Astronomy

How to calculate rest-frame luminosity in a specific wavelength band?

How to calculate rest-frame luminosity in a specific wavelength band?


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For instance, I have a set of templates whose y is in Jy and x is in Angstroms. I have both of their rest-frame and obs-frame wavelength (i.e. redshift). I want to know their luminosity in one specific waveband like rest-frame UKIRT.H (i.e.,~1.6um, L1.6) or IRAC2 (i.e.,~4.5um, L4.5).

I'm confused about how to calculate the rest-frame luminosity. For now, I convolve the rest wavelength and response curve so get an average flux density and times effective wavelength of the filter to get flux. And using redshift to get luminosity.

Am I right? I'm a little concern that maybe I should convolve obs wavelength and response curve?


How to calculate rest-frame luminosity in a specific wavelength band? - Astronomy

Aims: We study the evolution of the galaxy population up to z˜ 3 as a function of its colour properties. In particular, luminosity functions and luminosity densities were derived as a function of redshift for the blue/late and red/early populations.
Methods: We use data from the GOODS-MUSIC catalogue, which have typical magnitude limits z 850 ≤ 26 and K_s≤ 23.5 for most of the sample. About 8% of the galaxies have spectroscopic redshifts the remaining have well calibrated photometric redshifts derived from the extremely wide multi-wavelength coverage in 14 bands (from the U band to the Spitzer 8

μm band). We have derived a catalogue of galaxies complete in the rest-frame B-band, which has been divided into two subsamples according to their rest-frame U-V colour (or derived specific star formation rate) properties.
Results: We confirm a bimodality in the U-V colour and specific star formation rate of the galaxy sample up to z˜ 3. This bimodality is used to compute the luminosity functions of the blue/late and red/early subsamples. The luminosity functions of the blue/late and total samples are well represented by steep Schechter functions evolving in luminosity with increasing redshifts. The volume density of the luminosity functions of the red/early populations decreases with increasing redshift. The shape of the red/early luminosity functions shows an excess of faint red dwarfs with respect to the extrapolation of a flat Schechter function and can be represented by the sum of two Schechter functions. Our model for galaxy formation in the hierarchical clustering scenario, which also includes external feedback due to a diffuse UV background, shows a general broad agreement with the luminosity functions of both populations, the larger discrepancies being present at the faint end for the red population. Hints on the nature of the red dwarf population are given on the basis of their stellar mass and spatial distributions.


How to calculate rest-frame luminosity in a specific wavelength band? - Astronomy


Aims: We measure and study the evolution of the UV galaxy luminosity function (LF) at z = 3-5 from the largest high-redshift survey to date, the Deep part of the CFHT Legacy Survey. We also give accurate estimates of the SFR density at these redshifts.
Methods: We consider

100 000 Lyman-break galaxies at z ≈ 3.1, 3.8 and 4.8 selected from very deep ugriz images of this data set and estimate their rest-frame 1600 Å luminosity function. Due to the large survey volume, cosmic variance plays a negligible role. Furthermore, we measure the bright end of the LF with unprecedented statistical accuracy. Contamination fractions from stars and low-z galaxy interlopers are estimated from simulations. From these simulations the redshift distributions of the Lyman-break galaxies in the different samples are estimated, and those redshifts are used to choose bands and calculate k-corrections so that the LFs are compared at the same rest-frame wavelength. To correct for incompleteness, we study the detection rate of simulated galaxies injected to the images as a function of magnitude and redshift. We estimate the contribution of several systematic effects in the analysis to test the robustness of our results.
Results: We find the bright end of the LF of our u-dropout sample to deviate significantly from a Schechter function. If we modify the function by a recently proposed magnification model, the fit improves. For the first time in an LBG sample, we can measure down to the density regime where magnification affects the shape of the observed LF because of the very bright and rare galaxies we are able to probe with this data set. We find an increase in the normalisation, ϕ*, of the LF by a factor of 2.5 between z ≈ 5 and z ≈ 3. The faint-end slope of the LF does not evolve significantly between z ≈ 5 and z ≈ 3. We do not find a significant evolution of the characteristic magnitude in the studied redshift interval, possibly because of insufficient knowledge of the source redshift distribution. The SFR density is found to increase by a factor of

2 from z ≈ 5 to z ≈ 4. The evolution from z ≈ 4 to z ≈ 3 is less eminent.

Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.


How to calculate rest-frame luminosity in a specific wavelength band? - Astronomy

Context. The star formation rate density (SFRD) evolution presents an area of great interest in the studies of galaxy evolution and reionization. The current constraints of SFRD at z > 5 are based on the rest-frame UV luminosity functions with the data from photometric surveys. The VIMOS UltraDeep Survey (VUDS) was designed to observe galaxies at redshifts up to ∼6 and opened a window for measuring SFRD at z > 5 from a spectroscopic sample with a well-controlled selection function.
Aims: We establish a robust statistical description of the star-forming galaxy population at the end of cosmic HI reionization (5.0 ≤ z ≤ 6.6) from a large sample of 49 galaxies with spectroscopically confirmed redshifts. We determine the rest-frame UV and Lyα luminosity functions and use them to calculate SFRD at the median redshift of our sample z = 5.6.
Methods: We selected a sample of galaxies at 5.0 ≤ z spec ≤ 6.6 from the VUDS. We cleaned our sample from low redshift interlopers using ancillary photometric data. We identified galaxies with Lyα either in absorption or in emission, at variance with most spectroscopic samples in the literature where Lyα emitters (LAE) dominate. We determined luminosity functions using the 1/V max method.
Results: The galaxies in this redshift range exhibit a large range in their properties. A fraction of our sample shows strong Lyα emission, while another fraction shows Lyα in absorption. UV-continuum slopes vary with luminosity, with a large dispersion. We find that star-forming galaxies at these redshifts are distributed along the main sequence in the stellar mass vs. SFR plane, described with a slope α = 0.85 ± 0.05. We report a flat evolution of the specific SFR compared to lower redshift measurements. We find that the UV luminosity function is best reproduced by a double power law, while a fit with a Schechter function is only marginally inferior. The Lyα luminosity function is best fitted with a Schechter function. We derive a logSFRD UV (M ⊙ yr -1 Mpc -3 ) = -1.45 +0.06 -0.08 and logSFRD Lyα (M ⊙ yr -1 Mpc -3 ) = -1.40 +0.07 -0.08 . The SFRD derived from the Lyα luminosity function is in excellent agreement with the UV-derived SFRD after correcting for IGM absorption.
Conclusions: Our new SFRD measurements at a mean redshift of z = 5.6 are ∼0.2 dex above the mean SFRD reported in Madau & Dickinson (2014, ARA&A, 52, 415), but in excellent agreement with results from Bouwens et al. (2015a, ApJ, 803, 34). These measurements confirm the steep decline of the SFRD at z > 2. The bright end of the Lyα luminosity function has a high number density, indicating a significant star formation activity concentrated in the brightest LAE at these redshifts. LAE with equivalent width EW > 25 Å contribute to about 75% of the total UV-derived SFRD. While our analysis favors low dust content in 5.0 < z < 6.6, uncertainties on the dust extinction correction and associated degeneracy in spectral fitting will remain an issue, when estimating the total SFRD until future surveys extending spectroscopy to the NIR rest-frame spectral domain, such as with JWST.

Based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, under Large Program 185.A-0791.


2. Relativistic Astronomy

2.1. Relativistic Effects

When a camera travels in space with a speed close to c, some interesting relativistic effects would occur. For example, Christian & Loeb (2017) suggested that an interferometer moving at a relativistic speed offers a sensitive probe of acceleration making use of the temporal Terrell effect (Penrose 1959 Terrell 1959).

Here we focus on the observational distortions of emission from distant astronomical objects. In the comoving frame of the probe, all astronomical objects undergo a unique Doppler boost (Doppler factor ) or deboost ( ) depending on the Lorentz factor of the probe and the angular between the object with respect to the direction of probe motion. For the problem involving a flying probe, one can define two rest frames 4 : the Earth rest frame or the laboratory frame (which is also the rest frame of astronomical objects), K, and the probe comoving frame, K'. Let us define the Lorentz factor of the probe as , where β = v/c is the normalized speed of the probe. In Frame K', all the astronomical objects move with the same Lorentz factor Γ, but with different angles with respect to the opposite direction of the probe motion. The Doppler factor of the source is defined as (Rybicki & Lightman 1979)

where the angle between the object moving direction and the line of sight in two different frames are related through

Some characteristic angles and Doppler factors are

The Doppler factor connects the quantities of the source rest frame (frame K in our convention) and those in the probe comoving frame (frame K' in our convention). In particular, for the camera on board the probe all the emission is blueshifted (redshifted) for ( ), i.e.,

For an isotropic, point source, the specific flux and flux transformations are (Appendix A)

For an extended source, the specific flux and flux transformations for an emission pixel reads (Appendix A)

These salient relativistic effects provide a unique opportunity to study the universe and to test the principle of special relativity, which may be discussed under the broad umbrella of "relativistic astronomy."

2.2. Universe Seen from the Probe's Frame K'

For a wide-field camera moving with a Lorentz factor Γ, all the objects in the field of view undergo relativistic distortions, including shift of position (Equation (2)), shift of frequency (Equation (3)), and change of specific flux and flux (Equations (4) and (5)). The degree of distortion, characterized by the Doppler factor (Equation (1)), solely depends on the angle θ with respect to the direction of motion for a constant Γ. In general, an extended object is bluer and more compact as observed in Frame K'. Since astronomers already have a detailed view in Frame K, a measurement of the differences in the observed properties between Frames K and K' offers valuable information about the astronomical sources.

