Determination of Stellar Radii and Masses


Goals of this project:

Grading Policy
There are two parts to this lab. The IUE data analysis must be done. The 14" observations must be attempted, but things can go wrong. You need not do the 14" observations if:

If it appears that you will have difficulty completing the optical observations, you must discuss the situation with the instructor as soon as possible, and before the last week of the lab. We have previously-obtained data which you can analyze. The grade will be based on the quality of the data analysis and interpretation. In the absence of extenuating circumstances, the archival data analysis, or the optical data analysis, alone can be worth no more than a B grade.

I. Introduction

Newtonian gravity is a powerful diagnostic tool. Given that gravity is a 1/r2 central force, one can derive Kepler's Laws, conservation of energy, and conservation of angular momentum. One can then use these laws to determine the masses of stars. Mass is the fundamental parameter which drives stellar evolution, unfortunately stellar masses can only be determined when you have gravity - i.e., in a binary star system.

To determine the stellar masses, you need to know the orbital period P and the semi-major axis a of the orbit. You can determine a from P and the orbital velocity.

The point of this exercise is to apply Newton's laws and Kepler's laws. You will verify the orbital period by obtaining optical photometry using the 14" telescope. You will determine the orbital velocity from archival IUE spectra. This will yield the masses. In addition, you will use the timing of the eclipse to determine the stellar radii.

II. How to begin this lab

You must take care of the following details during the first lab session:
  1. Select a suitable target to observe. Read section III.. You will need to know what targets are observable at this time of year. Read the tutorials on times and coordinate systems.
  2. Arrange with your TA to learn how to use the 14" telescope.
  3. Examine the IUE database (Section V), and ensure that there are LWP-HI or LWR-HI observations you can use. If there are not enough (and this is the case for some good optical targets), then you may do the IUE data analysis on another target.
  4. Inform the instructor or the TA what your target(s) will be, and why you chose it (them).

III. Suggested Targets

Of the hundreds of thousands of stars visible from Stony Brook through the 14" telescope, you need to select one. Your constraints are:

There are two catalogs you can search for targets. Both are available as IDL databases.

  1. The Catalog of Chromospherically Active Binary Stars (CABS) (Strassmeier, Hall, Feckel, and Scheck, 1993, A&A Supp 100, 173). This catalog exclusively contains late-type stars with active chromospheres. The name of the database is CABS.
  2. The Eighth Catalog of Orbital Elements of Spectroscopic Binary Systems (Batten, A. H., Fletcher, J. M., MacCarthy, D. G. 1989, Publ. DAO, 17). This catalog is all-inclusive, but suffers from a lack of recent updates to the orbital elements. The name of the database is BATTEN.

I searched for eclipsing binaries with periods longer than 8 hours (to exclude contact binaries) and less than 2 days (so there's an eclipse every day), with the further constraints that the declination is greater than -10o and the primary spectral type is F or later. I found 13 candidates in the CABS database, and 16 candidates in the Batten database (there is overlap). Select a star from one of these lists, or from someplace else if you have another target. Then

  1. Make sure your star is visible during this semester. Need a hint?
  2. Look up the orbital elements, and predict the times of eclipse. Note that telescopic observations are best done during nighttime hours. You can find orbital elements in either the CABS or Batten databases. Note that these orbital elements are old, and extrapolation to the present epoch can lead to significant errors (if the uncertainty on a 1 day period is +/- 1 minute (0.07% accuracy), and the last epoch of minimum was in 1980, the uncertainty on the extrapolated phase is now about 200%!). Therefore ...
  3. You must double-check the period and epoch using recent references. You can find these in the ADS or SIMBAD databases. The IBVS (Information Bulletin on Variable Stars) may have the most recent lightcurves.

    To download issues of the IBVS, do the following:

    The IBVS is also available in the Science and Engineering Library.

  4. Go to the IUE catalog, and make sure there are LWP-HI or LWR-HI exposures near the times of quadratures. If there are, request these data from NDADS. If not, try another star.
  5. Figure out how to find this star with the 14" telescope. Because the telescope has no good setting circles, we use a star hopping technique. First, find a nearby bright star. You can search the Yale Bright Star Catalog, which is available as an on-line IDL database. You can use the Digitized Sky Survey (available on-line from the HST/MAST archives) to make a finding chart, if necessary. Note that the positions in the Batten catalog are equinox 1900; IUE used equinox 1950, and the HST uses equinox 2000 coordinates. You may have to precess your coordinates.
  6. Because star hopping can be tedious, it may take up to an hour or two to find the star. It would be best if the eclipse occurred in the second half of the night, at least until you are adept at finding the target. WARNING: it may be very difficult to find your target when the Moon is nearly full. If possible, try not to observe near the full Moon.

To figure out whether your star is observable at this time of year, you need to know the local sidereal time at midnight. You can use the IDL procedure LMST (type print,lmst(/help) for on-line help), or you can use the SKYCAL utility (click on the SKYCAL icon on the PC).

Sample images of stars that have already been observed as part of this lab are:

If you are curious about why these stars have the names that they do, check the naming of variable stars

IV. The Mount Stony Brook Observatory

The Mount Stony Brook Observatory (MtSB) is the name our undergraduate observers have given to the University's 14" telescope atop the ESS building. Details of the telescope and its operation are given in a postscript document. You must read this manual. Download it and print it out for reference. You can review the basics of CCD data reduction Read the basic telescope safety document.

