The escape speed from a star

Think first about the properties of our own sun. Its mass M = 1.99 * 10³⁰ kg and radius R = 6.96 * 10⁸ m are much larger than those of any planet, but compared to other stars, our sun is not exceptionally massive. You can find the sun’s average density r in the same way we found the average density of the earth in Section 13.2:

The sun’s temperatures range from 5800 K (about 5500°C or 10,000°F) at the surface up to 1.5 * 10⁷ K (about 2.7 * 10⁷ °F) in the interior, so it surely contains no solids or liquids. Yet gravitational attraction pulls the sun’s gas atoms together until the sun is, on average, 41% denser than water and about 1200 times as dense as the air we breathe

Now think about the escape speed for a body at the surface of the sun. In Example 13.5 (Section 13.3) we found that the escape speed from the surface of a spherical mass M with radius R is v = (2)^1/2GM/R. Substituting M = rV = r( 4/3 pR³) into the expression for escape speed gives

Using either form of this equation, you can show that the escape speed for a body at the surface of our sun is v = 6.18 * 10⁵ m/s (about 2.2 million km/h, or 1.4 million mi/h). This value, roughly 1 500 the speed of light in vacuum, is independent of the mass of the escaping body; it depends on only the mass and radius (or average density and radius) of the sun.

Now consider various stars with the same average density r and different radii R. Equation (13.29) shows that for a given value of density r, the escape speed v is directly proportional to R. In 1783 the Rev. John Mitchell, an amateur astronomer, noted that if a body with the same average density as the sun had about 500 times the radius of the sun, its escape speed would be greater than the speed of light in vacuum, c. With his statement that “all light emitted from such a body would be made to return toward it,” Mitchell became the first person to suggest the existence of what we now call a black hole.

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