For a probe with a constant velocity, based on the position shifts of three point sources (background stars) close to the direction of motion (where the sources are squeezed), one can uniquely determine the direction of motion and the Lorentz factor Γ (or dimensionless velocity β) of the probe (Appendix B).

Once the direction of motion and Γ are determined, one can calculate the Doppler factor of all the celestial objects. Given the same observing frequency of the camera, the intrinsic frequencies of different astronomical objects are different (corrected by the respective of the source). For a direction with θ < θc, one has , so that the intrinsic frequency of the source that the camera records is redder than the observed frequency. Conversely, for θ > θc ( ), the intrinsic frequency of the source of the camera records is bluer than the observed frequency. As a result, for a camera in a particular band (e.g., R band), one may study other frequencies (e.g., infrared (IR) or ultraviolet (UV)) of different sources without the need of using technically challenging IR/UV cameras. In a sense, a relativistically moving camera is a natural spectrograph.

In the regime, the fluxes of the sources are enhanced. So a relativistically moving camera is also a natural lens. So, in practice, astronomical objects are better studied in the regime, where the source is studied in an intrinsically redder band.

If a camera continuously observes an astronomical source as the camera is accelerated, it would record emission of the source in a span of frequencies, so that one can obtain a detailed spectrum of the source in the frequency range between νcamera and . The higher the achievable , the wider the spectrograph is. For different frequencies, one should properly correct for the respective flux Doppler boosting factors to get the intrinsic flux at those frequencies in order to retrieve the intrinsic spectrum of the source.

The camera itself may be designed to have a grism spectrograph. In this case, after determining the direction of motion and the probe Lorentz factor using photometry observations, one may turn on the grism mode to capture the fine spectra of the sources in a different (redder) spectral regime.

Due to the light aberration effect, the objects in the moving direction are more packed. The entire hemisphere in Frame K is combed into a cone defined by the angle . Given the same field of view, the camera can observe more objects. This effectively increases the field of view of the camera.

In order to visually show how the astronomical objects look differently in Frame K', we carry out some simulations based on true observational images. The upper panel of Figure 1 shows an observed image of the nearby galaxy M51. We use the HST images in three filters: F435W, F555W, and F814W. 5 The filter information is from the SVO filter profile service website. 6 In our simulation, we adopt β = 0.5 (Γ 1.7321). The simulation code loops through all pixels of the input image, and uses the counts of a pixel in different bands to generate a simple spectrum for that particular pixel. Then the spectrum undergoes the relativistic transformation according to Equations (3) and (5). Integration of the product of transformed spectrum and the response functions of the filters give rise to the new counts in different bands of that pixel in Frame K'. For each pixel, the polar coordinate angles are calculated relative to the direction of motion, which is set to (1075, 1525) pixel in the image. Using Equation (2) the observed polar angle from the moving direction in the probe frame (θ') can be calculated. With the image scale (0.2 arcsec/pixel), we then obtain the new position of the pixel in the simulated image. In order to simulate the color scheme of the image, we use an image editor named GIMP (GNU Image Manipulation Program) that enables the simulation with an image in more than three bands. The color of each layer of the image is represented by a hue value in GIMP that can be set manually by the user. However, the hue value of images in the same band should be kept the same to show the true color difference between the field of view in Frames K and K' due to the relativistic effects. The hue values of F435W, F555W, and F658N images after relativistic transformation are set to 42, 159, and 232, which are the same as the hue values in the original image. Due to the relativistic effects, the spectrum of M51 is shifted toward blue, meaning that the IR band spectrum is shifted into the optical band. In our simulation, we have used the M51 image in the F814W filter as the IR data, and shifted it to the Hα (F658N) filter range in the simulated image.

Figure 1. Comparison of the image of a nearby galaxy M51 in Frames K and K'. Top: false color HST image of M51 in three bands (F435W, F555W, and F814W) observed in the rest frame of Earth, i.e., Frame K. Bottom: simulated false color image of M51 as observed in the probe's comoving frame K'. The dimensionless velocity is set to β = 0.5 (Γ 1.7321), and the pixel scale of the image is 0.2 arcsec/pixel. The direction of motion is set to the (1075, 1525) pixel in the image (cross in both images).

The simulated image is presented in the lower panel of Figure 1. One can see that the simulated image is indeed bluer, brighter, and more compact (i.e., each extended source becomes smaller, and the two extended sources have a smaller spatial separation).

2.3. Examples of Astronomical Applications

The impacts of transrelativistic cameras on astronomy should be multi-fold and may not be fully appreciated until they become available. In the following, we discuss several examples that suggest these cameras may make important contributions to astrophysics.

2.3.1. Reionization History of the Universe

The first example is to study the reionization history of the universe (Loeb & Barkana 2001 Fan et al. 2006). The hot Big Bang theory predicts that the universe became neutralized around z

1100, when electrons and protons recombined to produce neutral hydrogen. This is the epoch of the cosmic microwave background radiation. The universe was later reionized by the first objects (first stars, galaxies, and quasars) in the universe that shine in UV and X-rays (above 13.6 eV). The epoch between recombination and reionization is sometimes called the "cosmic dark ages." Quasar observations suggest that reionization was nearly complete around z

6 (Fan et al. 2006). However, the exact detail of the reionization history is not known. Different models predict different neutron fractions as a function of redshift (e.g., Holder et al. 2003 and references therein). In order to map the reionization history in detail, one needs to populate bright beacons in the redshift range from 6 to

20, and measure the spectrum blueward of the redshifted Lyα line, i.e., λ ≤ (1 + z)1215.67 Å (or ≥ 10.2/(1+z) eV). A mostly neutral intergalactic medium would essentially absorb emission in this regime, forming the so-called Gunn–Peterson trough (Gunn & Peterson 1965). The shape of the red damping wing carries the important information of the neutral fraction of the IGM in that epoch (Miralda-Escudé 1998). However, such studies are hindered by the fact that the feature is moved to progressively more infrared bands as z increases, and that the sources are typically fainter at higher z as well.

Sending high-β probes toward high-z galaxies or quasars would make it more convenient to measure their redshifts and to study the detailed red damping wing of these objects. This is because the Gunn–Peterson trough is shifted to the bluer bands in the probe's frame and the source flux is enhanced.

Table 1 presents the relevant parameters for relativistic astronomy for different β values. The parameters include Lorentz factor Γ, maximum Doppler factor , its third power , and the relevant redshift, , for the Lyα wavelength (1215.67 Å) to be at a particular wavelength λ in Frame K'. For example, with β

10 can be probed with a 1 μm camera, and the source is brighter by

10 can even be probed with an R-band (λ

0.658 μm) camera, and the source is brighter by more than 2 mag. Since an optical grism is much easier to build, one can use relativistic cameras with different termination speeds to probe a range of redshifts to most thoroughly study the reionization history of the universe.

Table 1. Relevant Parameters for Relativistic Astronomy for Different β Values

β Γ zLyα (λ)
0 1 1 1 8.2259 (λ/1 μm) −1
0.1 1.0050 1.1055 1.3512 9.0941 (λ/1 μm) −1
0.2 1.0206 1.2247 1.8371 10.0746 (λ/1 μm) −1
0.3 1.0483 1.3628 2.5309 11.2100 (λ/1 μm) −1
0.4 1.0911 1.5275 3.5642 12.5653 (λ/1 μm) −1
0.5 1.1547 1.7321 5.1962 14.2477 (λ/1 μm) −1
0.6 1.2500 2.0000 8.0000 16.4518 (λ/1 μm) −1
0.7 1.4003 2.3805 13.4894 19.5816 (λ/1 μm) −1
0.8 1.6667 3.0000 27.0000 24.6777 (λ/1 μm) −1
0.9 2.2942 4.3589 82.8191 35.8559 (λ/1 μm) −1
0.95 3.2026 6.2450 243.555 51.3708 (λ/1 μm) −1
0.99 7.0888 14.1067 2807.20 116.0408 (λ/1 μm) −1

2.3.2. Redshift Desert

The redshift interval 1.4 z 2.5 has been described by some authors as the "redshift desert" due the lack of strong spectral lines in the optical band (4300–9000) Å. Since this redshift range coincides with the epoch of significant star formation, the lack of a large sample of galaxies in this redshift range hinders an unbiased mapping of the star formation history of the universe (Steidel et al. 2004 and references therein). Observations with transrelativistic cameras can easily fill this gap. One does not need a very high β in order to achieve this goal. For example, with β = 0.2, one has a Doppler factor range from to (depending on the observational direction θ'). This is already enough to remove the redshift desert. In particular, for the redshift range (1.9, 2.5), one can use the θ' = 0 mode, so that the effective redshift range in frame K' is changed to 1.2247 × (2.9, 3.5) −1 = (2.55, 3.29), which is outside the desert. Similarly, for the redshift range (1.4, 1.9), one can use the θ' = π mode, so that the effective redshift range is 0.8165 × (2.4, 2.9) −1 = (0.96, 1.37), again outside the desert.

