Astronomy 162: Professor Barbara Ryden

``The sun and the moon and the stars would have disappeared long ago...had they happened to be within the reach of predatory human hands.''

- Havelock Ellis

- The radius of very few stars can be found from their angular size and distance.
- The radius of other stars can be deduced from their luminosity and temperature.
- Stars have a wide range of radii.

If the angular size `a' of a star and the distance `d' to that star are known, then the radius `R' of the star can be computed.

For trigonometry fans only: the relevant equation is

R = d tan (a/2)

- a = 1920 arcseconds
- d = 1 AU = 150,000,000 kilometers
- We compute R = 700,000 kilometers

- a = 0.125 arcseconds
- d = 427 light years = 4 thousand trillion kilometers
- We compute R = 1.2 BILLION kilometers

In the above equation, E is the luminosity of a square meter of the blackbody's surface, the Greek letter sigma stands for a constant which has been measured in the laboratory, and T is the temperature of the blackbody's surface.

Double the temperature of a blackbody, and you increase its luminosity by a factor of 2 x 2 x 2 x 2 = 16.

To find the total luminosity of a blackbody, multiply the luminosity per square meter by the number of square meters on its surface. A star is well approximated by a SPHERICAL blackbody (surface area = 4 pi R^2), so the formula which gives the total luminosity of a star is the following:

Thus, if we know the luminosity L_{*} of a star (found from its intensity and
its distance) and if we know the temperature T_{*} of a star, we can compute
its radius R_{*}.

As an example, consider Sirius, the brightest star in the night sky. It is actually a binary system. The more luminous of the two stars in the system is called Sirius A; the less luminous (which can only be seen in a large telescope) is called Sirius B. Let's examine the two stars in the Sirius system individually.

Sirius A:

- L = 26 L
_{sun} - T = 10,000 Kelvin = 1.72 T
_{sun} - R is computed to be 1.7 R
_{sun}(Work this out for yourself!)

Sirius B:

- L = 0.0024 L
_{sun} - T = 15,000 Kelvin = 2.59 T
_{sun} - R is computed to be 0.007 R
_{sun}(Work this out for yourself!)

- Betelgeuse = 1800 R
_{sun} - Sirius = 1.7 R
_{sun} - Sun = 1 R
_{sun}, by definition - Companion of Sirius = 0.007 R
_{sun}

Because the radius of Betelgeuse is larger than that of the
Sun by a factor of 1800, its volume is larger by a factor
of 1800^{3} (about 6 billion). If the density of
Betelgeuse is comparable to that of the Sun, then its mass
is billions of times greater than that of the Sun. On the
other hand, if the mass of Betelgeuse is comparable to that
of the Sun, then its density is less than a billionth the
Sun's density. So which is it? Is Betelgeuse fat, or is it
just fluffy?

Prof. Barbara Ryden (ryden@astronomy.ohio-state.edu)

Updated: 2003 Jan 16

Copyright © 2003, Barbara Ryden