Astronomy 162: Professor Barbara Ryden

# HOW BIG IS A STAR?

``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

## Key Concepts

• 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.

#### (1) The radius of very few stars can be found from their angular size and distance.

The Sun's angular size is 1920 arcseconds (about half a degree, or the size of your little fingernail held at arm's length). Most stars are so far away from the Sun that their angular size is not resolved, even in large telescopes. They are indistinguishable from points of light. Among the few stars that have been resolved is Betelgeuse. The Hubble Space Telescope made a direct image of the star Betelgeuse. The image has an angular size of only 0.125 arcseconds.

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)

#### Astronomical applications

The Sun:
• a = 1920 arcseconds
• d = 1 AU = 150,000,000 kilometers
• We compute R = 700,000 kilometers
Betelgeuse:
• a = 0.125 arcseconds
• d = 427 light years = 4 thousand trillion kilometers
• We compute R = 1.2 BILLION kilometers
If the star Betelgeuse were superimposed on the Solar System, it would stretch nearly as far as the orbit of Saturn.

#### (2) The radius of stars can be deduced from their luminosity and temperature.

A star is approximately a blackbody. The luminosity of a blackbody depends very strongly on temperature. Consider a square meter of the surface of a blackbody. The rate at which it radiates light is given by the following equation: 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 Lsun
• T = 10,000 Kelvin = 1.72 Tsun
• R is computed to be 1.7 Rsun (Work this out for yourself!)

Sirius B:

• L = 0.0024 Lsun
• T = 15,000 Kelvin = 2.59 Tsun
• R is computed to be 0.007 Rsun (Work this out for yourself!)
One square meter of the companion of Sirius is 2.59 x 2.59 x 2.59 x 2.59 = 45 times more luminous than one square meter of the Sun. The reason why the companion to Sirius has such a low total luminosity is that it's tiny - even smaller than the Earth.

#### (3) Stars have a wide range of radii.

• Betelgeuse = 1800 Rsun
• Sirius = 1.7 Rsun
• Sun = 1 Rsun , by definition
• Companion of Sirius = 0.007 Rsun
The largest known star has a radius 3700 times that of the Sun. On the other hand, the objects known as `neutron stars' have radii of only 10 kilometers, although they are even more massive than the 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 18003 (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