest.

Especially with the advent of the telescope, this kind of informal approach proved inadequate, and in the middle of the nineteenth century, standards for the calculation of magnitudes were introduced. According to these standards, which are still in use today, a difference in brightness of five magnitudes is equivalent to factor of 100. For example, Alpha Centauri or Rigil Kentaurus (with a magnitude of very nearly 0.0) is one hundred times brighter than Marsic in Hercules, one of the faintest named stars with a magnitude of almost exactly +5.0. It follows from this that a difference of one magnitude corresponds to a difference in brightness of a little over 2½ times.

The magnitude scale is calibrated to the brightnesses of about a hundred specific stars, all lying within a few degrees of the Northern Celestial Pole and thus termed the North Polar Sequence. By coincidence, the North Polar constellation, Ursa Minor, provides a useful 'key' to stellar magnitudes. The four stars that form the rectangular 'bowl' of the Little Dipper each represent a different level of brightness. Kochab, the brightest, has a magnitude of +2.1 Pherkad is magnitude +3.0, Alifa al Farkadain is +4.3, and the faintest, Anwar al Farkadain, has a magnitude of +5.0.

+6.5 is conventionally regarded as the limit of visibility with the naked eye. The actual limit, of course, will depend on the observer, but few people can see objects fainter than this value. Since the magnitude scale works by the relative brightness objects, though, it can continue below this threshold of visibility. Objects of magnitude +6.0 are 2½ times fainter than those of magnitude +5.0, and so on. At the time of writing, for example, the planet Pluto has a magnitude of +13.8, far far too faint to be seen with the naked eye, but calculable as about 400,000 times fainter than Alpha Centauri.

A few very bright stars and certain objects within the Solar System exceed magnitude 0.0, and therefore need negative numbers to describe their brightness. There are four stars with negative magnitudes: Sirius (-1.4), Canopus (-0.6), Arcturus (-0.1) and Alpha Centauri, with a magnitude that comes extremely close to the zero mark at -0.01. Within the Solar System, all of the planets visible to the naked eye have negative magnitudes for at least some of the time. The brightest objects of all are the Moon (which can reach -13.0 or brighter) and, of course, the Sun, whose magnitude value varies around an extreme -26.7.

To a great extent, the brightness of objects in the sky depends on their position in space. Sirius is the brightest star in the sky, but it is a dwarf star, and not particularly luminous in stellar terms - it appears brilliant because it is less than nine light years away. By comparison, the star Deneb in Cygnus appears much fainter than Sirius in the sky, but actually a supergiant thousands of times more luminous than Sirius - it appears fainter because it is more than three thousand light years away.

For this reason, the brightness of objects as they appear in the sky is properly referred to as their apparent magnitude. A separate scale exists - absolute magnitude - to describe the intrinsic brightness of an object, irrespective of the location of its observer. This concept is discussed in its own entry on this site.

## NATIONAL RESEARCH COUNCIL

The physical universe [1] was anthropocentric to primitive man. At a subsequent stage of intellectual progress it was centered in a restricted area on the surface of the earth. Still later, to Ptolemy and his school, the universe was geocentric but since the time of Copernicus the sun, as the dominating body of the solar system, has been considered to be at or near the center of the stellar realm. With the origin of each of these successive conceptions, the system of stars has ever appeared larger than was thought before. Thus the significance of man and the earth in the sidereal scheme has dwindled with advancing knowledge of the physical world, and our conception of the dimensions of the discernible stellar universe has progressively changed. Is not further evolution of our ideas probable? In the face of great accumulations of new and relevant information can we firmly maintain our old cosmic conceptions?

[1: The word "universe" is used in this paper in the restricted sense, as applying to the total of sidereal systems now known to exist.]

As a consequence of the exceptional growth and activity of the great observatories, with their powerful methods of analyzing stars and of sounding space, we have reached an epoch, I believe, when another advance is necessary our conception of the galactic system must be enlarged to keep in proper relationship the objects our telescopes are finding the solar system can no longer maintain a central position. Recent studies of clusters and related subjects seem to me to leave no alternative to the belief that the galactic system is at least ten times greater in diameter - at least a thousand times greater in volume - than recently supposed.

Dr. Curtis [1], on the other hand, maintains that the galactic system has the dimensions and arrangement formerly assigned it by students of sidereal structure - he supports the views held a decade ago or so by Newcomb, Charlier, Eddington, Hertzprung, and other leaders in the stellar astronomy. In contrast to my present estimate of the diameter of at least three hundred thousand light-years Curtis outlines his position as follows: [2]

[1: See Part II of this article, by Heber D. Curtis.]
[2: Quoted from a manuscript copy of his Washington address.]

"As to the dimensions of the galaxy indicated by our Milky Way, till recently there has been a fair degree of uniformity in the estimates of those who have investigated the subject. Practically all have deduced diameters from 7,000 to 30,000 light years. I shall assume a maximum diameter of 30,000 light years as representing sufficiently well this older view to which I subscribe though this is pretty certainly too large."

I think it should be pointed out that when Newcomb was writing on the subject some twenty years ago, knowledge of those special factors that bear directly on the size of the universe was extremely fragmentary compared with our information of to-day. In 1900, for instance, the radial motions of about 300 stars were known now we know the radial velocities of thousands. Accurate distances were then on record for possibly 150 of the brightest stars, and now for more than ten times as many. Spectra were then available for less than one-tenth of the stars for which we have the types to-day. Practically nothing was known at that time of the photometric and spectroscopic methods of determining distance nothing of the radial velocities of globular clusters or of spiral nebulae, or even of the phenomenon of star streaming.

As a further indication of the importance of examining anew the evidence on the size of stellar systems, let us consider the great globular cluster in Hercules - a vast sidereal organization concerning which we had until recently but vague ideas. Due to extensive and varied researches, carried on during the last few years at Mount Wilson and elsewhere, we now know the positions, magnitudes, and colors of all its brightest stars, and many relations between color, magnitude, distance from the center, and star density. We know some of these important correlations with greater certainty in the Hercules cluster than in the solar neighborhood. We now have spectra of many of the individual stars, and the spectral type and radial velocity of the cluster as a whole. We know the types and periods of light variation of its variable stars, the colors and spectral types of these variables, and something also of the absolute luminosity of the brightest stars of the cluster from the appearance of their spectra. Is it surprising, therefore, that we venture to determine the distance of Messier 13 and similar systems with more confidence than was possible ten years ago when none of these facts was known, or even seriously considered in cosmic speculations?

If he were writing now, with knowledge of these relevant developments, I believe Newcomb would not maintain his former view on the probable dimensions of the galactic system.

For instance, Professor Kapteyn has found occasion, with the progress of his elaborate studies of laws and of stellar luminosity and density, to indicate larger dimensions of the galaxy than formerly accepted. In a paper just appearing as Mount Wilson Contribution, No. 188, [1] he finds, as a result of the research extending over some 20 years, that the density of stars along the galactic plane is quite appreciable at a distance of 40,000 light-years - giving a diameter of the galactic system, exclusive of distant star clouds of the Milky Way, about three times the value Curtis admits as a maximum for the entire galaxy. Similarly Russell, Eddington, and, I believe, Hertzprung, now subscribe to larger values of galactic dimensions and Charlier, in a recent lecture before the Swedish Astronomical Association, has accepted the essential feature of the larger galaxy, though formerly he identified the local system of B stars with the whole galactic system and obtained distances of the clusters and dimensions of the galaxy only a hundredth as large as I derive.

[1: The Contribution is published jointly with Dr. van Rhijn.]

SURVEYING THE SOLAR NEIGHBORHOOD

Let us first recall that the stellar universe, as we know it, appears to be a very oblate spheroid or ellipsoid - a disk-shaped system composed mainly of stars and nebulae. The solar system is not far from the middle plane of this flattened organization which we call the galactic system. Looking away from the plane we see relatively few stars looking along the plane, through a great depth of star-populated space, we see great numbers of sidereal objects constituting the band of light we call the Milky Way. The loosely organized star clusters, such as the Pleiades, the diffuse nebulae, of which the ring nebula is Lyra is a good example, the dark nebulosities - all these sidereal types appear to be a part of the great galactic system, and they lie almost exclusively along the plane of the Milky Way. The globular clusters, though not in the Milky Way, are also affiliated with the galactic system the spiral nebulae appear to be distant objects mainly if not entirely outside the most populous parts of the galactic region.