2.3.3. Gamma-Ray Bursts (GRBs)

GRBs are the most luminous astrophysical objects in the universe. In the case of observing transient relativistic events such as GRBs, relativistic astronomy would allow humankind to study a relativistically moving source by a relativistically moving observer for the first time, a scenario previously only imagined in a thought experiment. Catching the early laboratory-frame IR afterglow of a GRB using an optical camera on board a high-β probe would help to identify very high-z GRBs. Theoretical models suggest that GRBs might form as early as z

20 when the first-generation stars die (e.g., Toma et al. 2009 Mészáros & Rees 2010). A β

18.6 with a 1 μm camera. A systematic study of these explosions very early in the universe will help to probe the deep dark ages of the universe (Tanvir et al. 2011).

A good fraction of GRBs (30%–50%) are "optically dark" GRBs. Even though high-z GRBs may comprise a portion, most of them may be embedded in dusty star-forming regions, so that the optical emission is absorbed via dust extinction (e.g., Perley et al. 2009). If a relativistic camera is launched after the trigger of a GRB and an observation is carried out during the acceleration of the camera, given the same observational frequency, the camera would continuously observe a range of frequency toward the IR regime, which will catch the characteristic features of dust extinction. Combining afterglow modeling, one may also precisely map the extinction curve of the GRB host galaxy, which is currently poorly constrained (e.g., Scahdy et al. 2012).

2.3.4. Electromagnetic Counterpart of Gravitational Waves (GWs)

The new era of multi-messenger astrophysics just arrived recently with the detection of the first double neutron star (NS–NS) merger system GW170817 and its associated GRB 170817A and multiband electromagnetic counterpart (e.g., Abbott et al. 2017a, 2017b). One important phenomenon is the so-called "kilonova," a type of IR/optical transient arising from the r-process of neutron-rich materials dynamically ejected during the merger (Li & Paczyński 1998 Metzger et al. 2010). The kilonova associated with GW170817 appeared to have a "red" component and a "blue" component (e.g., Villar et al. 2017), with the former likely associated with the high-opacity ejecta, possibly involving heavy elements such as lanthanides. Understanding these events will greatly benefit from a careful study of the spectra in a broad range from IR to UV. In the future relativistic astronomy era, NS–NS and NS–BH mergers will be regularly detected by the next-generation GW detectors. Observations with the transrelativistic cameras in the directions toward and away from the GW trigger direction, together with ground-based observations, will help to uncover the broadband spectra of kilonovae, leading to an in-depth study of the NS–NS and NS–BH merger physics.

2.4. Testing Special Relativity

One can also use the observations of transrelativistic cameras to test the principles of spectral relativity. There are two ways to do so.

1. As seen from Figure 2 and discussed in Appendix B, the measurement of relative positions of three point sources in Frame K' as compared with those measured in K (on Earth) can uniquely solve the motion of the camera (the direction and the Lorentz factor). With this information, one can predict the positions of the fourth, fifth . point sources in the sky in Frame K'. If within the field of view of the camera there are more than three sources, a comparison between the observed and predicted positions in Frame K' offers a unique test of the effect of aberration of light in special relativity. So far, a direct test of aberration of light has been made via observing the parallaxes of distant stars (through the very nonrelativistic motion of Earth orbiting the Sun e.g., Hirshfeld 2001) or via Earth-based experiments to measure a small gravitational aberration of light (e.g., Kopeikin & Fomalont 2007). A transrelativistic camera will open the window to test this effect in the relativistic regime.

Figure 2. Geometry to solve the motion of the probe. The three bright point sources are marked as 1, 2, 3, respectively, in Frame K (left) and 1', 2', 3', respectively, in Frame K' (right). The direction of motion is marked as 0 and 0', respectively, in Frames K and K'. The relevant angular separations and opening angles are marked in Frame K but not fully in Frame K'.

2. A comparison of the observations of the same bright object in two different frames (K and K') at the same intrinsic frequency offers another way to test the principle of special relativity. In nonstandard theories, such as massive electrodynamics, the Doppler factor may take a form that slightly deviates from the simplest form Equation (1). A tight upper limit on the deviation of the measured (specific) flux at frequency in Frame K' and that at ν in Frame K (properly correcting for the Doppler boosting effect) would give a tight constraint on the deviation of from Equation (1), and hence, the violation of the principle of special relativity. No similar test has been performed so far.


Title: The Rest-frame Optical (900 nm) Galaxy Luminosity Function at z ∼ 4–7: Abundance Matching Points to Limited Evolution in the M/M Ratio at z ≥ 4

We present the first determination of the galaxy luminosity function (LF) at z ∼ 4, 5, 6, and 7, in the rest-frame optical at λ∼900 nm (z′ band). The rest-frame optical light traces the content in low-mass evolved stars (∼stellar mass—M ), minimizing potential measurement biases for M . Moreover, it is less affected by nebular line emission contamination and dust attenuation, is independent of stellar population models, and can be probed up to z ∼ 8 through Spitzer/IRAC. Our analysis leverages the unique full-depth Spitzer/IRAC 3.6–8.0 μm data over the CANDELS/GOODS-N, CANDELS/GOODS-S, and COSMOS/UltraVISTA fields. We find that, at absolute magnitudes where M> is fainter than ≳−23 mag, M> linearly correlates with M. At brighter M>, M presents a turnover, suggesting that the stellar mass-to-light ratio M/L could be characterized by a very broad range of values at high stellar masses. Median-stacking analyses recover an M/L> roughly independent on M> for M>≳−23 mag, but exponentially increasing at brighter magnitudes. We find that the evolution of the LF marginally prefers a pure luminosity evolution over a pure density evolution, with the characteristic luminosity decreasingmore » by a factor of ∼5× between z ∼ 4 and z ∼ 7. Direct application of the recovered M/L> generates stellar mass functions consistent with average measurements from the literature. Measurements of the stellar-to-halo mass ratio at fixed cumulative number density show that it is roughly constant with redshift for M≳10M. This is also supported by the fact that the evolution of the LF at 4≲z≲7 can be accounted for by a rigid displacement in luminosity, corresponding to the evolution of the halo mass from abundance matching. « less


How to calculate rest-frame luminosity in a specific wavelength band? - Astronomy

The flux within some specific wavelength range can be calculated if we consider equations (3), (5) and (6). Then we have (see Ellis 1971),

Therefore, the flux measured in the frequency range , + d by the observer may be written as

F is also called specific flux of the radiation.

The apparent magnitude in a specific observed frequency bandwidth is obtained from a different form than given by equation (7), which may written as

where W () is the function which defines the spectral interval of the observed flux (the standard UBV system, for instance). This is a sensitivity function of the atmosphere, telescope and detecting device.

From equations (14) and (15) the apparent magnitude in a specified spectral interval W may be written as

Some remarks about this equation are important to mention. Firstly, equation (16) calculates the apparent magnitude of a source whose intrinsic luminosity at a specific redshift is somehow known. Secondly, in a similar manner this equation can also be used to calculate the intrinsic luminosity of a cosmological source whose redshift and apparent magnitude are known from observations. Finally, since cosmological sources do evolve, the intrinsic luminosity L changes according to the evolutionary stage of the source, and therefore, L is actually a function of the redshift L = L ( z ). Hence, in order to use equation (16) to obtain the apparent magnitude evolution of the source, some theory for luminosity evolution is also necessary. For galaxies, L ( z ) is usually derived taking into consideration the theory of stellar evolution, from where some simple equations for luminosity evolution can be drawn (see Binney & Tremaine 1987, p. 552 Peebles 1993, p. 330, and references therein). (4) Finally, since J[(1 + z )] is a property of the source at a specific redshift, this function must be known in order to calculate the apparent magnitude, unless the K-correction approach is used (see below).

For magnitude limited catalogues, the luminosity distance and the observer area distance have both an upper cutoff, which is a function of the apparent magnitude, the frequency bandwidth used in the observations and the luminosity of the sources. Considering equation (1), the luminosity distance of flux limited sources may be written as

The relations above demand the knowledge of both the source spectrum and the redshift. However, when the source spectrum is not known, it is necessary to introduce a correction term in order to obtain the bolometric flux from observations. This correction is known as the K-correction, and it is a different way for allowing the effect of the source spectrum.

The method that will be presented next for deriving the K-correction follows the classical work of Humason, Mayall and Sandage (1956, appendix B see also Oke & Sandage 1968, Sandage 1988, 1995). We start by calculating the difference in magnitude produced by the bolometric flux F and the flux F W measured by the observer, but at the bandwidth W () in any redshift z . Therefore, I shall write both quantities as F ( z ) and F W( z ) respectively. Since, by definition, we know that

the difference in magnitude m ( z ) will be given by

The rate between the observed flux F W( z ) at a given redshift and at z = 0 defines the K-correction. Then, considering equation (19), we have that

which means that once we know the K-term and the observed magnitude m W, the bolometric magnitude is know within a constant m (0). If we now substitute equation (14) into equation (20), it is easy to show that

Remembering that by equation (5) we know that we can have the source spectrum transformed from the rest frame of the source to the rest-frame of the observer by a factor of (1 + z ), that is, J[(1 + z )] d = [J(G) d G] / (1 + z ), then we may also write equation (23) as

Note that the equations above allow us to write theoretical K-correction expressions for any given spacetime geometry, provided that the line element dS 2 is known beforehand. These theoretical expressions for observables like the K-correction could, in principle, be directly compared with observations.

As a final remark, it is obvious that if the source spectrum is already known, all relevant observational relations can be calculated without the need of the K-correction.

