Astronomy

Radial velocity curves

Radial velocity curves


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When I measure RV in a program for more spectral lines, will it be equivalent? I mean I measure RVs of a line in blue spectral region, RVs of a line in red spectral region… Many thanks.


It depends what lines you are measuring and in what kind of star.

When you measure the RV from a spectral line, you are measuring an intensity-weighted average RV over the region where the line is formed.

For a star like the Sun, the photospheric lines are all formed within a layer no thicker than about a 1000 km and the differences in RV with depth in the atmosphere are small (though not negligible if you are interested in finding low mass exoplanets using the RVs). On the other hand, if you were to start measuring RVs from lines originating in the chromosphere or Corona then you could find different RVs caused by mass motions of the emitting gas.

Similarly, hot stars are surrounded by fast moving winds with a temperature structure. Different lines can originate at different heights from gas with different velocity. This can lead to different RVs from different lines.

A better way to measure RV is by cross-correlation with a synthetic or observed template of a similar star with a known RV.


Circular orbit—radial-velocity shift¶

The radial velocity is the radial component of the velocity of a source relative to an observer and is usually inferred spectroscopically. Light from an object being part of a binary system and orbiting the common center of mass will be subject to the Doppler effect. Given a circular orbit, its radial velocity shift will show sinusoidal variations determined by the object’s orbital elements.

class PyAstronomy.modelSuite.radVel. SinRadVel ¶

  • P - float, Orbital period [d]
  • T0 - float, Central transit time [d]
  • K - float, radial velocity semi-amplitude [km/s]
  • rv0 - float, constant offset in radial velocity [km/s]

By default all parameters remain frozen.

The convention is that at phase zero, also the orbital radial velocity is zero. With increasing phase the object becomes bluer first.

MCMCautoParameters (ranges[, picky, …]) Convenience function to generate parameters for MCMC fit.
addConditionalRestriction (*args) Define a conditional restriction.
assignValue (specval) Assign new values to variables.
assignValues (specval) Assign new values to variables.
autoFitMCMC (x, y, ranges[, picky, stepsize, …]) Convenience function to using auto-generated sampling parameters in MCMC.
availableParameters () Provides a list of existing parameters.
delRestriction (parName) Delete restriction
description ([parenthesis]) Returns a description of the model based on the names of the individual components.
errorConfInterval (par[, dstat, statTol, …]) Calculate confidence interval for a parameter.
evaluate (x) Calculates and returns radial velocity shift according to current model parameters.
fit (x, y[, yerr, X0, minAlgo, mAA, …]) Carries out a fit.
fitEMCEE ([x, y, yerr, nwalker, priors, …]) MCMC sampling using emcee package.
fitMCMC (x, y, X0, Lims, Steps[, yerr, …]) Carry out MCMC fit/error estimation.
freeParamNames () Get the names of the free parameters.
freeParameters () Get names and values of free parameters.
freeze (specifiers) Consider variables free to float.
frozenParameters () Get names and values of frozen parameters.
getRelationsOf (specifier) Return relations of a variable.
getRestrictions () Get all restrictions.
hasVariable (specifier) Determine whether the variable exists.
numberOfFreeParams () Get number of free parameters.
parameterSummary ([toScreen, prefix, sorting]) Writes a summary of the parameters in text form.
parameters () Obtain parameter names and values.
relate (dependentVar, independentVars[, func]) Define a relation.
removeConditionalRestriction (*args) Remove an existing conditional constraint.
renameVariable (oldName, newName) Change name of variable.
restoreState (resource) Restores parameter values from file or dictionary.
saveState (*args, **kwargs) Save the state of the fitting object.
setObjectiveFunction ([miniFunc]) Define the objective function.
setPenaltyFactor (penalFac) Change the penalty factor.
setRestriction (restricts) Define restrictions.
setRootName (root[, rename]) Define the root name of the model.
showConditionalRestrictions (**kwargs) Show conditional restrictions.
steppar (pars, ranges[, extractFctVal, quiet]) Allows to step a parameter through a specified range.
thaw (specifiers) Consider variables fixed.
untie (parName[, forceFree]) Remove all relations of parameter parName, i.e., the parameter is not dependend on other parameters.
updateModel () Recalculate the model using current settings.
evaluate ( x ) ¶

Calculates and returns radial velocity shift according to current model parameters.