This conception of the galactic system, as a flattened, watch-shaped organization of stars and nebulae, with globular clusters and spiral nebulae as external objects, is pretty generally agreed upon by students of the subject but in the matter of the distances of the various sidereal objects - the size of the galactic system - there are, as suggested above, widely divergent opinions. We shall, therefore, first consider briefly the dimensions of that part of the stellar universe concerning which there is essential unanimity of opinion, and later discuss in more detail the larger field, where there appears to be a need for modification of the older conventional view.

Possibly the most convenient way of illustrating the scale of sidereal universe is in terms of our measuring rods, going from terrestrial units to those of stellar systems. On the earth's surface we express distances in units such as inches, feet, or miles. On the moon, as seen in the accompanying photograph made with the 100-inch reflector, the mile is still a usable measuring unit a scale of 100 miles is indicated on the lunar scene.

Our measuring scale must be greatly increased, however, when we consider the dimensions of a star - distances on the surface of our sun, for example. The large sun-spots shown in the illustration cannot be measured conveniently in units appropriate to earthly distance - in fact, the whole earth itself is none too large. The units for measuring the distances from the sun to its attendant planets, is, however, 12,000 times the diameter of the earth it is the so-called astronomical unit, the average distance from earth to sun. This unit, 93,000,000 miles in length, is ample for the distances of planets and comets. It would probably suffice to measure the distances of whatever planets and comets there may be in the vicinity of other stars but it, in turn, becomes cumbersome in expressing the distances from one star to another, for some of them are hundreds of millions, even a thousand million, astronomical units away.

[Text - processor's editor's note: Technical limitations may have caused the three figures cited in Dr. Shapley's text not to be reproduced in this version. The figure captions are included however, and it is believed that the line of argument is clear.]

Fig. 1. - The region of the Apennines on the surface of the moon as photographed with the 100- inch reflector. Photograph by F. G. Pease

Fig. 2. - A group of sun-spots first appearing in February 1920 and lasting for 100 days. The shaded and unshaded regions indicate magnetic polarities of opposite signs. Drawing by S. B. Nicholson.

Fig. 3. - Two successive photographs on the same plate of the diffuse nebula N. G. C. 221, made with the 100-inch reflector to illustrate the possibility of greatly increasing the photographic power of a large reflector though the use of accessory devices. The exposure time for the picture on the left was fifteen minutes it was five minutes for the picture on the right, which was made with the aid of the photographic intensifier described in Proc. Nat. Acad. Sci., 6., 127, 1920. In preparing the figure the two photographs were enlarged to the same scale.

This leads us to abandon the astronomical unit and to introduce the light-year as a measure for sounding the depth of stellar space. The distance light travels in a year is something less than six million million miles. The distance from the earth to the sun is, in these units, eight light minutes. This distance to the moon is 1.2 light-seconds. In some phases of our astronomical problems (studying photographs of stellar spectra) we make direct microscopic measures of a ten- thousandth of an inch and indirectly we measure changes in the wave-length of light a million times smaller than this in discussing the arrangement of globular clusters in space, we must measure a hundred thousand light-years. Expressing these large and small measures with reference to the velocity of light, we have an illustration of the scale of the astronomer's universe - his measures range from the trillionth of a billionth part of one light-second, to more than a thousand light centuries. The ratio of the greatest measure to the smallest is as 10^33 to 1.

It is to be notices that light plays an all-important role in the study of the universe we know the physics and chemistry of stars only through their light, and their distance from us we express by means of the velocity of light. The light-year, moreover, has a double value in sidereal exploration it is geometrical, as we have seen, and it is historical. It tells us not only how far away an object is, but also how long ago the light we examine was started on its way. You do not see the sun where it is, but where it was eight minutes ago. You do not see faint stars of the Milky Way as they are now, but more probably as they were when the pyramids of Egypt were being built and the ancient Egyptians saw them as they were at a time still more remote. We are, therefore, chronologically far behind events when we study conditions or dynamical behavior in remote stellar systems the motions, light emissions, and variations now investigated in the Hercules cluster are not contemporary, but, if my value of the distance is correct, they are the phenomena of 36,000 years ago. The great age of these incoming pulses of radiant energy is, however, no disadvantage in fact, their antiquity has been turned to good purpose in testing the speed of stellar evolution, in indicating the enormous ages of stars, in suggesting the vast extent of the universe in time as well as in space.

Taking the light-year as a satisfactory unit for expressing the dimensions of sidereal systems, let us consider the distances of neighboring stars and clusters, and briefly mention the methods of deducing their space positions. For nearby stellar objects we can make direct trigonometric measures of distance (parallax), using the earth's orbit or the sun's path through space as a base line. For many of the more distant stars spectroscopic methods are available, using the appearance of the stellar spectra and the readily measurable apparent brightness of the stars. For certain types of stars, too distant for spectroscopic data, there is still a chance of obtaining the distance by means of the photometric method. This method is particularly suited to studies of globular clusters it consists first in determining, by some means, the real luminosity of a star, that is, its so-called absolute magnitude, and second, in measuring its apparent magnitude. Obviously, if a star of known real brightness is moved away to greater and greater distances, its apparent brightness decreases hence, for such stars of known absolute magnitude, it is possible, using a simple formula, to determine the distance by measuring the apparent magnitude.

It appears, therefore, that although space can be explored for a distance of only a few hundred light-years by direct trigonometric methods, we are not forced, by our inability to measure still smaller angles, to extrapolate uncertainly or to make vague guesses relative to farther regions of space, for the trigonometrically determined distances can be used to calibrate the tools of newer and less restricted methods. For example, the trigonometric methods of measuring the distance to moon, sun, and nearer stars are decidedly indirect, compared with the linear measurement of distance on the surface of the earth, but they are not for that reason inexact or questionable in principle. The spectroscopic and photometric methods of measuring great stellar distance are also indirect, compared with the trigonometric measurement of small stellar distance, but they, too, are not for that reason unreliable or of doubtful value. These great distances are not extrapolations. For instance, in the spectroscopic method, the absolute magnitudes derived from trigonometrically measured distances are used to derive the curves relating spectral characteristics to absolute magnitude and the spectroscopic parallaxes for individual stars (whether near or remote) are, almost without exception, interpolations. Thus the data for nearer stars are used for purposes of calibration, not as a basis for extrapolation.

By one method or the other, the distance of nearly 3,000 individual stars in the solar neighborhood have now been determined only a few are within ten light-years of the sun. As a distance of about 130 light-years we find the Hyades, the well known cluster of naked eye stars at a distance of 600 light-years, according to Kapteyn's extensive investigations, we come to the group of blue stars in Orion - another physically -organized cluster composed of giants in luminosity. At distances comparable to the above values we also find the Scorpio-Centaurus group, the Pleiades, the Ursa Major system.

There nearby clusters are specifically referred to for two reasons.

In the first place I desire to point out the prevalence through-out all the galactic system of clusters of stars, variously organized as to stellar density and total stellar content. The gravitational organization of stars is a fundamental feature in the universe - a double star is one aspect of a stellar cluster, a galactic system is another. We may indeed, trace the clustering motive from the richest of isolated globular clusters such as the system in Hercules, to the loosely organized nearby groups typified in the bright stars of Ursa Major. At one hundred times its present distance, the Orion cluster would look much like Messier 37 or Messier 11 scores of telescopic clusters have the general form and star density of the Pleiades and the Hyades. The difference between bright and faint clusters of the galactic system naturally appears to be solely a matter of distance.

In the second place I desire to emphasize the fact that the nearby stars we use as standards of luminosity, particularly the blue stars of spectral type B, are members of stellar clusters. Therein lies a most important point in the application of photometric methods. We might, perhaps, question the validity of comparing the isolated stars in the neighborhood of the sun with stars in a compact cluster but the comparison of nearby cluster stars with remote cluster stars is entirely reasonable, since we are now so far from the primitive anthropocentric notions that it is foolish to postulate that distance from the earth has anything to do with the intrinsic brightness of stars.