With the calculations above we can obtain the theoretical expression for the colour of the sources for any given spacetime. Let us consider two bandwidths W and W' . From equation (16) we can find the difference in apparent magnitude for these two frequency bands in order to obtain an equation for the colour of the source in a specific redshift. Let us call this quantity C WW' . Thus,

Considering that cosmological sources do evolve, they should emit different luminosities in different redshifts due to the different evolutionary stages of the stellar contents of the sources, and this is reflected in the equation above by the source spectrum function which may be different for different redshifts. Note, however, that in the equation above the source is assumed to have the same bolometric luminosity in a specific redshift and, therefore, we can only use equation (25) to compare observation of objects of the same class and at similar evolutionary stages in certain z , since L = L ( z ). This often means galaxies of the same morphological type. In other words, equation (25) is assuming that a homogenous populations of cosmological sources do exist, and hence, the evolution and structure of the members of such a group will be similar.

Equation (25) also gives us a method for assessing the possible evolution of the source spectrum. For instance, by calculating B - V and V - R colours for E galaxies with modern determinations of the K-correction, Sandage (1995, p. 50) reported that no colour evolution was found to at least z = 0.4. However, for z 0.3 it was found that rich clusters of galaxies tend to be bluer (the Butcher-Oemler effect) than at lower redshifts (Peebles 1993, p. 202 see also Kron 1995, p. 299). Therefore, if we start from a certain metric, we can calculate the theoretical redshift range where colour evolution would be most important for the assumed geometry of the cosmological model.

Another point worth mentioning, from equation (25) we see that colour is directly related to the intrinsic characteristics of the source, its evolutionary stage, as given by the redshift and the assumptions concerning the real form of the source spectrum function at a certain z . However, this reasoning is valid for point sources whose colours are integrated and, therefore, we are not considering here structures, like galactic disks and halos, which in principle may emit differently and then will produce different colours. If we remember that cosmological sources are usually far enough to make the identification and observation of source structures an observational problem for large scale galaxy surveys, this hypothesis seems reasonable at least as a first approximation.

As a final remark, it is clear that in order to obtain a relationship between apparent magnitude and redshift we need some knowledge about the dependence of the intrinsic bolometric luminosity L and the source spectrum function J with the redshift. It seems that such a knowledge must come from astrophysically independent theories about the intrinsic behaviour and evolution of the sources, and not from the underlying spacetime geometry.

In any cosmological model if we consider a small affine parameter displacement dy at some point P on a bundle of past null geodesics subintending a solid angle d 0, and if n is the number density of radiating sources per unit proper volume at P, then the number of sources in this section of the bundle is (Ellis 1971, p. 159)

where, as before, k a is the propagation vector of the radiation flux and u a is the 4-velocity of the observer. Equation (26) considers the counting of all sources at P with number density n . Consequently, if we want to consider the more realistic situation that only a fraction of galaxies in the proper volume dV = ( r 0 ) 2 d 0 dl = ( r 0 ) 2 d 0 (- k a u a ) dy is actually detected and included in the observed number count, we have to write dN in terms of a selection function which represents this detected fraction of galaxies. Then equation (26) becomes (Ellis et al. 1985)

where dN 0 is the fraction number of sources actually observed in the unit proper volume dV with a total of dN sources.

In principle can be estimated from a knowledge of the galactic spectrum, the observer area distance, the redshift, and the detection limit of the sample as given by the limiting flux in a certain frequency bandwidth. The other quantities in equation (27) come from the assumed cosmological model itself, and inasmuch as equation (27) is general, it is valid for any cosmological model, either homogeneous or inhomogeneous.

In order to determine we need to remember that in any spacetime geometry the observed flux in bandwidth W is given by equations (14) and (18),

Then, if a galaxy at a distance r 0 is to be seen at flux F W, its luminosity L ( z ) must be bigger than <4 ( r 0 ) 2 (1+ z ) 3 F W> / <0 W () J[(1 + z )] d >. Therefore, the probability that a galaxy at a distance r 0 and with redshift z is included in a catalog with maximum flux F W is,

where this integral's lower limit is

L * is a parameter, and ( w ) is the luminosity function . model L * is a characteristic luminosity at which the luminosity function exhibits a rapid change in its slope.

Now, if we assume spherical symmetry, then equation (27) becomes

Thus, the number of galaxies observed up to an affine parameter y at a point P down the light cone, may be written as

All quantities in the integrand above are function of the past null cone affine parameter y , and, in principle, they must be explicitly calculated before they can be entered into equation (32). In some cases one may avoid this explicit determination and use instead the radial coordinate, a method which turns out to be easier than finding these expressions in terms of y (Ribeiro 1992). Then, once N 0 ( y ) is obtained, it becomes possible to relate it to other observables, since they are all function of the past null cone affine parameter. For example, if one can derive an analytic expression for the redshift in a given spacetime, say z = z ( y ), and if this expression can be analitically inverted, then we can write N 0 as a function of z .

It is important to mention that the local number density n is given in units of proper density and, therefore, in order to take a proper account of the curved spacetime geometry, one must relate n to the local density as given by the right hand side of Einstein's field equations. If, for simplicity, we suppose that all sources are galaxies with a similar rest-mass M G , then

An indication on how to use the expressions above can be grasped for the Einstein-de Sitter model, where it is well-known that the local density may be written as

If we remember that from a relativistic viewpoint astronomical observations are actually made along the past light cone, where dS 2 = 0, we must calculate a ( t ) and find its expression along the backward null cone,

before we can use equation (33) back into equation (32).

From the discussion above it is clear that the theoretical determination of N 0 depends critically on the spacetime geometry and the luminosity function . For the latter, in the Schechter (1976) model it has the form

where * and are constant parameters. One must not forget that this luminosity function shape was originally determined from local measurements (Schechter 1976), and there is now a controversy about the change of shape and parameters of the luminosity function in terms of evolution (Lonsdale & Chokshi 1993 Gronwall & Koo 1995 Ellis et al. 1996), that is, as we go down the light cone. In any case the Schechter's function above can be used at least as a starting point. In addition, since the luminosity function is being used as a probability in equation (29), it must be properly normalized. However, considering equation (32) one can choose the number density n to agree with the normalization of .

As a final remark, one must note that gravitational lensing magnification can also affect the counting of point sources, because weak sources with low flux might appear brighter due to lensing magnification. Such an effect will not be treated here, since its full treatment demands more detailed information about the sources themselves, such as considering them as extended ones, and is considered to be most important for QSO's (see Schneider et al. 1992).

4 Note that equation (16) also indicates that the source spectrum function J might evolve and change its functional form at different evolutionary stages of the source. Back. *****


How to calculate rest-frame luminosity in a specific wavelength band? - Astronomy

T HE A STRONOMICAL J OURNAL , 118:603-612, 1999 August
© 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.

EVIDENCE FOR A GRADUAL DECLINE IN THE UNIVERSAL REST-FRAME ULTRAVIOLET LUMINOSITY DENSITY FOR z < 1

L ENNOX   L .  C OWIE , 1, 2 A NTOINETTE   S ONGAILA , AND A MY   J .  B ARGER 1, 2
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822

Received 1998 November 17 accepted 1999 April 22

We have utilized various magnitude-limited samples drawn from an extremely deep and highly complete spectroscopic redshift survey of galaxies observed in seven colors in the Hawaii Survey Fields and the Hubble Deep Field to investigate the evolution of the universal rest-frame ultraviolet luminosity density from z = 1 to the present. The multicolor data ( U , B , V , R , and I , J , HK ) enable the sample selection to be made in the rest-frame ultraviolet for the entire redshift range. Because of the large sample size and depth ( U AB = 24.75, B AB = 24.75, I AB = 23.5), we are able to accurately determine the luminosity density to z = 1. We do not confirm the very steep evolution reported by Lilly et al. but instead find a shallower slope, approximately (1 + z ) 1.5 for q 0 = 0.5, which would imply that galaxy formation is continuing smoothly to the present time rather than peaking at z = 1. Much of the present formation is taking place in smaller galaxies. Detailed comparisons with other recent determinations of the evolution are presented.

Key words: galaxies: evolution galaxies: formation galaxies: luminosity function, mass function

      1 Visiting Astronomer, W. M. Keck Observatory, jointly operated by the California Institute of Technology and the University of California.
      2 Visiting Astronomer, Canada-France-Hawaii Telescope, operated by the National Research Council of Canada, the Centre National de la Recherche Scientifique of France and the University of Hawaii.

     Over the past several years it has become widely accepted that the global star formation rate at least as seen in the optical and ultraviolet light had a strong peak at around z = 1 and then fell very steeply at lower redshifts (e.g., Madau, Pozzetti, & Dickinson 1998, and references therein). This would mean that most of the integrated total star formation producing optically visible light would have occurred at roughly half the present age of the universe. However, the argument for the steep falloff at z < 1 relieson a single important analysis by Lilly et al. (1996) based on the Canada-France Redshift Survey (CFRS) sample. This sample has two weaknesses when it is used to determine the evolution of rest-frame UV light. The first of these is that it is a red ( I -band) selected sample with V and I photometry, primarily, and only partial B and K coverage. At low redshifts this requires a very substantial extrapolation across the 4000 Å break to obtain a 2800 Å rest-frame luminosity. The second weakness is that the CFRS is slightly too shallow for this problem ( I AB = 22.5, or I 22.1 in the Kron-Cousins system). At the highest redshifts, near z = 1, the sample does not probe deep enough in the luminosity function to allow a reliable extrapolation to a total luminosity density.

