Does anyone know how to find the size of stars?

Question

I need sizes of specific stars including Betelgeuse, Sirius, Antares, Pollux and other ones.

Answer 1

If you know that the star is on the main sequence, you can use its color temperature (obtained from observations with telescope, U-B-V filters, photomultiplier tube, and chart recorder) to infer its luminosity (HR diagram). Then you use the luminosity and the mass-luminosity relationship (see C.W. Allen's "Astrophysical Quantities" and make your own interpolations) to get the star mass. Then you raise the mass in suns to the power of 0.72 to get a reasonably decent estimate of its radius in solar radii. One solar radius is 696,000 kilometers.

Come to think of it, you can approximate the effective temperature with the color temperature, and then...

R = 6.96E8 meters (T / 5770 K)^{72 / (25a - 36)}
[Main sequence stars only!]

where (a) is the exponent from the mass luminosity relation, which is about 3.5 for most main sequence stars, so

R = 6.96E8 meters (T / 5770 K)^1.4
[Main sequence stars only!]

If you know the distance to the star, and the apparent magnitude, you can calculate the absolute magnitude. Then with the color temperature (assumed as a close approximation to the effective temperature) you can get the radius from the Stefan-Boltzmann law.

M = m - 5 log (d / 32.616 lightyears)
L/Ls = 10^{ (4.75 - M) / 2.5 }
Ls = 3.826E+26 watts
R = { 1184.6 m W^(-1/2) K^2 } sqrt(L) / T^2

where M = star's absolute magnitude
where m = star's apparent magnitude
where L = star's luminosity in watts
where Ls = sun's luminosity in watts
where T = star's effective temperature (approximated by color temperature) in Kelvins
where R = star's radius in meters
where the log function has a base of 10

However, if the star isn't on the main sequence (Betelgeuse, Antares, and Pullox are not) and if you have no way to tell how far away it is, then you will need to use a spectroscope to determine the star's rotational speed and radius from the doppler broadening of absorption or emission lines. Or you might be lucky and have a binary star whose companion eclipses it, which will let you estimate the size from the variation in the light curve.

There's a class of variable stars called 'Cepheids' whose period of brightness fluctuation is related to their peak absolute magnitudes. If the star is one of these, you don't need to know the distance to it to get its absolute magnitude. You can use the period of the star, instead, to get the absolute magnitude. Then from the peak absolute magnitude you get the peak luminosity, which with the color temperature (U-B-V filters, etc.) you can get the radius of the Cepheid variable star at peak.

M = -2.81 log P - 1.43
[Cepheid variable stars only!]
L/Ls = 10^{ (4.75 - M) / 2.5 }
Ls = 3.826E+26 watts
R = { 1184.6 m W^(-1/2) K^2 } sqrt(L) / T^2

where P = the Cepheid's period in days
where M = the Cepheid's absolute magnitude
where L = Cepheid's luminosity in watts
where Ls = sun's luminosity in watts
where T = Cepheid's effective temperature (approximated by color temperature) in Kelvins
where R = Cepheid's radius in meters
where the log function has a base of 10

Radii of stars....
Betelgeuse, 4.2E+11 meters
Antares, 5.7E+11 meters
Sirius, 1.2E+9 meters
Pollux, 7.0E+9 meters

Answer 2

All but a few stars appear as mere pinpoints in even the largest telescopes. They are much too far away to derive their diameters from measuring their angular diameters and distances. Eclipsing binaries are used to determine indirectly the diameters of stars. These are two stars orbiting each other in a plane that is parallel to your line of sight so you see their orbits edge-on. This means that one star will periodically cover up the other star. During the eclipse the total brightness measured for the binary will decrease. The amount of the dip in brightness depends on the luminosity and relative size of the two stars.

A star's diameter is found from speed = (distance travelled)/(time it takes). The speed comes from the doppler shift and the time is the length of the eclipse. The distance travelled during the eclipse is equal to the diameter of the star = 2 × radius. The light curve---plot of brightness vs. time---is used to derive the star diameters. Here is an example of two stars orbiting each other in circular orbits seen edge-on with one star small and hot and the other large and cool:

deriving the diameter of stars in an eclipsing binary system

When the small star moves from position 1 to position 2 (or from position 3 to position 4), it has moved a distance equal to its diameter. When the small star moves from position 1 to position 3 (or from position 2 to position 4), it has moved a distance equal to the diameter of the large star.

Star sizes can also be found (less accurately) from the luminosity and the flux. Recall from the magnitude section above that the luminosity = [4p×(star radius)2] × [sigma×(star's surface temperature)4], where sigma is the Stefan-Boltzmann constant. If you compare the star with the Sun, you can cancel out the constants to get (star's radius)/(Sun's radius) = (Sun's temperature/star's temperature)2 × Sqrt[star's luminosity/Sun's luminosity]. See the ``How do you do that?'' box below for an example. The sizes of different types of stars are summarized in the Main Sequence Star Properties table below.

How do you do that?
Antares is 9120 times more luminous than the Sun (Antares' luminosity/Sun's luminosity)= 9120) and has a temperature of only 3340 K and the Sun's temperature is 5840 K.

Antares' size/Sun's size = (5840/3340)2 × Sqrt[9120] = 3.057 × 95.5 = 292. Antares is almost 300 times the size of the Sun! If the Sun were replaced by Antares, the inner planets Mercury, Venus, and Earth would be inside Antares! It is a red giant star---a star close to death.
Vocabulary
light curve eclipsing binary
Formulae

* Eclipsing binary: diameter = speed × time of eclipse.
* Size from luminosity: star's radius/Sun's radius = (Sun's temperature/star's temperature)2 Sqrt[star's luminosity/Sun's luminosity].

Review Questions

1. How do you use the light curve to find the diameters of stars?
2. What special type of binary star system is used to find the diameters of stars?
3. Use the light curve in the figure in the section above. Assume that when star A is behind star B, the small dip in brightness is seen. When star B is behind star A, the big dip in brightness is seen. Which star is more luminous?
4. From the previous problem, if t1 = 45 minutes, t2 = 60 minutes, t3 = 105 minutes, t4 = 120 minutes, what is (star A diameter)/(star B diameter)? [Hint: find which star is brighter and in this circular orbit system (t8 - t6) = (t4 - t2).]
5. From the previous problem, if the velocity is 750 kilometers/second, what is the diameter of the larger star?
6. The white dwarf Sirius B has a temperature of 14,000 K and a luminosity only 0.00794 times the Sun's luminosity. What is the diameter of Sirius B in kilometers? (The Sun's radius = 696,000 kilometers.)

Answer 3

Usually, the size of star is determined by determining the absolute magnitude of the star (one needs a good estimate of the distance to do that) and its spectra type. With the stellar theory, one can derive an estimate of the mass and thus of the size.
For the specific stars you mention, I suggest you check the entry for the star names in Wikipedia. The size and mass is provided, based on astronomer's best estimates.

Answer 4

Hi. The rotation of stars causes a 'smearing' of the spectral lines due to the approaching side and the receding side. This value, tied into the stars brightness (main stream stars) can be interpreted into relative sizes. Only the very largest stars can be measured directly using interferometry.

Answer 5

I don' know those stars, but I think Halle Berry is a size 0 and Rosie ODonnel must be about a size 20

Answer 6

http://www.rasnz.org.nz/Stars/BrightStars.htm