ON THE DISTANCES OF GLOBULAR CLUSTERS

1. As stated above, astronomers agree on the distances to the nearby stars and stellar groups - the scale of the part of the universe that we may call the solar domain. But as yet there is lack of agreement relative to the distances of remote clusters, stars, and star clouds - the scale of the total galactic system. The disagreement in this last particular is not a small difference of a few percent, an argument on minor detail it is a matter of a thousand percent or more.

Curtis maintains that the dimensions I find for the galactic system should be divided by ten or more (see quotation on page 172) therefore, that galactic size does not stand in the way of interpreting spiral nebulae as comparable galaxies (a theory that he favors on other grounds but considers incompatible with the larger values of galactic dimensions). In his Washington address, however, he greatly simplified the present discussion by accepting the results of recent studies on the following significant points:

Proposition A. - The globular clusters form a part of our galaxy therefore the size of the galactic system proper is most probably not less than the size of subordinate system of globular clusters.

Proposition B. - The distances derived at Mount Wilson for globular clusters relative to one another are essentially correct. This implies among other things that (1) absorption of light in space has not appreciably affected the results, and (2) the globular clusters are much alike in structure and constitution, differing mainly in distance. (These relative values are based upon apparent diameters, integrated magnitudes, the magnitude of individual giants or groups of giants, and Cepheid variables Charlier has obtained much the same results from apparent diameters alone, and Lundmark from apparent diameters and integrated magnitudes.)

Proposition C. - Stars in clusters and in distant parts of the Milky Way are not peculiar - that is, uniformity of conditions and of stellar phenomena naturally prevails throughout the galactic system.

We also share the same opinion, I believe, on the following points:

a. The galactic system is an extremely flattened stellar organization, and the appearance of a Milky Way is partly due to the existence of distinct clouds of stars, and is partly the result of depth along the galactic plane.

b. The spiral nebulae are mostly very distant objects, probably not physical members of our galactic system.

c. If our galaxy approaches the larger order of dimensions, a serious difficultly at once arises for the theory that spirals are galaxies of stars comparable in size with our own: it would be necessary to ascribe impossibly great magnitudes to the new stars that have appeared in the spiral nebulae.

2. Through approximate agreement on the above points, the way is cleared so that the outstanding difference may be clearly stated: Curtis does not believe that the numerical value of the distance I derive for any globular cluster is or the right order of magnitude.

3. The present problem may be narrowly restricted therefore, and may be reformulated as follows: Show that any globular cluster is approximately as distant as derived at Mount Wilson then the distance of the other clusters will be approximately right (see Proposition B), the system of clusters and the galactic system will have dimensions of the order assigned (see Proposition A), and the "comparable galaxy" theory of spirals will have met with a serious, though perhaps not insuperable difficulty.

In other words, to maintain my position it will suffice to show that any one of the bright globular clusters has roughly the distance in light-years given below, rather than a distance one tenth of this value or less [1]:

[1: In the final draft of the following paper Curtis has qualified his acceptance of the foregooing propositions in such a manner that in some numerical details the comparisons given below are no longer accurately applicable to his arguments I believe, however, that the comparisons do correctly contrast the present view with that generally accepted a few years ago.]

Similarly it should suffice to show that the bright objects in clusters are giants (cf. last column above), rather than stars of solar luminosity.

4. From observation we know that some or all of these four clusters contain:

a. AN interval of at least nine magnitudes (apparent and absolute) between the brightest and faintest stars.

b. A range of color-index from -0.5 to +2.0, corresponding to the whole range of color commonly found among assemblages of stars.

c. Stars of types B, A, F, G, K, M (from direct observations of spectra), and that these types are in sufficient agreement with the color classes to permit the use of the latter for ordinary statistical considerations where spectra are not yet known.

d. Cepheid and cluster variables which are certainly analogous to galactic variables of the same types, in spectrum, color change, length of period, amount of light variation, and all characters of the light curve.

e. Irregular, red, small-range variables of the Alpha Orionis type, among the brightest stars of the cluster.

f. Many red and yellow stars of approximately the same magnitude as the blue stars, in obvious agreement with the giant star phenomena of the galactic system, and clearly in disagreement with all we know of color and magnitude relation for dwarf stars.

First, as globular cluster is a pretty complete "universe" by itself, with typical and representative stellar phenomena, including several classes of stars that in the solar neighborhood are recognized as giants in luminosity.

Second, we are very fortunately situated for the study of distant clusters - outside rather than inside. Hence we obtain a comprehensive dimensional view, we can determine relative real luminosities in place of relative apparent luminosities, and we have the distinct advantage that the most luminous stars are easily isolated and the most easily studied. None of the brightest stars in a cluster escapes us. If giants or super-giants are there, they are necessarily the stars we study. We cannot deal legitimately with the average brightness of stars in globular clusters, because the faintest limits are apparently far beyond our present telescopic power. Out ordinary photographs record only the most powerful radiators - encompassing a range of but three or four magnitudes at the very top of the scale of absolute luminosity, whereas in the solar domain we have a known extreme range of 20 magnitudes in absolute brightness, and a generally studied interval of twelve magnitudes or more. 6. Let us examine some of the conditions that would exist in the Hercules Cluster (Messier 13) on the basis of the two opposing values for its distance:

a. The blue stars. - The colors of stars have long been recognized as characteristic of spectral types and as being of invaluable aid in the study of faint stars for which spectroscopic observations are difficult or impossible. The color-index, as used at Mount Wilson, is the difference between the so-called photographic (pg) and photovisual (pv.) magnitudes - the difference between the brightness of objects in blue-violet and in yellow-green light. For a negative color-index (C. I. = pg. - pv.

Definition of units employed. - The distance traversed by light in one year, 9.5x10^12 km., or nearly six trillion miles, known as the light-year, has been in use for about two centuries as a means of visualizing stellar distances, and forms a convenient and easily comprehended unit. Throughout this paper the distances of the stars will be expressed in light-years.

The absolute magnitude of a star is frequently needed in order that we may compare the luminosities of different stars in terms of some common unit. It is the apparent magnitude which the star would have if viewed from the standard distance of 32.6 light-years (corresponding to a parallax of 0".1).

Knowing the parallax, or the distance, of a star, the absolute magnitude may be computed from one of the simple equations:

Limitations in studies of galactic dimensions. - By direct methods the distances of individual stars can be determined with considerable accuracy out to a distance of about two hundred light-years.

At a distance of three hundred light-years (28 x 10^14 km.) the radius of the earth's orbit (1.5 x 10^8 km) subtends an angle slightly greater than 0".01, and the probable error of the best modern photographic parallax determinations has not yet been reduced materially below this value. The spectroscopic method of determining stellar distance through the absolute magnitude probably has, at present, the same limitations as the trigonometric method upon which the spectroscopic method depends for its absolute scale.

A number of indirect methods have been employed which extend out reach into space somewhat farther for the average distances of large groups or classes of stars, but give not information as to the individual distances of the stars of the group or class. Among such methods may be noted as most important the various correlations which have been made between the proper motions of the stars and the parallactic motion due to the speed of our sun in space, or between the proper motions and the radial velocities of the stars.

The limitations of such methods of correlation depend, at present, upon the fact that accurate proper motions are known, in general, for the brighter stars only. A motion of 20 km/sec. across our line of sight will produce the following annual proper motions:

The average probable error of the proper motions of Boss is about 0".006. Such correlation methods are not, moreover, a simple matter of comparison of values, but are rendered difficult and to some extent uncertain by the puzzling complexities brought in by the variation of the space motions of the stars with spectral type, stellar mass (?), stellar luminosity (?), and still imperfectly known factors of community star drift.

[Text editor's comment: The question marks appearing in the previous paragraph appear in the published document.]

It will then be evident that the base-line available in studies of the more distant regions of our galaxy in woefully short, and that in such studies we must depend largely upon investigations of the distribution and of the frequency of occurrence of stars of the different apparent magnitudes and spectral types, on the assumption that the more distant stars, when taken in large numbers, will average about the same as known nearer stars. This assumption is a reasonable one, though not necessarily correct, as we have little certain knowledge of galactic regions as distant as five hundred light-years.

Were all the stars of approximately the same absolute magnitude, or if this were true even for the stars of any particular type or class, the problem of determining the general order of the dimensions of our galaxy would be comparatively easy.