     In this paper we utilize a large, extremely deep, and highly complete spectroscopic redshift survey of galaxies observed in seven colors ( U [3400 ± 150 Å], B , V , R , I , J , HK the limits in the blue and red are B AB = 24.75 and I AB = 23.5, respectively) in the Hawaii Survey Fields and the Hubble Deep Field to investigate the rest-frame UV luminosity density evolution from z = 1 to the present. With this sample we are able to avoid the above problems encountered in the CFRS study. First, the availability of both ultraviolet and blue data means that we can select objects based on their rest-frame ultraviolet magnitudes at all redshifts and hence avoid the serious problem of selecting at redder wavelengths and extrapolating to obtain the UV colors. Second, the substantial additional depth of our sample allows us to probe to the flat segments of the luminosity function, thereby giving a more accurate determination of the luminosity density function. It also enables us to extend the results to redshifts beyond z = 1. This uniform sample selection strongly minimizes possible errors due to dust in the galaxies, unless there is substantial evolution of galaxy dust properties with redshift. Thus, the shape of the present UV luminosity density evolution should be an accurate representation of the shape of the star formation history to within the one residual uncertainty of differential dust evolution.

     We find that the z < 1 UV luminosity density falls with a shallower slope roughly (1 + z ) 1.5 for q 0 = 0.5 than reported by Lilly et al. (1996). Our low-redshift UV luminosity density value is considerably higher than that found by Lilly et al. and agrees with the local UV luminosity density recently determined by Treyer et al. (1998). Consequently, much of the integrated total of optically visible stars is still being assembled at the present time.

     The outline of the paper is as follows: In § 2 we provide abrief, self-contained analysis of the distribution of light with redshift in the Hubble Deep Field. If this small field is representative, then the analysis can be used to demonstrate theprimary conclusion of the paper, namely, that the falloff in the rest-frame ultraviolet luminosity density below z = 1 is relatively shallow. We then proceed to the main analysis. We discuss the data in § 3 the photometry and the statistical and systematic errors are presented in the Appendixfor textual clarity. In § 4 we construct the rest-frame ultraviolet luminosity functions and the UV luminosity density as a function of redshift. We then present comparisons with the results of other analyses. In order to be fully self-consistent, we specifically restrict ourselves to the evolution of the ultraviolet luminosity density and do not attempt to compare with studies of emission-line evolution. In § 5 we briefly summarize our conclusions.

     The Hubble Deep Field (HDF Williams et al. 1996) provides our deepest view of the faint galaxy populations. The color information in the HDF has been extensively used to estimate the high- z star formation history of optically selected galaxies (e.g., Madau et al. 1996 Connolly et al. 1997 Sawicki, Lin, & Yee 1997). As large numbers of spectroscopic redshifts have accumulated for HDF galaxies (Steidel et al. 1996 Cohen et al. 1996 Lowenthal et al. 1997 Phillips et al. 1997 Barger et al. 1999 3 ), it is now possible to use these spectroscopic data to examine the history of star formation at z < 1 and to show directly that, within the HDF itself, a substantial fraction of the star formation is occurring at low redshift. A similar conclusion is reached by Pascarelle, Lanzetta, & Fernandez-Soto (1998) based on a very different photometric redshift analysis.

     The sky surface brightness of the integrated galaxy light computed from the Williams et al. objects in the deeper areas of the WFC chips an area of 4.9 arcmin 2 is shown by the solid line in Figure 1. There are 96 objects in this region of the HDF that now have spectroscopic identifications. These identified objects, which are generally the brighter galaxies, contain most of the optical and roughly half of the 3000 Å light. We have subdivided the contributions by redshift, showing the light from known z < 1 galaxies as filled boxes and that from known z < 0.5 galaxies as filled diamonds. The dashed line, which shows the combined light from all known z > 1 galaxies and all unidentified or unobserved objects, represents an extreme upper bound to the fraction of light arising from galaxies at z > 1.

F IG . 1. Sky surface brightness of the integrated galaxy light in the Hubble Deep Field. The solid line shows the integrated light from all the galaxies, the filled squares show that of known z < 1 galaxies, and the filled diamonds show that of z < 0.5 galaxies. The dashed line shows the light from galaxies known to be at z > 1 plus all unidentified galaxies.

     Since some (possibly substantial) fraction of the fainter unidentified galaxies will lie at z < 1, the ratio of the z > 1 light to that at lower redshift is overestimated here but can already be used to make the argument that a large fraction of star formation is local. As was pointed out in Cowie (1988), Songaila, Cowie, & Lilly (1990), and Madau et al. (1996), the integrated massive star formation, which can be characterized by the local metal density, is directly proportional to the observed-frame ultraviolet sky brightness, independent of cosmology, through the relation

Estimates of the normalizing factor can be found in Songaila et al. (1990) and Madau et al. (1996). Applying this relation to the 3000 Å light in Figure 1, we see immediately that comparable amounts of metals have been produced by z < 0.5 galaxies and by z > 1 galaxies, since the ultraviolet sky brightness of the known z < 0.5 galaxies is already 54% of the maximum possible z > 1 light. However, in this simple form, the argument depends on the assumption that the light from star-forming galaxies is roughly flat in f .

     A more conservative approach is to compare the known z < 0.5 3000 Å light with the possible z > 1 6020 Å light. This comparison at equivalent rest-frame wavelengths allows for the shapes of the spectral energy distributions (SEDs) and for dust extinction. It also takes into account the disappearance of very high-redshift galaxies from the 3000 Å band. The ratio R SF light( z < 0.5)/light( z > 1) from Figure 1 is then 26%. This represents a very extreme lower bound on the ratio of the integrated star formation at z < 0.5 to that at z > 1.

     The implications for the evolution of the low- z star formation rate [ * ( z )] can be shown with a simple model,

which can be integrated analytically. For q 0 = 0.5, R SF =18% for = 4, 30% for = 3, and 48% for = 2. Thus, even comparing the candidate z > 1 visual light with the known z < 0.5 ultraviolet light requires 3.2. Provided only that the small HDF is a representative field, even the very conservative and extremely robust version of the argument requires a shallower slope than has been inferred from the CFRS data (viz., 3.9 ± 0.75). The remainder of the paper verifies this conclusion using our wide-field ultraviolet and optically selected samples.

     The primary data sets used in the present study are two 6 × 2 5 areas crossing the HDF and the Hawaii Survey Field SSA22 (Lilly, Cowie, & Gardner 1991). In each case the field has been imaged in seven colors U (3400 ± 150 Å), B , V , R , I , J , and HK where R and I are Kron-Cousins and the HK (1.9 ± 0.4 m) filter is described in Wainscoat & Cowie (1999). The images were obtained onthe Keck II Telescope using LRIS (Oke et al. 1995) and onthe University of Hawaii 2.2 m and the Canada-France-Hawaii 3.6 m telescopes using QUIRC (Hodapp et al. 1996), ORBIT, and the UH8K CCD Mosaic Camera built by Metzger, Luppino, and Miyazaki. The fields, whose sizes were set by the usable LRIS area, are fully covered in all colors. All magnitudes were measured in 3 diameter apertures and corrected to total magnitudes following the procedures in Cowie et al. (1994). A further two similarly sized areas covering the Hawaii Survey Fields SSA4 (Cowie et al. 1994) and SSA13 (Lilly et al. 1991) with B , V , I , and HK data were used to augment the SSA22 and HDF B , I , and HK -selected samples. The fields, areas, magnitude limits, and numbers of spectroscopically identified galaxies, stars, and unknowns (unobserved or unidentified objects) for each of the color-selected samples are given in Table 1. A more detailed discussion and analysis of the photometric and spectroscopic samples used in this paper can be found in Barger et al. (1999).

TABLE 1      C OLOR - SELECTED S AMPLES

4. REST-FRAME ULTRAVIOLET LUMINOSITYDENSITY EVOLUTION

4.1. Construction of the Luminosity Functions

     For redshifts, z , for which the rest-frame UV wavelength, U , maps directly to the wavelength of one of the observed color bands, the absolute rest-frame UV magnitude, M U , in the AB system ( M U = 0 corresponds to log F = -48.57) is given by

where m AB is the observed magnitude at the redshifted wavelength U (1 + z ) in the AB system and d L ( z ) is the luminosity distance. For redshifts such that U (1 + z ) differs by a small amount from this central wavelength, we can apply the same formula but with the addition of a small differential k correction dk ( z ), defined as

where f is the SED of the galaxy, and z C is the redshift corresponding to the center of the band. Because of the frequent, regularly spaced color sampling, we can simply construct dk ( z ) by interpolation from the neighboring color bands. The value of dk ( z ) is generally small (less than 0.2 mag) for the redshift intervals that are used.

     The 2500 Å rest-frame absolute magnitudes, computed in the appropriate redshift intervals from the U , B , V, and I samples, are shown versus redshift in Figure 2. In the overlapping redshift intervals the absolute magnitudes compare well. The rise in the maximum absolute UV rest-frame luminosity with redshift mapped by the upper envelope of the distribution is partly a reflection of the larger volume sampled at high redshift but also partly reflects the known evolution that causes the maximum UV luminosity to increase with redshift (Lilly et al. 1996 Cowie et al. 1996) we will discuss this in more detail below.

F IG . 2. Absolute 2500 Å rest-frame magnitudes vs. redshift, computed from the U ( filled squares ), B ( open diamonds ), V ( stars ), and I ( filled triangles ) samples.