Radial velocity curves - Astronomy

Stars with the strongest hydrogen lines (designated with H in the above picture) were originally classed as type A, and stars with the weakest hydrogen lines were classed as type O. Originally, there were 15 types (A-O), but later some of the classes were omitted, because they had been invented to fit some of the poor quality spectra. In 1920, an Indian astronomer named Saha figured out that the spectra could be sorted in order of temperature, and the classification scheme was re-ordered:
OBAFGKM
O stars are the hottest stars and M stars are the coolest stars. A stars have the strongest hydrogen lines (which is why Annie classified them this way!).

The spectral classes are subdivided into 10 subclasses, 0-9. An O0 star is the hottest, followed by an O1, O2, etc. O9 is just a little bit hotter than a B0, and so on.

Most stars can be classified in this way. There are a few that just don't fit, sort of like the duck-billed platypus, which doesn't fit into zoology classifications. With those, we just do what we can.

So the strength of the lines indicates the temperature, as well as the abundance of the element. In a way, this is bad, because it makes it difficult to figure out if a star has strong lines because there is a lot of that atom there, or because it's just the right temperature. So we need an INDEPENDENT way to determine the temperature.

Temperature: Objects have colors for one of two reasons. Think about your clothes. Your clothes are all different colors, because they are made of all different things. They have intentionally been dunked in dye so that they are red or yellow or blue or some combination of these. But objects can have color for another reason. Think about the burner on an electric stove. As you turn up the temperature, the burner changes from black to red. This tells you that the temperature can also change the color.

This is also true for stars. In this case, we find the temperature of a star by comparing the continuum spectrum of the star (the part that's NOT lines) with a blackbody spectrum. Here are a set of blackbody curves:

From these curves, you can see that hot objects are bluer, and cool objects are redder. So red stars are cooler than blue stars. "What!?" You say, "Stars have colors?!" Absolutely they do. If you go out at night and find Orion in the sky, you will see Betelgeuse and Rigel, a red star and a blue star, in one constellation. It's quite spectacular, and once you've noticed it, you'll see color in lots of stars.

The other important thing that you should notice about this figure is that the lines do not have exactly the same shape. If you shift them up or down, they will not peak in the same place, nor will they line up if you shift them left or right! This means that you can find the temperature of a star just by measuring the continuum spectrum in a couple of places, and comparing to the slopes of the blackbody curves for different temperatures.

Motion: All this time, I've been talking about the stars as though they are fixed in the sky. But that's not quite true. They DO move, just very slowly! Stars move in all kinds of directions in the sky relative to the Earth. When astronomers try to figure out how a star is moving, they generally do it in two steps, because each part has to be determined in a different way. The motion across the sky is called the proper motion, and the motion towards or away from us is called the radial motion.

    Proper Motion: This is the motion across the sky, and is measured as a change in the Right Ascension and Declination of a star. Proper Motion was discovered by Halley (of Halley's comet). He compared the positions of Arcturus, Sirius and Aldebaran with positions noted by the ancient Greeks, and with that several thousand year baseline, was able to determine that they had moved 0.5 degrees, which is not very far, if you think about it.

  1. These three stars are among the brightest.
  2. This means that they are perhaps also among the closest.
  3. These closest stars are the only stars that I can see moving.
  4. Therefore, perhaps ALL stars move, and if my measurements were more accurate, I could see that.

Barnard's star is the fastest mover in the sky, and moves about 10".25 per year. (Remember how small an arcsecond is---a tennis ball eight miles away!)

To understand this, we need a digression.

The Doppler Effect: Imagine that you are lying on the sofa, watching soaps. A fire engine comes down the street. While it's approaching you, the pitch of the siren is higher. While it's receding from you, the pitch of the siren is lower. If you could hear stars, they would do the same thing. The stars that are approaching you would have a higher pitch, and the ones that were going away from you would have a lower pitch.

Unfortunately, noise does not travel through space. Fortunately, the same effect also applies to light, if we just replace the word pitch with the word frequency.

The Doppler effect shifts the light's frequency, depending on whether the object is moving towards you or away from you. If the object moves towards you, it is catching up a little bit to the light as it is emitted, and the frequency gets higher. This is called "blue-shifting", and the light is bluer. If the object is moving away from you, the light gets stretched out, and the frequency gets lower. This is called "red-shifting", and the light is redder. The source in the following image is the yellow dot. It is moving to the left.