But the problem is complicated by the fact that, taking the stars of all spectral types together, the dispersion in absolute luminosity is very great. Even with the exclusion of a small number of stars which are exceptionally bright or faint, this dispersion probably reaches ten absolute magnitudes, which would correspond to a hundred-fold uncertainty in distance for a given star. However, it will be seen later that we possess moderately definite information as to the average absolute magnitude of the stars of the different spectral types.

Dimensions of our galaxy. - Studies of the distribution of the stars and of the ratio between the numbers of stars of successive apparent magnitudes have led a number of investigators to the postulation of fairly accordant dimensions for the galaxy a few may be quoted:
Wolf about 14,000 light-years in diameter.
Eddington about 15,000 light-years.
Shapley (1915) about 20,000 light-years.
Newcomb not less than 7,000 light-years later - perhaps 30,000 light-years in diameter and 5,000 light-years in thickness.
Kapteyn about 60,000 light-years. [1]

[1: A complete bibliography of the subject would fill many pages. Accordingly, references to authorities will in general be omitted. An excellent and nearly complete list of references may be found in Lundmark's paper, - "The Relations of the Globular Clusters and Spiral nebulae to the Stellar System," in K. Svenska Vet. Handlingar, Bd. 60, No. 8, p. 71, 1920.]

General structure of the galaxy. - From the lines of investigation mentioned above there has been a similar general accord in the deduced results as to the shape and structure of the galaxy:

1. The stars are not infinite in number, nor uniform in distribution.

2. Our galaxy, delimited for us by the projected contours of the Milky Way, contains possibly a billion suns.

3. This galaxy is shaped much like a lens, or a thin watch, the thickness being probably less than one-sixth of the diameter.

4. Our Sun is located fairly close to the center of figure of the galaxy.

5. The stars are not distributed uniformly through the galaxy. A large proportion are probably actually within the ring structure suggested by the appearance of the Milky Way, or are arranged in large and irregular regions of greater star density. The writer believes that the Milky Way is at least as much a structural as a depth effect.

A spiral structure has been suggested for our galaxy the evidence for such a spiral structure is not very strong, except as it may be supported by the analogy of the spirals as island universes, but such a structure is neither impossible nor improbable. The position of our Sun near the center of figure of the galaxy is not a favorable one for the precise determination of the actual galactic structure.

Relative paucity of galactic genera. - Mere size does not necessarily involve complexity it is a remarkable fact that in a galaxy of a thousand million objects we observe, not ten thousand different types, but perhaps not more than five main classes, outside the minor phenomena of our own solar system.

1. The stars. - The first and most important class is formed by the stars. In accordance with the type of spectrum exhibited, we may divide the stars into some eight or ten main types even when we include the consecutive internal gradations within these spectral classes it is doubtful whether present methods will permit us to distinguish as many as a hundred separate subdivisions in all. Average space velocities vary from 10 to 30 km/sec., there being a well- marked increase in average space velocity as one proceeds from the blue to the redder stars.

2. The globular star clusters are greatly condensed aggregations of from ten thousand to one hundred thousand stars. Perhaps one hundred are known. Though quite irregular in grouping, they are generally regarded as definitely galactic in distribution. Space velocities are of the order of 300 km/sec.

3. The diffuse nebulae are enormous, tenuous, cloud-like masses fairly numerous always galactic in distribution. They frequently show a gaseous spectrum, though many agree approximately in spectrum with their involved stars. Space velocities are very low.

4. The planetary nebulae are small, round, or oval, and almost always with a central star. Fewer than one hundred and fifty are known. They are galactic in distribution spectrum is gaseous space velocities are about 80 km/sec.

5. The spirals. - Perhaps a million are within reach of large reflectors the spectrum is generally like that of a star cluster. They are emphatically non-galactic in distribution, grouped about the galactic poles, spiral in form. Space velocities are of the order of 1200 km/sec.

Distribution of celestial genera. - With one, and only one, exception, all known genera of celestial objects show such a distribution with respect to the plane of our Milky Way, that there can be no reasonable doubt that all classes, save this one, are integral members of our galaxy. We see that all the stars, whether typical, binary, variable, or temporary, even the rarer types, show this unmistakable concentration toward the galactic plane. So also for the diffuse and the planetary nebulae and, though somewhat less definitely, for the globular star clusters.

The one exception is formed by the spirals grouped about the poles of our galaxy, they appear to abhor the regions of the greatest star density. They seem clearly a class apart. Never found in our Milky Way, there is no other class of celestial objects with their distinctive characteristics of form, distribution, and velocity in space.

The evidence at present available points strongly to the conclusion that the spirals are individual galaxies, or island universes, comparable with our own galaxy in dimensions and in number of component units. While the island universe theory of the spirals is not a vital postulate in a theory of galactic dimensions, nevertheless, because of its indirect bearing on the question, the arguments in favor of the island universe hypothesis will be included with those which touch more directly on the probable dimensions of our own galaxy.

Other theories of galactic dimensions. - From evidence to be referred to later Dr. Shapley has deduced very great distances for the globular star clusters, and holds that our galaxy has a diameter comparable with the distances which he has derived for the clusters, namely, a galactic diameter of about 300,000 light-years, or at least ten times greater than formerly accepted. The postulates of the two theories may be outlined as follows:

Our galaxy is probably not more than 30,000 light-years in diameter, and perhaps 5,000 light-years in thickness

The clusters, and all other types of celestial objects except the spirals, are component parts of our own galactic system.

The spirals are a class apart, and not intra- galactic objects. As island universes, of the same order of size as our galaxy, they are distant from us 500,000 to 10,000,000, or more, light years.

The galaxy is approximately 300,000 light- years in diameter, and 30,000 or more, light- years in thickness.

The globular clusters are remote objects, but a part of our own galaxy. The most distant cluster is placed about 220,000 light-years away.

The spirals are probably of nebulous constitution, and possibly not members of our own galaxy, driven away in some manner from the regions of greatest star density.

EVIDENCE FURNISHED BY THE MAGNITUDE OF THE STARS

The "average" star. - It will be of advantage to consider the two theories of galactic dimensions from the standpoint of the average star. What is the "average" or most frequent type of star of our galaxy or of a globular cluster, and if we can with some probability postulate such an average star, what bearing will the characteristics of such a star have upon the question of its average distance from us?

No adequate evidence is available that the more distant stars of our galaxy are in any way essentially different from stars of known distance nearer to us. It would seem then that we may safely make such correlations between the nearer and the more distant stars, en masse. In such comparisons the limitations of spectral type must be observed as rigidly as possible, and results based upon small numbers of stars must be avoided, if possible.

Many investigations notably Shapley's studies of the colors of stars in the globular clusters, and Fath's integrated spectra of these objects and of the Milky Way, indicate that the average star of a star cluster or of the Milky Way will, in the great majority of cases, be somewhat like our Sun in spectral type, i. e., such an average star will be, in general, between spectral types F and K.

Characteristics of F-K type stars of known distance. - The distances of stars of type F-K in our own neighborhood have been determined in greater number, perhaps, than for the stars of any other spectral type, so that the average absolute magnitude of stars of this type seams fairly well determined. There is every reason to believe, however, that our selection of stars of these or other types for direct distance determinations has not been a representative one. Our parallax programs have a tendency to select stars either of great luminosity or of great space velocity.

Kapteyn's values for the average absolute magnitudes of the stars of the various spectral types are as follows:

The same investigator's most recent luminosity-frequency curve places the maximum of frequency of the stars in general, taking all the spectral types together, at absolute magnitude +7.7.

A recent tabulation of about five hundred modern photographically determined parallaxes places with average absolute magnitude of stars of type F-K at about +4.5.

The average absolute magnitude of five hundred stars of spectral types F to M is close to +4, as determined spectroscopically by Adams.

It seems certain that the two last values of the average absolute magnitude are too low, that is, - indicate too high an average luminosity, due to the omission from our parallax programs of the intrinsically fainter stars. The absolute magnitudes of the dwarf stars are, in general, fairly accurately determined the absolute magnitudes of many of the giant stars depend upon small and uncertain parallaxes. In view of these facts we may somewhat tentatively take the average absolute magnitude of F-K stars of known distance as not brighter than +6 some investigators would prefer a value of +7 or +8.