     We adopt the traditional V max method of Felten (1977) for constructing the luminosity functions. The number density of galaxies in the redshift range [ z 1 , z 2 ] with magnitude M is given by

where the sum is over all galaxies with magnitude M ± dM /2. V max ( M ) is the maximum total volume in all the samples where galaxies with absolute magnitude M are observable in the appropriate apparent magnitude range and lie in the redshift range. (A very complete description of the procedure may be found in Ellis et al. 1996.) The ultraviolet luminosity density, l , is then

where the sum is over all the observed objects. We use Schechter function fits with = -1 to extend l from the brighter absolute magnitudes observed to M = -16, which we then use as our total luminosity density.

     In the computation of the luminosity functions, we assign errors based on the Poisson distribution corresponding to the number of galaxies contributing to the absolute magnitude bin. However, a larger potential source of error is the missing or unidentified galaxies in each sample. Following the standard procedure, we compute the luminosity function after assuming that these objects follow the redshift distribution of the identified objects, but because of systematic effects, this assumption may not be valid. We have therefore also computed the luminosity functions (LFs) with all, and also with none, of the missing objects allocated into the redshift bin, which should provide extremal estimates of this error. In the case where we allocated the missing objects into the redshift bin, the allocated redshifts were randomly distributed uniformly within the redshift interval. We are able to use this very robust procedure because of the extremely high completeness of the spectroscopic samples.

4.2. UV Luminosity Functions versus Rest Wavelength

     A key issue is the optimum wavelength at which to compute the rest-frame LFs and the UV light densities. Previous efforts, such as those by the CFRS, have computed the light densities at a relatively long wavelength of 2800 Å, which is optimal for a red sample and minimizes dust extinction but may be a poorer measure of massive star formation rates than shorter wavelengths. The local UV sample of Treyer et al. (1998), which provides an invaluable low-redshift comparison, is at 2000 Å. However, our own sample gives the best combination of wide redshift range and the largest galaxy sample at a rest wavelength around 2500 Å.

     In the following discussion we shall use all three of these rest wavelengths for specific comparisons and analyses. Before proceeding, however, we use the various color samples to show that at z = 1 the luminosity functions and light densities are only weakly sensitive to the wavelength choice. At z = 1 the U , B , and V samples correspond to rest-wavelengths of 1700, 2250, and 2750 Å. In Figure 3 we compare the luminosity functions constructed at each of these rest wavelengths in the redshift interval 0.7 < z < 1.3. It is apparent that the luminosity functions are quite similar, with only a slight fading of M U with decreasing wavelength. The corresponding luminosity densities for galaxies with luminosities greater than M U = -17.5 ( squares ), or extrapolated to M = -16 assuming an = -1 Schechter function ( dashed line ), are shown in Figure 4. The total UV luminosity density above M = -16 in this redshift interval is fitted by a power law,

where h 65 is the Hubble constant in units of 65 km s -1 Mpc -1 . We shall generally avoid comparing light densities at different wavelengths. Where necessary, however, we use the weak wavelength dependence of equation (7), assuming, somewhat arbitrarily, that this is valid for other redshifts.

F IG . 3. Luminosity functions in the redshift interval 0.7 < z < 1.3 constructed at rest wavelengths of 1700 Å ( open diamonds ), 2250 Å ( filled squares ), and 2750 Å ( open triangles ) from the U , B , and V samples, respectively.

F IG . 4. Luminosity density in the redshift interval 0.7 < z < 1.3 computed from the luminosity functions of Fig. 3. The squares are the luminosity density for galaxies with luminosity M AB > -17.5. The thin dashed line represents a power-law fit to the data. The solid line is 2.6 × 10 26 ( /3000 Å) 1.1 ergs cm -2 s -1 Hz -1 , which represents the power-law fit to the luminosity densities extrapolated to M AB = -16, assuming an = -1 Schechter function.


4.3. The Redshift Evolution of the2500Å Rest-Frame Luminosity Density

     We first construct the 2500 Å rest-frame luminosity function in three redshift bands: z = 0.20 0.50, using the U -selected sample (Fig. 5) z = 0.6 1.0, using the B -selected sample (Fig. 6) and z = 1.0 1.5, using the V -selected sample (Fig. 7). In each case the solid curve is the incompleteness-corrected luminosity function with ۫ Poisson error bars on the symbols, the dotted curve is the luminosity function obtained from only the observed objects (the minimal function), and the dashed curve is the luminosity function when all the unidentified objects are taken to lie within the redshift interval (the maximal function).

F IG . 5. The 2500 Å rest-frame luminosity function ( diamonds and solid line ) in the redshift interval 0.2 < z < 0.5 constructed from the U sample ۫ error bars are shown. The dotted line is the luminosity function computed from only the observed objects (the "minimal" function), whereas the dashed line is computed after placing all of the unidentified objects into the redshift bin (the "maximal" function).

F IG . 6. Same as in Fig. 5, but for the redshift interval 0.6 < z < 1.0, constructed from the B sample.


F IG . 7. Same as in Fig. 5, but for the redshift interval 1.0 < z < 1.5, constructed from the V sample.


     We next compute the luminosity density for galaxies more luminous than M U = -16. This is a directly determined quantity in the lowest redshift interval, but for the higher redshift intervals it requires an extrapolation that was made with an = -1 Schechter function fit. For z = 0.6 1.0 the observed light density for M AB -17.5 is 1.7 × 10 26 h 65 ergs s -1 Hz -1 Mpc -3 , and the extrapolated light density is 21% higher. For z = 1.0 1.5 the observed light density for M AB -18.25 is 1.4 × 10 26 h 65 ergs s -1 Hz -1 Mpc -3 , and the extrapolated light density is 34% higher.

     The incompleteness-corrected M AB -16 light density as a function of redshift is shown in Figure 8 as the filled squares the redshift range is shown as the thin horizontal lines. The statistical and systematic errors (see Appendix) are shown as the thicker portions of the error bar. We have performed similar calculations for both the minimal and maximal functions. These represent the extremal range of the possible light density at a given redshift and are shown as the open squares joined by the vertical thin lines. The solid line shows a simple (1 + z ) 1.5 evolution law that would result in equal amounts of star formation per unit redshift interval and would constitute an acceptable fit to the data. The dashed line shows a redshift evolution of (1 + z ) 4 matched to the second data point this is clearly too steep, even allowing for the maximum incompleteness correction.

F IG . 8. M U < -16 light density at 2500 Å as a function of redshift, directly determined in the interval 0.2 < z < 0.5 and incompleteness-corrected via an = -1 Schechter function for the intervals 0.6 < z < 1.0 and 1.0 < z < 1.5. In each redshift interval the light density is shown as a filled square and the redshift range as a thin horizontal line. Statistical and systematic errors (see Appendix) are shown as the thick portions of the error bar. Light densities computed using the minimal and maximal functions (see text and Figs. 5 7) are shown as open squares joined by thin vertical lines. The solid line is a (1 + z ) 1.5 evolution law. The dashed line shows an evolution of (1 + z ) 4 , normalized to the second data point.

     The shape and normalization of the rest-frame UV luminosity function is of course a consequence of a complex mix of the evolution of the star formation rates in galaxies of various types and masses and the decrease in M U * with declining redshift, which most likely reflects a preferential drop in the star formation rates of the most massive galaxies (Cowie et al. 1996). Nevertheless, it is interesting to compare the functions to determine if there is evidence for a change of shape (Fig. 9). For q 0 = 0.5 the functions can be quite well overlaid with a simple shift in the absolute magnitude (pure luminosity evolution). Though there is a hint that the higher redshift functions might be steeper, there is no statistically significant difference in the shapes as a function of redshift. The absolute magnitude shifts used in constructing Figure 9 (+0.65 mag at z = 0.6 1.0 and +0.75 mag at z = 1.0 1.5, both relative to z = 0.2 0.5) can also be used to check the 2500 Å rest-frame luminosity density evolution and would imply a ratio of 1:1.8:2.0 in these redshift bins the best-fit power law corresponds to (1 + z ) 1.4 , which is quite similar to the estimates given above.

F IG . 9. 2500 Å rest-frame luminosity functions ( q 0 = 0.05) for theredshift intervals 0.20 < z < 0.50 ( filled diamonds ), 0.60 < z < 1.00 ( filledtriangles offset by +0.65 mag relative to the [0.20, 0.50] function),and 1.00 < z < 1.50 ( open triangles offset by +0.75 mag relative to the [0.20, 0.50] function).

4.4. Comparison with Other Samples

     The most directly comparable sample is the low-redshift, rest-frame ultraviolet (2000 Å) selected sample of Treyer et al. (1998). To make a direct comparison with this sample we have computed the 2000 Å luminosity functions at z = 0.5 0.9 using the U -selected sample and at z = 1.0 1.5 using the B -selected sample. We compare these with the luminosity function determined by Treyer et al. in Figures 10 and 11. The solid line with the symbols shows the presently determined luminosity function. The dashed line shows the Treyer et al. function for H 0 = 65 km s -1 Mpc -1 and converting the magnitude system used by Treyer et al. to AB magnitudes with a 2.29 mag offset. Here we have used a renormalization rather than a magnitude offset to overlay the functions. For z = 0.5 0.9 the normalization of the Treyer et al. function is multiplied by 1.7. For z = 1.0 1.5 the normalization is multiplied by 2.8. The corresponding luminosity density changes would be an increase of l (1 + z ) 1.3 .