How can you tell the difference between an object which has been blue-shifted, and one that is hotter?

The answer is to use the 'lines' discussed above, which show what the star is made of. These have a particular pattern, and also a particular set of wavelengths when they are at rest. We look at the spectra of stars, and measure how much the lines have shifted.

The shift in frequency, the rest frequency and the velocity are all related:

where c is the speed of light.

So. Finally. What did we just figure out? Oh right. The radial velocity, or the motion of the star towards or away from you. To figure this out, you need to know that an atom's signature lines get shifted when the object moves, and the amount of the shift is determined by the speed of the star.

So those are all the things that we can observe about stars. Next, we'll start talking about what we can determine ABOUT stars from these observable features.


Theoretical Radial Velocity Curves

Okay. so here's the problem I'm working on. Suppose you're observing a celestial body in space that is following a (relatively small) elliptical orbit around a larger distant celestial body (let's say it's at a distance d such that d>>a, where a is the semi-major axis of the orbit). For simplicity let's treat both the earth and the larger celestial body as fixed, so the only moving object is the smaller body.

I would like to find a way of expressing the radial velocity (i.e. the component of the object's velocity that lies along the line of sight from the earth) as a function of time, expressed in terms of the angle of inclination i, the semi-major axis a (or perhaps a sin i as one parameter), the eccentricity e, the period P, the longitude of the ascending node capital omega, and the argument of the perihelion lower-case omega.

I am relatively familiar with all of the equations relating each of the parameters to each other, and I can even calculate for a specific time t what you would expect the radial velocity to be (with given initial conditions), but if it's possible I would like some kind of an equation for it (again, "it" being the radial velocity) even if it contains inverse trig functions, so that I can plot my own radial velocity curves to look at, without a large amount (or rather, any amount) of programming knowledge. In other words, I'd like to be able to do it with Mathmematica, if possible.

I don't expect anyone to hand me an equation or anything, but any information, insight, references, etc. relating to this matter would help.

I've been reading some literature on binary star systems, which seems to include some of this kind of stuff. It's been helpful, and I've seen a number of places where people have plotted these kinds of curves, but they don't do the greatest job of explaining how it was done.


The component of velocity along the line of sight to the observer. Objects with a negative radial velocity are travelling towards the observer whereas those with a positive radial velocity are moving away.

In astronomy, radial velocities can be determined by examining the redshift of spectral lines in a star or galaxy’s spectrum. This allows astronomers to compute the distance to galaxies using the Hubble expansion law and also study the orbits of stars in binaries.

Because of the expansion of the Universe almost all galaxies (except some of the very nearest) have positive radial velocities.

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Fig. 2

Schematic depiction of the instantiation path of all EXOSIMS modules. The arrows represent instantiation of an object, and object references to each module are always passed up directly to the top calling module so that a given module has access to any other module connected to it by a direct path of instantiations. For example, TargetList has access to both PostProcessing and BackgroundSources, whereas Observatory does not have access to any other modules. The typical entry point to EXOSIMS is the construction of a MissionSimulation object, which causes instantiation of SurveySimulation, which in turn instantiates all the other modules.

Yield is calculated from a generated survey ensemble, which contains the outputs of N survey simulations, each for a given simulated universe, all other mission and instrument parameters being constant. The simulated universe is regenerated for each individual mission simulation by resampling the planet population distributions and synthesizing entirely new planets. The execution of a single mission simulation occurs within a top-level loop, which selects each subsequent target from the pool of “currently” (at the current simulated mission time) available targets, calculates the required integration time, simulates the observation (and followup characterization, if called for by the selected mission rules), and then simulates the outcome, which can be a true positive (detection), false positive (misidentified speckle or background object), true negative (null detection), or false negative (missed detection). The functionality of the default survey execution is extended and augmented in multiple additional survey simulation implementations designed specifically for analyzing HabEx and LUVOIR. Below we detail two of these, which were used extensively to produce the results in this report.