Comparison of Milky Way stars with the "average" stars. - We may take, without serious error, the distances of 10,000 and 100,000 light-years respectively, as representing the distance in the two theories from our point in space to the central line of the Milky Way structure. Then the following short table may be prepared:

It will be seen from the above table that the stars of apparent magnitudes 16 to 20, observed in our Milky Way structure in such great numbers, and, from their spectrum, believed to be predominantly F-K in type, are of essentially the same absolute luminosity as known nearer stars of these types, if assumed to be at the average distance of 10,000 light-years. The greater value postulated for the galactic dimensions requires, on the other hand, an enormous proportion of giant stars.

Proportion of giant stars among stars of known distance.- All existing evidence indicates that the proportion of giant stars in a given region of space is very small. As fairly representative of several investigations we may quote Schouten's results, in which he derives an average stellar density of 166,000 stars in a cube 500 light-years on a side, the distribution in absolute magnitude being as follows:

Comparison of the stars of the globular clusters with the "average" star.- From a somewhat cursory study of the negatives of ten representative globular clusters I estimate the average apparent visual magnitude of all the stars in these clusters as in the neighborhood of the eighteenth. More powerful instruments may eventually indicate a somewhat fainter mean value, but it does not seem probable that this value is as much as two magnitudes in error. We then have:

Here again we see that the average F-K star of a cluster, if assumed to be at a distance of 10,000 light-years, has an average luminosity about the same as that found for known nearer stars of this type. The greater average distance of 100,000 light-years requires a proportion of giant stars enormously greater than is found in those regions of our galaxy of which we have fairly definite distance data.

While it is not impossible that the clusters are exceptional regions of space and that, with a tremendous spatial concentration of suns, there exists also a unique concentration of giant stars, the hypothesis that cluster stars are, on the whole, like those of known distance seems inherently the more probable.

It would appear, also, that galactic dimensions deduced from correlations between large numbers of what we may term average stars must take precedence over values found from small numbers of exceptional objects, and that, where deductions disagree, we have a right to demand that a theory of galactic dimensions based upon the exceptional object of class shall not fail to give an adequate explanation of the usual object or class.

The evidence for greater galactic dimensions. - The arguments for a much larger diameter for our galaxy than that hitherto held, and the objections which have been raised against the island universe theory of the spirals rest mainly upon the great distances which have been deduced for the globular star clusters.

I am unable to accept the thesis that the globular clusters are at distances of the order of 100,000 light-years, feeling that much more evidence is needed on this point before it will be justifiable to assume that the cluster stars are predominantly giants rather than average stars. I am also influenced, perhaps unduly, by certain fundamental uncertainties in the data employed. The limitations of space available for the publication of this portion of the discussion unfortunately prevents a full treatment of the evidence. In calling attention to some of the uncertainties in the basal data, I must disclaim any spirit of captious criticism, and take this occasion to express my respect to Dr. Shapley's point of view, and my high appreciation of the extremely valuable work which he has done on the clusters. I am willing to accept correlations between large masses of stellar data, whether of magnitudes, radial velocities, or proper motions, but I feel that the dispersion in stellar characteristics is too large to permit the use of limited amounts of any sort of data, particularly when such data is of the same order as the probable errors of the methods of observation.

The deductions as to the very great distances of the globular clusters rest, in the final analysis, upon three lines of evidence:

1. Determination of the relative distances of the clusters on the assumption that they are objects of the same order of actual size.

2. Determination of the absolute distances of the clusters through correlations between Cepheid variable stars in the clusters and in our galaxy.

3. Determination of the absolute distances of the clusters through a comparison of their brightest stars with the intrinsically brightest stars of our galaxy.

Of these three methods, the second is given most weight by Shapley.

It seems reasonable to assume that the globular clusters are of the same order of actual size, and that from their apparent diameters the relative distances may be determined. The writer would not, however, place undue emphasis upon this relation. There would seem to be no good reason why there may not exist among these objects a reasonable amount of difference in actual size, say from three- to five-fold, differences which would not prevent them being regarded as of the same order of size, but which would introduce considerable uncertainty into the estimates of relative distance.

The evidence from Cepheid variable stars. - This portion of Shapley's theory rests upon the following three hypotheses or lines of evidence:

A. That there is a close coordination between absolute magnitude and length of period for the Cepheid variables of our galaxy, similar to the relation discovered by Miss Leavitt among Cepheids of the Smaller Magellanic Cloud.

B. That, if of identical periods, Cepheids anywhere in the universe have identical absolute magnitudes.

C. This coordination of absolute magnitude and length of period for galactic Cepheids, the derivation of the absolute scale for their distances and the distances of the clusters, and, combined with A) and B), the deductions therefrom as to the much greater dimensions of our galaxy, depend almost entirely upon the sized and the internal relationships of the proper motions of eleven Cepheid variables.

Under the first heading, it will be seen later that the actual evidence for such a coordination among galactic Cepheids is very weak. provided that the Smaller Magellanic Cloud is not in some way a unique region of space, the behavior of the Cepheid variables in this Cloud is, through analogy, perhaps the strongest argument for postulating a similar phenomenon among Cepheid variables of our galaxy.

Unfortunately there is a large dispersion in practically all the characteristics of the stars. That the Cepheids lack a reasonable amount of such dispersion is contrary to all experience for the stars in general. There are many who will regard the assumption made under B) above as a rather drastic one.

If we tabulate the proper motions of these eleven Cepheids, as given by Boss, and their probable errors as well, it will be seen that the average proper motion of these eleven stars is of the order of one second of arc per century in either coordinate that the average probable error is nearly half this amount, and that the probable errors of half of these twenty-two coordinates may well be described as of the same size as the corresponding proper motions.

Illustrations bearing on the uncertainty of proper motions of the order of 0".01 per year might be multiplied at great length. The fundamental and unavoidable errors in our star positions, the probable errors of meridian observations, the uncertainty in the adopted value of the constant of precession, the uncertainties introduced by the systematic corrections applied to different catalogues, all have comparatively little effect when use is made of proper motions as large as ten seconds or arc per century. Proper motions as small as one second of arc per century are, however, still highly uncertain quantities, entirely aside from the question of the possible existence of systematic errors. As an illustration of the differences in such minute proper motions as derived by various authorities, the proper motions of three of the best determined of this list of eleven Cepheids, as determined by Auwers, are in different quadrants for those derived by Boss.

There seems no good reason why the smaller coordinates of this list of twenty-two may not eventually prove to be different by once or twice their present magnitude, with occasional changes of sign. So small an amount of presumably uncertain data is insufficient to determine the scale of our galaxy, and many will prefer to wait for additional material before accepting such evidence as conclusive.

1. The known uncertainties of small proper motions, and,

2. The known magnitude of the purely random motions of the stars, the determination of individual parallaxes from individual proper motions can never give results of value, though the average distances secured by such methods of correlation from large numbers of stars are apparently trustworthy. The method can not be regarded as a valid one, and this applies whether the proper motions are very small or are of appreciable size.

As far as the galactic Cepheids are concerned, Shapley's curve of coordination between absolute magnitude and length of period, though found through the mean absolute magnitude of the group of eleven, rests in reality upon individual parallaxes determined from individual proper motions, as may be verified by comparing his values for the parallax of these eleven stars with [1] the values found directly from the upsilon component of the proper motion (namely, - that component which is parallel to the Sun's motion) and the solar motion. The differences in the two sets of values, 0".0002 in the mean, arise from the rather elaborate system of weighting employed.

[1: Mt. Wilson Contr. No. 151, Table V.]

The final test of a functional relation is the agreement obtained when applied to similar data not originally employed in deducing the relation. We must be ready to allow some measure of deviation in such a test, but when a considerable proportion of the other available data fails to agree within a reasonable amount, we shall be justified in withholding our decision.

If the curve of correlation deduced by Shapley for galactic Cepheids is correct in both its absolute and relative scale, and if it is possible to determine individual distances from the individual proper motions, the curve of correlation, using the same method as far as the proper motions are concerned (the validity of which I do not admit), should fit fairly well with other available proper motion and parallax data. The directly determined parallaxes are known for five of this group of eleven, and for five other Cepheids. There are, in addition, twenty-six other Cepheids or which proper motions have been determined. One of these was omitted by Shapley because of irregularity of period, one for irregularity of the light curve, two because the proper motions were deemed of insufficient accuracy, two because the proper motions are anomalously large the proper motions of the others have been recently investigated at the Dudley Observatory, but have less weight than those of the eleven Cepheids used by Shapley.