Prior to using the telescope, you will be given a lesson by a TA or other experienced observer. You may wish to assist others on a few observing sessions prior to your first solo night. Note that no one is allowed to observe alone: for safety reasons you must have an observing partner.

The telescope is heavily booked. You must sign up on the board outside room ESS 443C.

V. The IUE

The International Ultraviolet Explorer (IUE) was the little satellite that could. It operated for nearly 20 years before being turned off for lack of funding.

We will use IUE data because IUE offers a large and uniform spectroscopic archive. The IUE obtained spectra in two wavelength ranges: 1150-1950 (short wavelength, or SW) and 1900-3200 (long wavelength, or LW) Angstroms. It offered both low (6 Angstroms resolution) and high (10,000 resolution) dispersion. The high dispersion used an echelle spectrograph, which permited the full spectral range to be imaged simultaneously on a 2-dimensional detector. In particular, observations with the LWP or LWR (the P and R refer to the prime and redundant cameras) cameras in high dispersion reveal the Mg II h and k lines in emission in chromospherically-active stars. By measuring the radial velocities of the Mg II lines at opposite quadratures, you can determine the orbital velocities of the stars. You will search the archive for appropriate observations of your targets.

You will downoad the IUE data from the MAST archives. Instructions for searching MAST are here. Use the IUE Search Form.

Then click on the "Search" button. When the search results come up, mark the images you want (probably all of them), and click the "Download NEWSIPS MX files as a .tar file" button. A filke called iue.tar should be directly downloaded to you. Extract the files using wtar; then will be written to a subdirectory called "iue". These file have ".mxhi" extensions, but they are Fits format.

Click here to learn how to read IUE data.

VI. What to Measure

The IUE Data
You want to identify a stellar line (preferably one in each star), and measure its wavelength as a function of time. Use the Doppler effect to determine the radial velocity as a function of orbital phase, and determine the maximum velocity for each star. You will also need to determine the systemic, or <em>gamma</em>, velocity for the star. You can determine this directly by observing at phase 0 and 180.

This must be done using high dispersion spectra. The best line to use is the 2795/2802 Angstrom Mg II doublet (the resonance lines of singly ionized Magnesium). Mg II is in emission in all rapidly rotating late-type stars. The rest wavelengths of the line are 2795.523 and 2802.698 Angstroms. Note that these are air wavelengths. By convention, wavelengths greater than 2000 Angstroms are generally quoted in air, while wavelengths shorter than 2000 Angstroms are quoted in vacuum. When the IUE project reprocessed their data, they chose to use the vacuum wavelength scale for all their data. The ratio of the vacuum to air wavelengths is the index of refraction of air.

The Optical Data
You want to measure the flux of the star as a function of time through its eclipse. It is almost impossible to make an absolute flux measurement from Stony Brook, so we rely on differential photometry. Choose another star in the CCD field, and measure its brightness too. The other star is most likely not as variable as the eclipsing binary (Most stars are binaries, however, so you might check this one out just in case. Try the General Catalog of Variable Stars.).

If you are unable to do the observations because of inclement weather (or any other reason), you must still undertake the imaging analysis. There is a sequence of images of RT And available in the directory \phy515\rtand. Analyze these as though you took them. The readme.txt file describes the files; you will need to read the headers to learn the exposure times and times of observation.

0.4 times the logarithm of the ratio of counts is the magnitude difference. If there is no other star in the CCD image, or if the star is very faint (hence noisy), find a nearby star about as bright as the eclipsing binary, and observe it frequently to monitor trends in transparency.

Of course, photometry is not really this easy. Details can be found here.

Plot the brightness of the target as a function of time. If you did everything right, you should see a decrease in brightness followed by a return to the original brightness centered on the time of eclipse.

In order to do these reductions, you will have to learn how to display and analyze your images using IDL. Before you take your own data, you may practice reducing images and preparing your software using the files c:\idl\testdata\ (an image of RT And) and c:\idl\testdata\ (a flat field image).

You should start your preparations by reading the IDL primer. Note especially the sections on Plotting and Measuring Data and Using the Cursor to measure positions.

The basic data reduction involved extracting the subarray of data containing the star (star=image(x1:x2,y1:y2)) and a subarray containing the background. Use the total command to determine the number of counts. Scale the background counts to the area of the star subarray, and subtract the background. Do the same for the comparison star, and record the ratio of counts. If you have many images, consider writing a program to loop through the images, doing this automatically.

contacts in an eclipsing binary When determining the eclipse times, or the times of contact, it is sufficient to model the eclipse as four straight line segments (see figure). Assume the brightness is constant outside eclipse. Further assume, if the eclipse is total or annular, constant brightness between 2nd and 3rd contacts. Assume that the light curve between 1st and 2nd contacts, and between 3rd and 4th, can be modeled as a linear function of time. The intersections of these 4 line segments provide a good estimate of the times of contact.

VII. Questions to be answered

  1. Why must you use high dispersion spectra for this project?
  2. Why do we use differential photometry? What conditions do you need for absolute photometry?
  3. How do you use the orbital velocities and the length of the eclipse to determine the radii of the two stars?
  4. At primary minimum, which star is in front and which is behind? Answer this for your binary, not in general. What can you say about the relative temperatures of the two stars?
  5. What is the narrow absorption line in the IUE Mg II spectra, and why doesn't its wavelength change?
  6. What are your best estimates of the masses and radii of the stars? Include error estimates. Look up the true masses and radii in the CABS, and comment on the differences between your estimates and the truth, especially if your error bars are too small.

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