Observation Scheduling

The scheduler module is the backbone of EXOSIMS. It drives the dynamic observing sequence and contains the logic for autonomous target selection. Multiple schedulers exist for EXOSIMS, each tailored to an architecture and an observing scenario. The weighted linear cost function (WLCF) scheduler is used for architectures having a starshade (HabEx 4S and HabEx 4H). In the case of the HabEx 4H architecture, which uses both a starshade and a coronagraph, a tiered scheduler is further used. For coronagraph-only architectures, a simplified WLCF scheduler is used. Below we introduce the core scheduling approach used, how it is adapted to serve the coronagraph/starshade hybrid case, and how it accommodates starshade-only repeat observations for orbit determination.

Weighted linear cost function scheduler

The WLCF scheduler, implemented in EXOSIMS as “linearJScheduler,” is based on an observation scheduling methodology fully described by Savransky et al. 11 A starshade scheduler must find a compromise between high-completeness targets and low-slew targets. The problem of maximizing completeness of a series of observations subject to an overall slew constraint is related to the well-known TSP, but with key differences that make a time-windowed but dynamic approximation more effective. 11 A detailed description of the approach and cost function terms are in Appendix B.

Changing the weighting factors on the cost function terms changes mission priorities. Optimal weighting factors can be determined by wrapping the whole simulation in an optimization scheme with an appropriately defined objective function (e.g., the number of Earth-twin spectral characterizations within a fixed set of simulated universes). This approach was used several times to evaluate parameter choices for this scheduler under different operating conditions.

Tiered scheduler

The tiered scheduler was developed for the HabEx hybrid starshade and coronagraph architecture. It is a hierarchical scheduler that schedules the starshade at the top tier and fills in coronagraph observations at the lower tier. Observations are scheduled sequentially to allow for dynamic response to the discoveries and the success or failure of observations. The starshade observations have first priority, using the above WLCF scheduler to find an effective starshade path. During the starshade slews, coronagraph observations are scheduled within the second tier. Also during the slews, general observatory (GO) observation time is allocated at the specified rate (50% for the HabEx baseline mission concept). GO time is allocated whenever “owed” GO time accumulation exceeds 1 day, unless a starshade observation is occurring, in which case the owed time is allocated at the end of the starshade observation. This distributes the GO time evenly throughout the mission. Additional details are in the results section for the HabEx 4H hybrid case (Sec. 2.3.2).

Figure 3, available as a video, shows an observing sequence for one possible universe for the HabEx hybrid starshade and coronagraph scenario, scheduled with the tiered scheduler. (The still-image version shows the final tour executed by the mission.) The region of observability of the starshade given the solar keepout constraints is the white annulus within the figure, and starshade slews are indicated by the black arrows. While the starshade slews to its next target, the telescope rapidly repoints to allow the coronagraph to search for new exoplanets. Revisit observations are indicated by an increased marker size. Once sufficient observations determine the orbit, the target is promoted to the starshade for spectral characterization.


An Etymological Dictionary of Astronomy and AstrophysicsEnglish-French-Persian

A peculiar → Mira variable with a → pulsation period of 387 days surrounded by an extended → emission nebula. R Aqr is a → symbioticbinary star with a mass-losing, pulsating → red giant and an accreting hot companion with a jet → outflow which ionizes an emission nebula. The orbital period of the R Aqr binary is about 44 years as inferred from periodic phases of reduced brightness observed around 1890, 1933, and 1977. These phases are interpreted as partial obscurations of the mira by the companion with its → accretion disk and the associated gas and dust flows. The inferred orbital period is supported by → radial velocity measurements (Schmid et al., 2018, A&A 602, A53 and references therein).

A → stellar association containing a number of → reflection nebulae. The stars are of low or intermediate mass and young, less than a million years old. They are still surrounded by patches of dust that reflect and absorb light from the interstellar cloud in which they formed. This type of association was first suggested by Sidney van den Bergh (1966, AJ 71, 900).

A → nucleosynthesis process in which → chemical elements heavier than → zinc are created through the intense bombardment of other elements by → neutrons in rapid succession. The essential feature of the r-process is the release of great numbers of neutrons in a very short time (less than 100 seconds). The r-process is a "rapid" version of the → s-process, occurring in supernova → core collapse and possibly when a → neutron star merges with a → black hole in a → binary star.

r stands for rapid, since the process entails a succession of rapid neutron captures on iron seed nuclei → process.