[Text - processor's editor's note: Technical limitations may have caused the figure cited in Dr. Curtis' text not to be reproduced in this version. The figure caption is included however, and it is believed that the line of argument is clear.]

Fig. 1. - Agreement of other data with the luminosity - period correlation curve. Absolute magnitudes calculated from the upsilon component of the proper motion are indicated by circles the eleven employed by Shapley are marked with a bar. Blackdots represent directly determined parallaxes. The arrows attached to the circles at the upper edge of the diagram indicate that either the parallax or the upsilon component of the proper motion is negative, and the absolute magnitude indeterminate in consequence.

In Figure 1 the absolute magnitudes are plotted against the logarithm of the period the curve is taken from Mt. Wilson Contr. No. 151, and is that finally adopted by Shapley after the introduction of about twelve Cepheids of long period in clusters, twenty-five from the Smaller Magellanic Cloud, and a large number of short period cluster-type variables in clusters with periods less than a day, which have little effect on the general shape of the curve. The barred circles represent the eleven galactic Cepheids employed by Shapley, the black dots those Cepheids for which parallaxes have been determined, while the open circles indicate variables for which proper motions have since become available, or not employed originally by Shapley. For the stars at the upper edge of the diagram, the attached arrows indicate that either the parallax, or the upsilon component of the proper motion is negative, so that the absolute magnitude is indeterminate, and may be anything from infinity down.

From the above it would seem that available observational data lend little support to the fact of a period-luminosity relation among galactic Cepheids. In view of the large discrepancies shown by other members of the group when plotted on this curve, it would seem wiser to wait for additional evidence as to proper motion, radial velocity, and, if possible, parallax, before entire confidence can be placed in the hypothesis that the Cepheids and cluster-type variables are invariably super-giants in absolute luminosity.

Argument from the intrinsically brightest stars. - If the luminosity-frequency law is the same for the stars of the globular clusters as for our galaxy, it should be possible to correlate the intrinsically brightest stars of both regions and thus determine cluster distances. It would seem, a priori, that the brighter stars of the clusters must be giants, or at least approach that type, if the stars of the clusters are like the general run of stars. Through the application of a spectroscopic method Shapley has found that the spectra of the brighter stars in clusters resemble the spectra of galactic giant stars, a method which should be exceedingly useful after sufficient tests have been made to make sure that in this phenomenon, as is unfortunately the case in practically all stellar characteristics, there is not a large dispersion, and also whether slight differences in spectral type may at all materially affect the deductions.

The average "giant" star. - Determining the distance of Messier 3 from the variable stars which it contains, Shapley then derives absolute magnitude -1.5 as the mean luminosity of the twenty-five brightest stars in this cluster. From this mean value, -1.5, he then determines the distances of other clusters. Instead, however, of determining cluster distances of the order of 100,000 light-years by means of correlations - on a limited number of Cepheid variables, a small and possibly exceptional class, and from the distances thus derived deducing that the absolute magnitudes of many of the brighter stars in the cluster are as great as -3, while a large proportion are greater than -1, it would seem preferable to begin the line of reasoning with the attributes of known stars in our neighborhood, and to proceed from them to the clusters.

What is the average absolute magnitude of a galactic giant star? On this point there is room for honest difference of opinion, and there will doubtless be many who will regard the conclusions of this paper as ultra conservative. Confining ourselves to existing observational data, there is no evidence that a group of galactic giants, of average spectral type about G5, will have a mean absolute magnitude as great as -1.5 it is more probably in the neighborhood of +1.5, or three absolute magnitudes fainter, making Shapley's distances four times too large.

Russell's suggestion is worth quoting in this connection, written in 1913, when parallax data were far more limited and less reliable than at present:

"The giant stars of all the spectral classes appear to be about the same mean brightness, - averaging a little above absolute magnitude zero, that is, about a hundred times as bright as the Sun. Since the stars of this series . . . have been selectee by apparent brightness, which gives a strong preference to those of greatest luminosity, the average brightness of all the giant stars in a given region of space must be less than this, perhaps considerably so."

Some reference has already been made to the doubtful value of parallaxes of the order of 0".010, and it is upon such small or negative parallaxes that most of the very great absolute luminosity in present lists depend. It seems clear that parallax work should aim at using as faint comparison stars as possible, and that the corrections applied to reduce relative parallaxes to absolute parallaxes should be increased very considerably over what was thought acceptable ten years ago.

From a study of the plotted absolute magnitudes by spectral type of about five hundred modern direct parallaxes, with due regard to the uncertainties of minute parallaxes, and keeping in mind that most of the giants will be of types F to M, there seems little reason for placing the average absolute magnitude of such giant stars as brighter than +2.

The average absolute magnitude for the giants in Adams's list of five hundred spectroscopic parallaxes is +1.1. The two methods differ most in the stars of type G, where the spectroscopic method shows a maximum at +0.6, which is not very evident in the trigonometric parallaxes.

In such moving star clusters as the Hyades group, we have thus far evidently observed only the giant stars of such groups.

The mean absolute magnitude of forty-four stars believed to belong to the Hyades moving cluster is +2.3. The mean absolute magnitude of the thirteen stars of types F, G, and K, is +2.4. The mean absolute magnitude of the six brightest stars is +0.8 (two A5, one G, and three of K type).

The Pleiades can not fittingly be compared with such clusters or the globular clusters its composition appears entirely different as the brightest stars average about B5, and only among the faintest stars of the cluster are there any as late as F in type. The parallax of this group is still highly uncertain. With Schouten's value of 0".037 the mean absolute magnitude of the six brightest stars is +1.6.

With due allowance for the redness of the giants in clusters, Shapley's mean visual magnitude of the twenty-five brightest stars in twenty-eight globular clusters is about 14.5. Then, from the equation given in the first section of this paper we have, -
+2 = 14.5 + 7.6 - 5 x log distance,
or, log distance = 4.02 = 10,500 light-years as the average distance.

If we adopt instead the mean value of Adams +1.1, the distance becomes 17,800 light- years.

Either value for the average distance of the clusters may be regarded as satisfactorily close to those postulated for a galaxy of the smaller dimensions held in this paper, in view of the many uncertainties in the data. Either value, also, will give on the same assumptions a distance of the order of 30,000 light-years for a few of the faintest and apparently most distant clusters. I consider it very doubtful whether any cluster is really so distant as this, but find no difficulty in provisionally accepting it as a possibility, without thereby necessarily extending the main structure of the galaxy to such dimensions. While the clusters seem concentrated toward our galactic plane, their distribution in longitude is a most irregular one, nearly all lying in the quadrant between 270 degrees and 0 degrees. If the spirals are galaxies of stars, their analogy would explain the existence of frequent nodules of condensation (globular clusters) lying well outside of and distinct from the main structure of a galaxy. From the minuteness of their proper motions, most investigators have deduced vary great luminosities for such stars in our galaxy. Examining Kapteyn's values for stars of this type, it will be seen that he finds a range in absolute magnitude from +3.25 to -5.47. Dividing the 433 stars of his lists into two magnitude groups, we have:

Either the value for the brighter stars, -1.32, or the mean of all, -0.36, is over a magnitude brighter than the average absolute magnitude of the giants of the other spectral types among nearer galactic stars. Now this galactic relation is apparently reversed in such clusters as M. 3 or M. 13, where the B-type stars are about three magnitudes fainter than the brighter K and M stars and about a magnitude fainter than those of G type. Supposing that the present very high values for the galactic B-type stars are correct, if we assume similar luminosity for those in the clusters we must assign absolute magnitudes of -3 to -6 to the F to M stars of the clusters, for which we have no certain galactic parallel, with a distance of perhaps 100,000 light-years. On the other hand, if the F to M stars of the cluster are like the brighter stars of these types in the galaxy, the average absolute magnitude of the B-type stars will be only about +3, and too low to agree with present values for galactic B stars. I prefer to accept the latter alternative in this dilemma, and to believe that there may exist B-type stars of only two to five times the brightness of the Sun.