F IG . 10. 2000 Å rest-frame luminosity function in the redshift interval 0.5 < z < 0.9 ( filled diamonds and solid line ), computed from the U sample, compared with the luminosity function constructed from the local rest-frame UV-selected sample of Treyer et al. (1998) ( dashed line ) for H 0 = 65 km s -1 Mpc -1 . The latter has been renormalized upward by a factor of 1.7 to match the z = 0.7 luminosity function.

F IG . 11. Same as in Fig. 10, but for the redshift range 1.0 < z < 1.5, constructed from the B -selected sample. Here the local UV luminosity function of Treyer et al. (1998) has been multiplied by a factor of 2.8 to match the z = 1.25 function.


     The measured 2000 Å luminosity densities in the Treyer et al. (1998) sample are compared with the incompleteness corrected 2000 Å luminosity densities from this work in Figure 12, where once again we have shown the maximal and minimal luminosity densities with open squares. As was the case for the 2500 Å data, the combined data sets exhibit a slow evolution, as shown by the (1 + z ) 1.5 solid line. A steeper dependence, such as the (1 + z ) 4 evolution of the dashed line, is radically inconsistent with the data. Because of the steep rise in the Treyer et al. luminosity function at the faintest magnitudes, it is best fitted by an = -1.6 power law rather than the = -1.0 power law (characteristic of the optical luminosity functions) that we have adopted in extrapolating the luminosity functions to the fainter magnitudes. In order to investigate the dependence on we have fitted the 2000 Å luminosity functions at z = 0.5 0.9 and z = 1 1.5 with Schechter functions with indices = -1.0 and = -1.5. The fitted parameters and the luminosity densities above -16 are summarized in Table 2, where they are compared with the values derived by Treyer et al. Increasing the index from -1.0 to -1.5 has the effect of preferentially increasing the luminosity density at the higher redshifts where the extrapolation is larger. However, the effect is not large. Using the -1.5 derived luminosity density would increase the slope to (1 + z ) 1.7 from (1 + z ) 1.3 for the = -1.0 case.

F IG . 12. Incompleteness-corrected rest-frame 2000 Å luminosity densities to M AB = -16 in the redshift intervals 0.5 < z < 0.9 and 1.0 < z < 1.5 ( filled squares ), constructed from the luminosity functions of Figs. 10 and 11, compared with the measured 2000 Å luminosity density from the local Treyer et al. (1998) sample ( filled diamond ), for H 0 = 65 km s -1 Mpc -1 and q 0 = 0.5. This number has been computed for the range M AB < -16 only. Thin horizontal lines show the redshift ranges. For the present data, the open squares show the densities computed with the minimal and maximal functions (see text and Figs. 5 7), joined by thin vertical lines. Statistical and systematic errors (see Appendix) are shown by thick vertical lines. The solid line is a (1 + z ) 1.5 evolution line, while the dashed line is a (1 + z ) 4 evolution, normalized to the z = 0.7 point.

TABLE 2      F ITS TO 2000 Å L UMINOSITY F UNCTIONS

     We next compare the present sample with the Lilly et al. (1996) analysis of the CFRS data at a rest-frame wavelength of 2800 Å. In order to make the most direct comparison possible, we followed the Lilly et al. redshift intervals of [0.2, 0.5], [0.5, 0.75], and [0.75,1], constructing the luminosity functions with the U sample, the B sample, and the V sample, respectively. The luminosity densities constructed from these samples are shown in Figure 13, where the filled squares are again the incompleteness-corrected luminosity density, the thicker portions of the error bars show the systematic and statistical errors, and the open squares show the minimal and maximal light densities. The open diamonds show the Lilly et al. analysis their three higher redshift points are based on the CFRS data, and the lowest redshift point is based on the data of Loveday et al. (1992). The lowest redshift point is already contradicted by Treyer et al.'s (1998) analysis, shown as the filled diamond, corrected upward by a factor of 1.5 for the wavelength difference between 2000 and 2800 Å. The Treyer et al. point is also comparable with the [0.2, 0.5] Lilly point. The present data exhibit a much shallower slope than the Lilly et al. data in the z = 0.2 1 range. This results primarily from the [0.75, 1] point being about a factor of 2 lower than the highest redshift Lilly point. The two intermediate redshift points agree well with the CFRS analysis.

F IG . 13. Incompleteness-corrected rest-frame 2800 Å luminosity density as a function of redshift. Filled squares show the densities from the present work for the redshift intervals 0.2 < z < 0.5 (constructed from the U sample), 0.5 < z < 0.75 (from the B sample), 0.75 < z < 1 (from the V sample) and 1.0 < z < 1.5 (also from the V sample). The filled diamond is the density constructed from the Treyer et al. (1998) sample corrected to 2800, the open diamonds are Lilly et al.'s (1996) "LF-estimated" total 2800 Å densities from the CFRS survey, and the open triangles show the densities derived from Connolly et al. (1997) ( upright symbols ) and Sawicki et al. (1997) ( inverted symbols ), scaled downward by 20% as discussed in the text. H 0 = 65 km s -1 Mpc -1 and q 0 = 0.5. The dashed line shows a (1 + z ) 4 evolution law, similar to the (1 + z ) 3.9۪.75 law derived in Lilly et al. (1996). Thin horizontal lines show the redshift intervals in all cases. For the present data, open squares show the densities constructed from the minimal and maximal functions (see text and Figs. 5 7), which are connected by thin vertical lines. Thick vertical lines show the statistical and systematic errors (see Appendix).

     We also compare with the analyses of Connolly et al. (1997) and Sawicki et al. (1997), shown as open triangles and inverted open triangles, respectively, in Figure 13. These two analyses are based on photometrically estimated redshifts in the small HDF proper and are in broad general agreement with each other and with the work of Pascarelle et al. (1998). (We do not include a direct comparison to the latter work here as it is computed at 1500 Å.) Following the discussion in the Appendix, we have made a small 20% downward correction to allow for the slightly higher counts in the relevant magnitude range in the HDF proper versus the wide surrounding field. Both the [0.5, 1] and [1.0, 1.5] points from these analyses are higher than our incompleteness-corrected estimate but are consistent within the statistical and incompleteness errors. Our best estimate value may indeed be slightly low here, since there may be preferential incompleteness in these redshift ranges. However, there is also relatively little validation of photometrically estimated redshifts in the range z = 1 to 2, where the observed spectral range ( = 3000 22,000 Å) is quite featureless for blue galaxies. Thus, the high-redshift Connolly et al. and Sawicki et al. points could be overestimated if lower redshift blue irregulars have been assigned to the wrong bin. Since our maximal incompleteness-corrected estimate is not substantially different from the photometrically estimated values, the discussion of § 5 does not depend on the resolution of this issue.

4.5. The Relative Evolution of the Rest-Frame UV,Blue and Red Light

     As has been known for several years now (Lilly et al. 1996 Cowie et al. 1996), the rest-frame red luminosity has an extremely weak evolution with redshift out to z = 1. This slow evolution can be clearly seen in Figure 14, in which we compare the rest-frame 8000 Å luminosity at z = 0.2 0.5, obtained using the I -selected sample, with that at z = 0.7 1.3, computed using the K -selected sample. The two functions can be brought into full consistency with a very small (0.2 mag) pure luminosity shift corresponding to a decrease of only 20% in the red light from z = 0.8 to z = 0.35, thereby confirming with the present sample that the downward evolution of the UV light density by a factor of 2 between z = 1 and z = 0.35 does take place against an essentially invariant shape and normalization for the red light density.

F IG . 14. 8000 Å rest-frame luminosity functions in the redshift intervals 0.2 < z < 0.5 ( filled diamonds and dotted line ), constructed from the I -selected sample, and 0.7 < z < 1.3 ( filled squares and solid line ), constructed from the K -selected sample ۫ errors are shown at each point.

     We have investigated the evolution of the universal ultraviolet luminosity density from z = 1 to the present using a magnitude-limited sample selected on the basis of rest-frame ultraviolet colors at all redshifts from an extremely deep and highly complete spectroscopic redshift survey. Our uniform selection procedure avoids the serious problem of selecting the sample at redder wavelengths and then extrapolating to obtain ultraviolet colors, and it strongly minimizes possible errors due to dust in the galaxies, unless there is substantial evolution of galaxy dust properties with redshift. The depth of the current sample is also sufficiently deep to probe to the flat segments of the luminosity function and to extend the results to redshifts beyond z = 1, thereby enabling us to make an accurate determination of the luminosity density function.

     We find that our incompleteness-corrected rest-frame 2500 Å luminosity densities for M U -16 as a function of redshift are well fitted by an l (1 + z ) 1.5 evolution law for q 0 = 0.5. (The slope would be shallower for open geometries.) The decline in M U with decreasing redshift most likely reflects a preferential drop in the star formation rates of the most massive galaxies. A direct comparison of the measured low-redshift 2000 Å luminosity density of Treyer et al. (1998) with incompleteness-corrected 2000 Å luminosity densities computed from our sample also shows that the slow evolution law of l (1 + z ) 1.5 provides a good fit. When we compare the Lilly et al. (1996) CFRS analysis with our sample analyzed at rest-frame 2800 Å, we find that our sample gives a much shallower slope in the z = 0.2 1range. This most probably arises in part due to the relatively large extrapolation at low redshifts Lilly et al. needed to make to go from a primarily V - and I -based data sample to ultraviolet colors. The two lowest-redshift Lilly et al. points are also in disagreement with the Treyer et al. analysis. Interestingly, however, the present UV luminosity density analysis leads to a closer agreement with star formation rates found in an H analysis of the CFRS data by Tresse & Maddox (1998).

