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Curriculum: [ Sample Labs | Astronomy Courses @UI ] |
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Background and TheoryAlthough stars have a enormous range of masses, sizes, temperature, and luminosities, a simple measurements of the apparent magnitude in two filters is often sufficient information to determine these characteristics. Stars are relatively simple entities which can be described by a few elementary laws of thermodynamics and radiation. (This ignores details of the fusion engine in the core, which are not so simple!) In particular, the observed radiation has the same characteristic intensity—wavelength dependence that all thermal emitters (e.g. stovetops, people) have, creating a simple relation between observed color and surface temperature. If the star is on the main-sequence (90% of all stars are), the luminosity can be inferred from the color. Since the total luminosity of the star is proportional to surface area and we know that (most) stars are (mostly) spherical, the radius of the star can be inferred from the luminosity. Finally, it turns out that stellar models predict a simple mathematical relationship between luminosity and mass which is verified by observations. In this observing project, you will choose a young open cluster star system and observe several stars using visual (V) and infrared (I) colored filters. Using these measurements, it will be possible to determine the distance, temperature, luminosity, radius, and mass of each star. The V and I magnitudes for each star are obtained using differential photometry and a reference star in the same field with known magnitudes. Surface Temperature
Once the true V and I magnitudes are found, the color index V-I can be calculated. For V-I in the range [1] 0.0 < V-I < +1.5, the relationship between V-I and the surface temperature can be written as: The color index can also be used to determine
the spectral type using Table II at the end of this section. DistanceThe clusters chosen for this project are relatively young, so most cluster members are on the main sequence. Therefore, the spectral type can be used to determine the absolute visual magnitude MV using the HR diagram in the Appendix. The distance in parsec is related to the observed visual magnitude and absolute visual magnitude by LuminosityThe luminosity of the star depends on the absolute bolometric magnitude Mbol which represents the sum of the energy output at all wavelengths. This is given in Table I for main sequence stars as a function of spectral type. The luminosity can be determined by comparison with the sun’s bolometric luminosity
where Msun = 4.7 and Lsun = 3.8×1026 W. RadiusThe stellar radius is given by considering the formula for the Stefan-Boltzmann law for the star and for the Sun and dividing the two equations to eliminate constants. One obtains where Rsun =
7 x 105 km and Tsun = 5700 K are the radius
and surface temperature of the sun, respectively. Mass Determination
For stars on the main sequence, the mass-luminosity relation can be approximated by the following function [2] (in this case Msun is the mass of the Sun, not its absolute magnitude). where the sine argument is in radians. Solving for log (M/Msun), we get: Note: Be careful to convert arcsin (sin-1) to radians if your calculator returns angles in
degrees. ProcedureObservations1. Choose one of the open clusters from Table I. Make sure it is well placed in the sky (at least 30° elevation) during the available observing period. To check the elevation, use a sky display program such as Starry Night
2.
Schedule V and I filter exposures. For V and I images, 5 and 20 second
exposures will give you a good range of stars for most clusters. Warning: Longer exposures, while revealing more of
the fainter stars, will overexpose the bright stars. This is okay, as long as the brighter objects
are not used for photometry (i.e. the maximum ADU count of the center
of the star should not exceed 50,000). Image Analysis
1. Run MaxIm. Load your images, both V and I. Select at least five stars from the cluster, making sure the stars you select are visible in both images and are not overexposed. 2. A list of files to be used as finding charts can be found in Table I (the charts themselves are located in clusters). They include calibrator stars to be used in the absolute magnitude calculations. (Note: Do not use the calibrator star as one of your five.) You can load them into MaxIm or some other image inspection program and print them out. 3. Perform differential photometry to find the apparent magnitudes of the stars in both V and I. Click on Calibrate in the Information window. Find the reference star in the image, and click on it. Click in the Magnitude box, and replace the 0 with the magnitude of the reference star. Be careful to use the correct magnitude for each filter. Click on OK. Clicking with the right mouse button on any other star in the field will cause the magnitude (and the error in the magnitude calculation, estimated from the air mass, sky background, etc.) to appear in the photometry window. Tabulate your data in a chart such as the one given. Calculate (V - I) and enter the results for your stars into the chart. 4. Calculate the effective surface temperature of each star as in the Background and Theory section and enter it into the chart. 5. Using Table II and your (V- I) calculations, find the spectral type and absolute magnitude (MV) of each star. (Note: Because these clusters are relatively young, it is assumed that members of the clusters are main sequence stars. The values in Table II are derived from the H-R Diagram in Appendix D.) Use this information to calculate the distance d to each star. The distance to each star should be roughly the same, as they should all be members of the same cluster. If there is a star whose distance is much different than the others, the star may be a foreground or background star, and not actually a member of the cluster. Once the absolute magnitude is known, calculate the luminosity of each star, using the bolometric magnitude for Mbol. Enter this information in the data chart. 6. The luminosity and temperature determine the radius of a star. Use your values to determine the radius of each star in units of solar radii as discussed in the Background and Theory section. 7. Using the mass-luminosity relation for main sequence stars, calculate the mass of each star. 8. Discuss trends in your results.
Table 1: Finding charts for open clusters
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