Mechanical point

Escape velocity is defined as the minimal velocity (in the « astrocentric » reference frame) to give to a material point of mass \(m\) for it to escape the gravitational pull of the star.

In this reference frame (which we suppose to be inertial), the mechanical energy of a material point is (if we suppose only the conservative gravitational force of the star is applied to it) :

\(E_m=\frac{1}{2}mv^2-\frac{GmM}{r}\)

On the surface of the star, it is : (\(R\) and \(M\) are the radius and the mass of the star)

\(E_m=\frac{1}{2}mv_{lib}^2-GmM\frac{1}{R}\)

When the point escapes the gravitational pull of the star, its potential energy and its kinetic energy are equal to zero. Its mechanical energy is also zero, thus :

\(E_m=\frac{1}{2}mv_{lib}^2-GmM\frac{1}{R}=0\)

So :

\(v_{lib}=\sqrt{\frac{2GM}{R}}\)

We write the masses as :

\(M = (4/3)\pi {R^3}{\rho _T}\) and \({M_T} = (4/3)\pi R_T^3{\rho _T}\), \(M = {(R/{R_T})^3}{M_T}\)

Thus :

\({v_{lib}} = \sqrt {\frac{{2G{M_T}}}{R}{{\left( {\frac{R}{{{R_T}}}} \right)}^3}}\)

Numerical application :

\(R = 250\,{R_S} = {1,75.10^{11}}m\) and \({v_{lib}} = {3,06.10^8}\;m.{s^{ - 1}} > c\).

Laplace concluded that such a star cannot let light escape its surface, thus making it invisible.

In general relativity, such an « object » is nowadays called a « black hole ».

• Escape velocity is \( c\) if the radius is \(R_C\) :

  \(c = \sqrt {2GM/{R_c}}\) so \({R_c} = 2GM/{c^2}\)

  Numerically :

  \(R_C=3\;km\) for the Sun and \(R_C=8,9\;mm\) for the Earth.

• The gravitational pull exerted by the black-hole on the point P is :

  \({F_{TN}} = GmM/{r^2}\)

  Or :

  \(GM = {c^2}{R_c}/2\)

  Thus :

  \({F_{TN}} = m{c^2}{R_c}/2{r^2}\)

  Numerically :

  \(F_{TN}=29,4\;N\)

  This is the weight of an object placed on the surface of the Earth, since the Earth's radius is equal to the distance between P and the center of the black hole.

• The gravitational field on the surface is :

  \({g_{TN}} = GM/R_c^2\)

  So :

  \({g_{TN}} = {c^2}/2{R_c} = {5,1.10^{18}}\;m.{s^{ - 2}}\)

  It is huge : it is the field created by an object which has the same mass as the Earth, but contained in the volume of a small marble !

  The interior of the Schwarzschild sphere (radius \(R_C\)) is the critical volume from which light cannot escape.

  Light received by an exterior observer is at best emitted by the surface of the sphere, hence the name « event horizon ».

The reference frame is the one of the massive body, centered in O (same as the barycentric reference frame).

The second law of motion for a particle A in the ring of matter, having a circular motion of radius \(r_0\) and velocity \(v\) around the black hole (see figure) is, when projected along OA :

\(m\frac{{{v^2}}}{{{r_0}}} = \frac{{Gm{M_{OM}}}}{{r_0^2}}\)

So :

\({M_{OM}} = \frac{{{r_0}{v^2}}}{G}\)

Numerically, with \({r_0} = 7,5\,ly\; = \;{7,1.10^{16}}\;m\), \({M_{OM}} = {4,26.10^{37}}\;kg = {2,13.10^7}\;{M_S}\), more than \(20\) millions times the mass of the Sun !

The Schwarzschild radius of the black hole is : \({R_c} = 2GM/{c^2} = {6,3.10^{10}}\;m\), which is \(63\) millions of kilometers, or \(42\)% of the distance between the Earth and the Sun.

Such gigantic black holes have a mass of millions of solar mass and they are very dense.

They could be at the center of many galaxies.