     We can summarize our results in the form of the widely discussed plot of UV luminosity density versus redshift, our version of which is shown for the 2500 Å rest frame in Figure 8 and at 2000 Å in Figure 12. This plot is rather different from the now conventional "Madau" form, which has a strong rise to z = 1. Rather, we see a fairly slow decline in UV from z = 1.5 to z = 0, which is reasonably well described by a simple UV = 7 × 10 25 (1 + z ) 1.5 ergs s -1 Mpc -3 Hz -1 relation. This result has fairly profound philosophical implications in that the integrated star formation is continuing to rise smoothly at the present time and the bulk of the star formation has occurred at recent times. (This result holds irrespective of cosmological geometry.) Put succinctly, as seen in the optical, now may be the epoch of galaxy formation and not z = 1.

SYSTEMATIC AND STATISTICAL ERRORS AND COMPARISON WITH HDF PHOTOMETRY AND NUMBER COUNTS

     Because much of our information on the z > 1 rest-frame UV luminosity density is based on the small HDF, it is important to check the consistency of the photometry and number counts between the present data and the HDF tables of Williams et al. (1996).

     Since the wide-field area around the HDF fully overlaps the HDF proper, we can compare the photometric systems directly. We first identified the corresponding objects in the two tables whose centroids lie within 0 4 of each other and are not complex objects. We then compared our corrected aperture magnitudes with the total magnitudes in Williams et al. The color equations for the four bands are found to be,

where we have forced the color term in the B equation to zero. Allowing for the offsets in zero points between the color systems ( I - I AB = -0.33, B - B AB = 0.16), we can see that there is a maximum deviation of 0.15 mag in the V band versus AB 602 , where the color term is, however, large. We take this to be the maximum systematic uncertainty in the photometry.

     Using these photometric transformations we next constructed the V -band number counts (galaxies and stars) in the WFPC areas of the HDF itself (4.9 arcmin 2 ) and in the 81 arcmin 2 area of the HDF flanking field region. This comparison is shown in Figure 15. There is broad general agreement in the counts within the small number fluctuations. Over the range, 21 < V < 24.25, the HDF proper has a 17% higher total flux per unit area than the large flanking field area. In Figure 16 we show the HDF galaxy counts in B compared with the average B galaxy counts in the regions of the HDF and SSA22 used in the present analysis. (Only spectroscopically confirmed stars have been removed from both counts.) The total flux per unit area of objects in the HDF with 21 < B < 24.75 is 13% higher than in the average of the two fields.

F IG . 15. V -band number counts (galaxies plus stars) in the HDF proper ( small squares ) and in the HDF flanking field region ( large symbols ). The dashed line is a power-law fit to the bright-end data.

F IG . 16. B -band galaxy counts from the HDF proper (Williams et al. 1996 small squares ) compared with the average B -band galaxy counts in the regions of the HDF and SSA22 used in the present work ( large symbols ). Spectroscopically confirmed stars have been removed in both cases.


     Finally, in Figure 17 we show a comparison of the total HDF 0 < z < 0.5 and 0 < z < 1 HDF sky surface brightness in the ranges 21 < V < 24.5, 21 < B < 24.75 and 21 < U < 24.5 with the present samples. In the analysis the HDF lies about 25% higher in the total light in this range, with a disproportionately high amount of z < 0.5 light, which is between 25% and 50% higher in the various colors. The results emphasize that clustering can cause fluctuations at this level in fields of the size of the HDF, and they emphasize the importance of using multiple large areas. In order to compare with the HDF-derived data of Connolly et al. (1997) we have therefore revised their fluxes downward by 20%. However, the systematic error in the present much larger data sample should be considerably smaller than this and is dominated by the photometric errors. We will adopt a conservative systematic error limit of 䔸% of the rest-frame UV luminosity densities.

F IG . 17. Comparison of sky surface brightnesses of galaxies in the HDF proper ( filled symbols and solid line ) with those in the much larger flanking fields. The integration excludes known stars and is over the AB magnitude range 21 24.25 in the red, 21 24.5 in the visual, and 21 24.75 in the blue. The solid line shows the total sky surface brightness for the HDF proper for comparison with the large open squares, which show this for the flanking fields. The smaller squares are for known z < 1 galaxies and the smaller diamonds for z < 0.5 galaxies. The dashed line shows the possible z > 1 light for this magnitude range.

     In order to obtain a realistic estimate of the statistical error we divided the B sample into two parts one comprising the HDF and SSA22 data and one comprising the SSA13 and SSA4 data and constructed the rest-frame 2500 Å luminosity function in the 0.6 < z < 1 range from each subsample. The two luminosity functions are shown in Figure 18. The luminosity densities inferred from the two functions differ by 17%. Based on this comparison, we have assigned a statistical error of 0.2 ( N /40) -0.5 , where N is the total number of galaxies in the sample bin, and we have added this error in quadrature to the systematic error to determine the formal error bars in the rest-frame UV luminosity density plot.


How to calculate rest-frame luminosity in a specific wavelength band? - Astronomy

Aims: We want to study the IR (>8 μm) emission of galaxies selected on the basis of their rest-frame UV light in a very homogeneous way (wavelength and luminosity) from z = 0 to z=1. We compare their UV and IR rest-frame emission to study the evolution in dust attenuation with z as well as to check if a UV selection is capable of tracking all star formation. This UV selection will also be compared to a sample of Lyman break galaxies selected at z ≃ 1.
Methods: We select galaxies in UV (1500-1800 Å) rest-frame at z=0, z=0.6--0.8, z=0.8--1.2, and with as Lyman break galaxies at z=0.9--1.3, the samples are compiled to sample the same range of luminosity at any redshift. The UV rest-frame data come from GALEX for z<1 and the U-band of the EIS survey (at z=1). The UV data are combined with the IRAS 60 μm observations at z=0 and the Spitzer data at 24 μm for z>0 sources. The evolution in the IR and UV luminosities with z is analysed for individual galaxies as well as in terms of luminosity functions.
Results: The L_IR/L_UV ratio is used to measure dust attenuation. This ratio does not seem to evolve significantly with z for the bulk of our sample galaxies, but some trends are found for both galaxies with a strong dust attenuation and UV luminous sources: galaxies with L_IR/L_UV>10 are more frequent at z>0 than at z=0, and the largest values of L_IR/L_UV are found for UV faint objects in contrast, the most luminous galaxies of our samples (L_UV> 2 × 10 10

L_☉), detected at z=1, exhibit a lower dust attenuation than fainter ones. The value of L_IR/L_UV increases with the K rest-frame luminosity of the galaxies at all redshifts considered and shows a residual anticorrelation with L_UV. The most massive and UV luminous galaxies exhibit quite high specific star formation rates. Lyman break galaxies exhibit systematically lower dust attenuation than UV-selected galaxies of same luminosity, but similar specific star formation rates. The analysis of the UV + IR luminosity functions leads to the conclusion that up to z = 1, most of the star formation activity of UV-selected galaxies is emitted in IR. Although we are able to infer information about all the star formation from our UV selection at z=0.7, at z = 1 we miss a large fraction of galaxies more luminous than ≃10 11


The Faintest Dwarf Galaxies

Joshua D. Simon
Vol. 57, 2019

Abstract

The lowest luminosity ( L) Milky Way satellite galaxies represent the extreme lower limit of the galaxy luminosity function. These ultra-faint dwarfs are the oldest, most dark matter–dominated, most metal-poor, and least chemically evolved stellar systems . Read More

Supplemental Materials

Figure 1: Census of Milky Way satellite galaxies as a function of time. The objects shown here include all spectroscopically confirmed dwarf galaxies as well as those suspected to be dwarfs based on l.

Figure 2: Distribution of Milky Way satellites in absolute magnitude () and half-light radius. Confirmed dwarf galaxies are displayed as dark blue filled circles, and objects suspected to be dwarf gal.

Figure 3: Line-of-sight velocity dispersions of ultra-faint Milky Way satellites as a function of absolute magnitude. Measurements and uncertainties are shown as blue points with error bars, and 90% c.

Figure 4: (a) Dynamical masses of ultra-faint Milky Way satellites as a function of luminosity. (b) Mass-to-light ratios within the half-light radius for ultra-faint Milky Way satellites as a function.

Figure 5: Mean stellar metallicities of Milky Way satellites as a function of absolute magnitude. Confirmed dwarf galaxies are displayed as dark blue filled circles, and objects suspected to be dwarf .

Figure 6: Metallicity distribution function of stars in ultra-faint dwarfs. References for the metallicities shown here are listed in Supplemental Table 1. We note that these data are quite heterogene.

Figure 7: Chemical abundance patterns of stars in UFDs. Shown here are (a) [C/Fe], (b) [Mg/Fe], and (c) [Ba/Fe] ratios as functions of metallicity, respectively. UFD stars are plotted as colored diamo.

Figure 8: Detectability of faint stellar systems as functions of distance, absolute magnitude, and survey depth. The red curve shows the brightness of the 20th brightest star in an object as a functi.

Figure 9: (a) Color–magnitude diagram of Segue 1 (photometry from Muñoz et al. 2018). The shaded blue and pink magnitude regions indicate the approximate depth that can be reached with existing medium.