A luminous star with the same density as the Earth and a diameter 250 times that of the Sun would not, because of its attraction, allow any of its rays to reach us. It is therefore possible that the largest luminous bodies in the universe are, for that very reason, invisible.

— Pierre-Simon Laplace, Exposition of the System of the World (1796)

These sentences are in the first edition of the Exposition of the System of the World. Laplace wrote them at a time when people believed, like Newton, that light was made of particles. If light is a swarm of corpuscles, gravity must pull on them as it pulls on a stone. And then, if a star is compact enough, the light it emits will fall back, like a stone thrown upward.

Escape velocity

A stone thrown upward with speed $v$ from the surface of a body of mass $M$ and radius $R$ escapes for good only if its energy of motion beats the energy of gravity:

$$\frac{1}{2}mv^2 \ge \frac{GMm}{R} \quad\Longleftrightarrow\quad v \ge v_e = \sqrt{\frac{2GM}{R}}.$$

The stone’s mass cancels: the escape velocity $v_e$ depends only on the body. On Earth it is about 11.2 km/s; on the Sun, 618 km/s.

Density and size

For a body of density $\rho$, the mass is $M = \frac{4}{3}\pi R^3 \rho$. Substituting,

$$v_e = \sqrt{\frac{2G}{R}\cdot\frac{4}{3}\pi R^3\rho} = R\,\sqrt{\frac{8\pi G\rho}{3}}.$$

With the density fixed, the escape velocity grows in proportion to the radius. That is exactly Laplace’s reasoning. Starting from the Sun, with $v_e = 618$ km/s and mean density 1.41 g/cm³, he imagined a body 250 times larger and with the density of the Earth, 5.51 g/cm³:

$$v_e \approx 618 \times 250 \times \sqrt{\frac{5.51}{1.41}} \approx 618 \times 250 \times 1.98 \approx 305\,000 \text{ km/s},$$

a little more than the speed of light, $c \approx 300\,000$ km/s. The light of such a body would not escape.

The critical radius

One can also fix the mass and ask how much the body must shrink. Light is trapped when $v_e = c$:

$$\sqrt{\frac{2GM}{R}} = c \quad\Longleftrightarrow\quad R = \frac{2GM}{c^2}.$$

For the Sun, with $G = 6.67 \times 10^{-11}$, $M = 1.99 \times 10^{30}$ kg and $c = 3.0 \times 10^8$ m/s:

$$R = \frac{2 \times 6.67 \times 10^{-11} \times 1.99 \times 10^{30}}{(3.0 \times 10^8)^2} \approx 2\,950 \text{ m}.$$

The whole Sun squeezed into a ball 3 km in radius.

Exercise: the Earth as a black hole

The mass of the Earth is $5.97 \times 10^{24}$ kg. What would its critical radius $R = \frac{2GM}{c^2}$ be?

$R = \frac{2 \times 6.67 \times 10^{-11} \times 5.97 \times 10^{24}}{9.0 \times 10^{16}} \approx 8.8 \times 10^{-3}$ m. The whole Earth would fit in a ball 9 millimetres in radius, the size of a marble.

Exercise: proportionality

Use the formula $v_e = R\sqrt{\frac{8\pi G\rho}{3}}$ to answer: how many times the Sun’s radius would a body with the Sun’s density need to have in order to trap light?

With the same density, $v_e$ is proportional to $R$. We need to multiply 618 km/s until we reach 300 000 km/s: $\frac{300\,000}{618} \approx 485$. A body with the Sun’s density and about 485 times its radius would be dark. Laplace got 250 because he used a density four times larger, the Earth’s, and $\frac{485}{\sqrt{3.9}} \approx 246$.

Sources: Pierre-Simon Laplace, Exposition du système du monde (1796), book V, ch. VI · Pierre-Simon Laplace, “Beweis des Satzes, dass die anziehende Kraft bey einem Weltkörper so groß seyn könne, dass das Licht davon nicht ausströmen kann”, Allgemeine geographische Ephemeriden 4 (1799) · John Michell, “On the Means of Discovering the Distance, Magnitude, &c. of the Fixed Stars”, Philosophical Transactions 74 (1784) · Kip Thorne, Black Holes and Time Warps (1994).

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