A spherical → ionization front of → H II regions that moves radially outward from the → exciting star at a velocity much higher than → sound speed in the surrounding cold neutral gas of uniform density (ahead of the front). R-type ionization fronts corresponds to early evolution of H II regions, and will eventually transform into → D-type ionization fronts. If the motion of the front is supersonic relative to the gas behind as well as ahead of the front, the front is referred to as weak R. The strong R front correspond to a large density increase across the front.

R referring to a r arefied gas → typeionizationfront.

The central object of the → 30 Doradus nebula in the → Large Magellanic Cloud. Also known as HD 38268, it was thought to be a single star of several thousands → solar masses until → speckle interferometry techniques resolved it into a rich and compact star cluster. Recent high-resolution studies have shown that R136 contains 39 known O3 stars, which is more than known to be contained in the rest of the → Milky Way, → LMC, and → SMC combined. R136 is a prototype "super star cluster," with an estimated mass of 10 5 solar masses. Its most massive stars are less than 1-2 million yeas old, while its lower-mass stars formed 4-5 millions years ago.

The Radcliffe serial number 136 (Feast et al. 1960, MNRAS 121, 25).

1a) A group of persons related by common descent or heredity.
1b) A population so related.
2) A contest of speed, as in running, riding, driving, or sailing (Dictionary.com).

1) From M.Fr. race "race, breed, lineage, family," from It. razza, (cf. Sp. and Port. raza), of unknown origin.
2) M.E. ras(e), from O.N. ras "running, race," cognate with O.E. ræs "a running, a rush, a leap, jump."

1) Nežâd, literally "born," ultimately from Proto-Ir. *nizat-, cf. Av. nizənta- "born," from → ni- + *zan- "to give birth, to be born," cognate with âzâd, → free see also → generate.
2) Tâz, present stem of Tâxtan/tâz- "to run, rush upon, assault" → surge.

1) A belief or doctrine that inherent differences among the various human races determine cultural or individual achievement, usually involving the idea that one's own race is superior and has the right to rule others.
2) A policy, system of government, etc., based upon or fostering such a doctrine discrimination (Dictionary.com).

A unit of energy absorbed from ionizing radiation, equal to 100 → ergs per gram, or 0.01 → joules per kilogram, of irradiated material. Rad has been replaced by → gray (Gy).

Shortened form r oentgen a bsorbed d oseroentgen.

An emitting/receiving device in which the echo of a pulse of microwave radiation is used to detect and locate distant objects.

From ra (dio) d (etecting) a (nd) r (anging).

Graphic display of measurements by a → radar of mineral deposits on a planetary surface.

Emanating from a common central point arranged like the radii of a circle.

From L. radialis, from → radius-al.

The process whereby a → disk star changes its → galactocentric distance. Radial migration involves → angular momentum transfer, resulting from → resonances created by transient → density waves such as → bars or → spiral arms in → galactic disks. According to → galactic dynamics models, → churning is the main cause of radial migration. Radial migration of stars plays an important role in shaping the properties of galactic disks.

A motion away from or toward a central point or axis.

Any of short-lived (generally lasting less than 24 hours) radial features that periodically appear over the outer half of → Saturn's → B ring, when the ring tilt angle is small. These features revolve at the same rate as the planet's → magnetic field and maintain their shape over much of the course of their existence even though they extend tens of thousands of kilometers across the rings. It is believed that the tiny particles that make up these spokes are electrically charged and temporarily "frozen" into the planet's magnetic field (Ellis et al., 2007, Planetary Ring Systems, Springer).

The component of a three-dimensional velocity vector of an object directed along the line of sight. It is measured by examining the Doppler shift of lines in the spectrum of astronomical objects.

A curve describing the variation of the radial velocity of a star, due to the Doppler effect, under the gravitational effect of a secondary body (companion or exoplanet). The amplitude of these variations depends upon the mass of the secondary and its distance from the star.

The technique based on the analysis of the → radial velocity curve, used to detect the presence of an invisible secondary around a host star. This method holds the majority of exoplanet discoveries.

A unit of angular measure one radian is that angle with an intercepted arc on a circle equal in length to the radius of the circle.

From radi(us) + -an an originally adj. suffix.