While I hold to a theory of galactic dimensions approximately one-tenth of that supported by Shapley, if does not follow that I maintain this ratio for any particular cluster distance. All that I have tried to do is to show that 10,000 light-years is a reasonable average cluster distance. There are so many assumptions and uncertainties involved that I am most hesitant in attempting to assign a given distance to a given cluster, a hesitancy which is not diminished by a consideration of the following estimates of the distance of M. 13 (The Great Cluster in Hercules).

It should be stated here that Shapley's earlier estimate was merely a provisional assumption for computational illustration, but all are based on modern material, and illustrate the fact that good evidence may frequently be interpreted in different ways.

My own estimate, based on the general considerations outlined earlier in this paper, would be about 8,000 light-years, and it would appear to me, at present, that this estimate is perhaps within fifty per cent of the truth.

THE SPIRALS AS EXTERNAL GALAXIES

The spirals. - If the spirals are island universes it would seem reasonable and most probable to assign to them dimensions of the same order as our galaxy. If, however, their dimensions are as great as 300,000 light-years, the island universes must be placed at such enormous distances that it would be necessary to assign what seem impossibly great absolute magnitudes to the novae which have appeared in these objects. For this reason the island universe theory has an indirect bearing on the general subject of galactic dimensions, though it is, of course, entirely possible to hold both to the island universe theory and to the belief in the greater dimensions for our galaxy by making the not improbable assumption that our own island universe, by chance, happens to be several fold larger than the average.

Some of the arguments against the island universe theory of the spirals have been cogently put by Shapley, and will be quoted here for reference. It is only fir to state that these earlier statements do not adequately represent Shapley's present point of view, which coincides somewhat more closely with that held by the writer.

"With the plan of the sidereal system here outlined, it appears unlikely that the spiral nebulae can be considered separate galaxies of stars. In addition to the evidence heretofore existing, the following points seem opposed to the "island universe" theory (a) the dynamical character of the region of avoidance (b) the size of the galaxy (c) the maximum luminosity attainable by a star (d) the increasing commonness of high velocities among other sidereal objects, particularly those outside the region of avoidance . . . the cluster work strongly suggests the hypothesis that spiral nebulae . . . are, however, members of the galactic organization . . . the novae in spirals may be considered as the engulfing of a star by the rapidly moving nebulosity. " (Publ. Astron. Soc. of the Pacific, Feb. 1918, p. 53.)

"The recent work on star clusters, in so far as it throws some light on the probable extent and structure of the galactic system, justifies a brief reconsideration of the question of external galaxies, and apparently leads to the rejection of the hypothesis that spiral nebulae should be interpreted as separate stellar systems.

"Let us abandon the comparison with the galaxy and assume an average distance for the brighter spirals that will give a reasonable maximum absolute magnitude for the novae (and in a footnote- provisionally, let us say, of the order of 20,000 light-years). Further, it is possible to explain the peculiar distribution of the spirals and their systematic recession by supposing them repelled in some manner from the galactic system, which appears to move as a whole through a nebular field of indefinite extent. But the possibility of these hypotheses is of course not proposed as competent evidence against the "island universe" theory. . . Observation and discussion of the radial velocities, internal motions, and distribution of the spiral nebulae, of the real and apparent brightness of novae, of the maximum luminosity of galactic and cluster stars, and finally of the dimensions of our own galactic system, all seem definitely to oppose the "island universe" hypothesis of the spiral nebulae . . . (Publ. Astron. Soc. of the Pacific, Oct. 1919, pp. 261 ff.)

The dilemma of the apparent dimensions of the spirals. - In apparent size the spirals range from a diameter of 2 degrees (Andromeda), to minute flecks 5", or less, in diameter.

They may possibly vary in actual size, roughly in the progression exhibited by their apparent dimensions.

The general principle of approximate equality of size for celestial objects of the same class seems, however, inherently the more probable, and has been used in numerous modern investigations, e. g. by Shapley in determining the relative distances of the clusters.

On the principle of approximate equality of actual size:

Their probable distances range from about 500,000 light-years (Andromeda), to distances of the order 100,000,000 light- years.

At 500,000 light-years the Nebula of Andromeda would be 17,000 light-years in diameter, or of the same order of size as our galaxy.

If the Nebulae of Andromeda is but 20,000 light-years distant, the minute spirals would need to be at distances of the order of 10,000,000 light-years, or far outside the greater dimensions postulated for the galaxy.

If all are galactic objects, equality of size must be abandoned, and the minute spirals assumed to be a thousand-fold smaller than the largest.

The spectrum of the spirals. -

The spectrum of the average spiral is indistinguishable from that given by a star cluster.

It is approximately F-G in type, and in general character resembles closely the integrated spectrum of our Milky Way.

It is just such a spectrum as would be expected from a vast congeries of stars.

The spectrum of the spirals offers no difficulties on the island universe theory.

If the spirals are intragalactic, we must assume that they are some sort of finely divided matter, or of gaseous constitution.

In either case we have no adequate and actually existing evidence by which we may explain their spectrum.

Many diffuse nebulosities of our galaxy show a bright-line gaseous spectrum. Others associated with bright stars, agree with their involved stars in spectrum, and are well explained as a reflection or resonance effect.

Such an explanation seems untenable for most of the spirals.

The distribution of the spirals.- The spirals are found in greatest numbers just where the stars are fewest (at the galactic poles), and not at all where the stars are most numerous (in the galactic plane). This fact makes it difficult, if not impossible, to fit the spirals into any coherent scheme of stellar evolution, either as a point of origin, or as a final evolutionary product. No spiral has as yet been found actually within the structure of the Milky Way. This peculiar distribution is admittedly difficult to explain on any theory. This factor of distribution in the two theories may be contrasted as follows:

It is most improbable that our galaxy should, by mere chance, be placed about half way between the two great groups of island universes.

So many of the edgewise spirals show peripheral rings of occulting matter that this dark ring may well be the rule rather than the exception.

If our galaxy, itself a spiral on the island universe theory, possesses such a peripheral ring of occulting matter, this would obliterate the distant spirals in our galactic plane, and would explain the peculiar apparent distribution of the spirals.

There is some evidence for such occulting matter in our galaxy.

With regard to the observed excess of velocities of recession, additional observations may remove this. Part of the excess may well be due to the motion of our own galaxy in space. The Nebula of Andromeda is approaching us.

If the spirals are galactic objects, they must be a class apart from all other known types why none in our neighborhood?

Their abhorrence of the regions of greatest star density can only be explained on the hypothesis that they are, in some unknown manner, repelled by the stars.

We know of no force adequate to produce such a repulsion, except perhaps light-pressure.

Why should this repulsion have invariably acted essentially at right angles to our galactic plane?

Why have not some been repelled in the direction of our galactic plane?

The repulsion theory, it is true, is given some support by the fact that most of the spirals observed to date are receding from us.

The space velocities of the spirals. -

The spirals observed to date have the enormous average space velocity of 1200 km/sec.

In this velocity factor they stand apart from all galactic objects.

Their space velocity is one hundred times that of the galactic diffuse nebulosities, about thirty times the average velocity of the stars, ten times that of the planetary nebulae, and five times that of the clusters.

Such high speeds seem possible for individual galaxies.

Our own galaxy probably has a space velocity, relatively to the system of the spirals, of several hundred km/sec. Attempts have been made to derive this from velocities of the spirals, but are uncertain as yet, as we have the radial velocities of but thirty spirals.

Space velocities of several hundred km/sec. have been found for a few of the fainter stars.

It has been argued that an extension of radial velocities surveys to the fainter stars would possibly remove the discrepancy between the velocities of the stars and those of the spirals.

This is possible, but does not seem probable. The faint stars thus far selected for investigation have been stars of known large proper motions. They are exceptional objects through this method of selection, not representative objects.

High space velocities are still the exception, not the rule, for the stars of our galaxy.

Proper motions of the spirals.- Should the results of the next quarter-century show close agreement among different observers to the effect that the annual motions of translation or rotation of the spirals equal or exceed 0".01 in average value, it would seem that the island universe theory must be definitely abandoned.

A motion of 700 km/sec. across our line of sight will produce the following annual proper motions:

The older visual observations of the spirals have so large a probable error as to be useless for the determinations of proper motions, if small the available time interval for photographic determinations is less than twenty-five years.