1) Generally, the → radiant energy per unit → solid angle per unit of → projected area of the → source. It is usually expressed in → watt per → steradian per → squaremeter (W m -2 sr -1 ). Same as steradiancy.
2) Of any particular → wavelength within the interval covered by a → spectral line, the → energy per unit → surface per steradian, per wavelength denoted I λ . The term radiance is often loosely replaced by "→ intensity." The radiance of the whole line is given by I = ∫ I λ dλ. The radiance of an → emission line depends, among other things, upon the → number of → atoms per unit area in the → line of sight (the → column density) in the → upper level of the line.

From radia(nt), → radiant, + → -ance.

Tâbešmandi, noun from tâbešmand "possessing radiation," from tâbeš, → radiation, + -mand a suffix denoting possession Mid.Pers. -ômand suffix forming adjectives of quality.

1) Sending out rays of light bright shining.
See also: → radiant energy, → radiant flux, → radiant intensity.
2) The point in the sky from which → meteors in a → meteor shower appear to radiate or come. See also: → radiant drift.

M.E., from M.Fr. radiant, from L. radiantem (nominative radians) "shining," pr.p. of radiare "to shine, radiate," → radiation.

1) Tâbandé, tâbeši adj. from tâbidan, → radiate.
2) Tâbsar, from tâb "light, radiation," → radiation, + sar "head, top, summit, point," → head.


Radial Velocity of Galaxies

I suppose this thread might be an appropriate place for me to add my request for assistance on the question of rotating galactic systems. I am interested in the problem of the observed asymmetry in left side/right side rotational velocity of curves of spirals and barred galaxies, but I cannot seem to find any thing that provides a general/comprehensive treatment of the subject, or even an introductory report on the nature and scope of the analytical issues involved. The advanced papers I have read on the subject are highly focused and, it appears to me, do not fully explicate the underlying issues involved in interpretating the "data". For example, I have canvassed probably 75 papers dealing with the kinematics and asymmetric velocity curves of these systems and not one of them identifies the adjustments required for interpreting the rotational velocity data on account of the Hubble shift. I have ordered Binney and Merrifield as it appears to be the go to resource in the field, but, I find it very hard to believe that there isnt a generally available resource that is addressed to the specific problem of sorting out the data evidencing the asymmetry in the rotational velocities of these systems. At this point, I am specifically interested in the modeling of the effects of the Hubble Flow on the interpretation of the Left/Right rotation curves of these galaxies across the scale of low to high z systems. Any assistance in pointing me to helpful sources/resources addressing this subject or the broader subject of asymmetric kinematics (exclusive of "lopsided" systems--spatial asymmetry of mass distribution), of these systems would be much appreciated.


Radial Velocity Studies of Galactic Cepheids

This seemed like an appropriate time to summarize the results of the Cepheid radial velocity studies which I have been carrying out at DDO for nearly 30 years now. The main thrust of the program has been radial velocity measurements of Cepheids, particularly binaries. This has been combined with satellite (IUE and HST) velocity work to measure Cepheid masses. These are still the best fiducial for testing evolutionary tracks of evolved objects, as well as quantitative measurements of primary distance indicators. In addition the program has produced a definitive distribution of mass ratios in intermediate mass binaries, important information on star formation. Finally the combined ground-based and satellite studies leave us poised to make use of newly resolved binary systems. The one program which I am currently working on is to try to measure the velocities of the bright companions of 3 Cepheids near 4000 A. This is particularly well-suited to DDO, since it requires spectral subtraction. If we (Evans, Vinko, Kiss, and Beattie) succeed - and I did many years ago with SU Cyg--three systems will be strong candidates for both resolution and HST velocity work. The program started in the 1970's and is still continuing. In 1989, a new mode of service observing has been started when I was associated with the Institute for Space and Terrestrial Science in Toronto. The first papers from this mode started appearing in 1994 since then we have had 6 papers in refereed journals and 9 papers in conference proceedings. The following persons contibuted to the program: Tom Bolton, John Percy, Ben Sugars, Jozsef Vinko, Ron Lyons, Jim Thomson, Irina Dashevsky and Andrzej Udalski. Below I include a list of publications, those either containing DDO data, or very directly following from DDO results such as orbits. (A couple of Southern stars are thrown in because they complete the Cepheid information.) DDO has been a wonderful facility to work on the multi-year orbits of Cepheids, and without the orbits, the breakthrough satellite results (double-lined spectroscopic binaries) would not have been possible. Nancy