The first proper motion given above should inevitably have been detected by either visual or photographic methods, from which it seems clear that the spirals can not be relatively close to us at the poles of our flattened galactic disk. In view of the hazy character of the condensations measured, I consider the trustworthy determination of the second proper motion given above impossible by present methods without a much longer time interval than is at present available for the third and the fourth, we should need centuries.

New stars in the spirals.- Within the past few years some twenty-seven new stars have appeared in spirals, sixteen of these in the Nebula of Andromeda, as against about thirty-five which have appeared in our galaxy in the last three centuries. So far as can be judged from such faint objects, the novae in spirals have a life history similar to that of the galactic novae, suddenly flashing out, and more slowly, but still relatively rapidly, sinking again to a luminosity ten thousand-fold less intense. Such novae form a strong argument for the island universe theory and furnish, in addition, a method of determining the approximate distances of spirals.

With all its elements of simplicity and continuity, our universe is too haphazard in its details to warrant deductions from small numbers of exceptional objects. Where no other correlation is available such deductions must be made with caution, and with a full appreciation of the uncertainties involved.

It seems certain, for instance, that the dispersion of the novae in the spirals, and probably also in our galaxy, may reach at least ten absolute magnitudes, as is evidenced by a comparison of S Andromedae with faint novae found recently in this spiral. A division into two magnitude classes is not impossible.

Tycho's Nova, to be comparable in absolute magnitude with some recent galactic novae, could not have been much more than ten light-years distant. If as close to us as one hundred light-years it must have been of absolute magnitude -8 at maximum if only one thousand light- years away, it would have been of absolute magnitude -13 at maximum.

The distances and absolute magnitudes of but four galactic novae have been thus far determined the mean absolute magnitude is -3 at maximum, and +7 at minimum.

These mean values, though admittedly resting upon a very limited amount of data, may be compared with the fainter novae which have appeared in the Nebula of Andromeda somewhat as follows: where 500,000 light-years is assumed for this spiral on the island universe hypothesis and, for comparison, the smaller distance of 20,000 light-years.

It will be seen from the above that, at the greater distance of the island universe theory, the agreement in absolute magnitude is quite good for the galactic and spiral novae. If as close as 20,000 light years, however, these novae must be unlike similar galactic objects, and of unusually low absolute magnitude at minimum. Very few stars have thus far been found as low in luminosity as absolute magnitude +13, corresponding, at this distance, to apparent magnitude 27.

"The simple hypothesis that the novae in spirals represent the running down of ordinary galactic stars by the rapidly moving nebulosity becomes a possibility on the basis of distance (i. e., 20,000 light-years) for the brighter spirals are within the edges of the galactic system (Shapley)."

This hypothesis of the origin of the novae in spirals is open to grave objections. It involves:

1. That the stars thus overtaken are of smaller absolute luminosity than the faintest thus far observed, with very few exceptions.

2. That these faint stars are extraordinarily numerous, a conclusion which at variance with the results of star counts, which seem to indicate that there is a marked falling off in the number of stars below apparent magnitude 19 or 20.

As an illustration of the difficulties which would attend such a hypothesis, I have made a count of the stars in a number of areas about the Nebula of Andromeda, including, it is believed, stars at least as faint as magnitude 19.5, and find a star density, including all magnitudes, of about 6,000 stars per square degree.

If no more than 20,000 light-years distant this spiral will lie 7,000 light-years from the plane of the Milky Way, and if moving at the rate of 300 km/sec., it will sweep through 385 cubic light-years per year.

To make the case as favorable as possible for the hypothesis suggested, assume that none of the 6,000 stars per square degree are as close as 15,000 light-years, but that all are arranged in a stratum extending 5,000 light-years each way from the spiral.

Then the Nebula of Andromeda should encounter one of these stars every 520 years. Hence the actual rate at which novae have been found in this spiral would indicate a star density about two thousand times as great as that shown by the count each star would occupy about one square second of arc on the photographic plate.

The spirals as island universes: summary. -

1. On the theory we avoid the almost insuperable difficulties involved in an attempt to fit the spirals in any coherent scheme of stellar evolution, either as a point of origin, or as an evolutionary product.

2. On this theory it is unnecessary to attempt to coordinate the tremendous space velocities of the spirals with those of the average star.

3. The spectrum of the spirals is such as would be expected from a galaxy of stars.

4. A spiral structure has been suggested for our own galaxy, and is not improbable.

5. If island universes, the new stars observed in the spirals seem a natural consequence of their nature as galaxies. Correlations between the novae in the spirals and those in our galaxy indicate distances ranging from perhaps 500,000 light-years in the case of the Nebula of Andromeda, to 10,000,000 or more light-years for the more remote spirals.

6. At such distances, these island universes would be of the same order of size as our own galaxy.

7. Very many spirals show evidence of peripheral rings of occulting matter in their equatorial planes. Such a phenomenon in our galaxy, regarded as a spiral, would serve to obliterate the distant spirals in our galactic plane, and would furnish an adequate explanation of the otherwise inexplicable distribution of the spirals.

There is a unity and an internal agreement in the features of the island universe theory which appeals very strongly to me. The evidence with regard to the dimensions of the galaxy, on both sides, is too uncertain as yet to permit of any dogmatic pronouncements. There are many points of difficultly in either theory of galactic dimensions, and it is doubtless true that many will prefer to suspend judgement until much additional evidence is forthcoming. Until more definite evidence to the contrary is available, however, I feel that the evidence for the smaller and commonly accepted galactic dimensions is still the stronger and commonly accepted galactic dimensions is still the stronger and that the postulated diameter of 300,000 light-years must quite certainly be divided by five and perhaps by ten.

I hold, therefore, to the belief that the galaxy is probably not more than 30,000 light-years in diameter that the spirals are not intra-galactic objects but island universes, like our own galaxy, and that the spirals, as external galaxies, indicate to us a greater universe into which we may penetrate to distances of ten million to a hundred million light-years.

[Transcribed to text-processor format by Robert Nemiroff. Every attempt has been made to transcribe the text in original form, including typos. Much text in italics format did not remain in italics format in this version. New typos may have been added, however, so I cannot guarantee the above text is a perfect reproduction. For precise quotes, please refer to the original text: Curtis, H. 1921, Bull. Nat. Res. Coun., 2, 194 Shapley, H. 1921, Bull. Nat. Res. Coun., 2, 171]

## Question about total absolute magnitudes of galaxies - negative or not? - Astronomy

It has been stated that the Tully-Fisher (TF) relation is the workhorse'' of peculiar velocity surveys. One can anticipate a time in the not so distant future when more accurate techniques may supplant it, but for the next few years at least, the TF relation is likely to remain the most widely used distance indicator in cosmic velocity studies. Its role in such studies to date has been, in fact, too large to be reviewed here, and interested readers are referred to Strauss & Willick (1995) Section 7. Several recent developments are discussed in Section 8.

The TF relation is one of the most fundamental properties of spiral galaxies. It is the empirical statement of an approximately power-law relation between luminosity and rotation velocity. Specifically, it is found that

or, using the logarithmic formulation preferred by working astronomers,

In equation (4) M = -2.5 log(L) + const. is the absolute magnitude, and the velocity width parameter log(2vrot) - 2.5, where vrot is expressed in km s -1 , is a useful dimensionless measure of rotation velocity.

An important fact, not always sufficiently appreciated, is that the power law exponent does not have a unique value. The details of both the photometric and spectroscopic measurements affect it. A typical result found in contemporary studies is 3. The corresponding value of the TF slope,'' b, is

7.5. However, slight changes in the details of measurement can result in significant changes in b. This is illustrated in Figure 2, in which TF relations in four bandpasses are plotted. The optical bandpasses (B, R, and I) all represent data from the sample of Mathewson et al. (1992). In each case the slope is 0.2µ . It follows that an error A in the TF zero point thus corresponds to a fractional distance error f = 10 -0.2A . A Hubble constant inferred from such distances will then be off by a factor f -1 . For peculiar velocities, calibration of the TF relation consists in choosing A such that d 10 0.2[m - (A - b )] gives a galaxy's distance in km s -1 . There is no requirement that it yield the distance in Mpc. Nonetheless, as mentioned above, zero point calibration errors are still possible. Errors in A produce distances in km s -1 that differ by a fraction f from the true Hubble velocity, with resultant peculiar velocity errors vp = -fd.