Free fall time

This article is about free fall time in astronomy. For the meaning of skydiving, see Skydiving # General .

In astronomy, the freefall time describes the time it takes an extended gas cloud or a star to collapse on a point under the action of its own gravity if all forces except gravity are disregarded. Based on its assumptions, the model is more suitable for describing the formation of a star from a gas cloud (see “First collapse” in Star Formation) than the formation of a black hole from a star (see gravitational collapse ). ${\displaystyle t_{\mathrm {ff} }}$

In particular, the model neglects

• that the collapse releases binding energy that leads to radiation pressure ,
• that, due to the Pauli principle, the atoms of the star cannot come as close as desired, which leads to degeneracy pressure ,
• and that the approximation of the classical description of gravity breaks down at high matter densities and a description by the general theory of relativity is required.

If these aspects are neglected, the free fall time according to a classic calculation for a spherical, homogeneous gas cloud without internal energy results :

${\displaystyle t_{\mathrm {ff} }={\sqrt {\frac {\pi ^{2}R^{3}}{8GM}}}={\sqrt {\frac {3\pi }{32G\rho }}},}$

Are there

• ${\displaystyle G}$ the gravitational constant,
• ${\displaystyle R}$ the radius of the star,
• ${\displaystyle M}$ its mass and
• ${\displaystyle \rho }$ its density.

Derivation

For an infinitesimal mass element that is located in the gas cloud, Newton's law of gravitation applies in conjunction with Newton's shell theorem

${\displaystyle {\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}=-G{\frac {M_{<}}{r^{2}}}=-{\frac {4\pi }{3}}G\rho {\frac {r_{0}^{3}}{r^{2}}}}$,

with the mass that is located within a sphere with a radius smaller than the original distance of the mass element to the center of the gas cloud . ${\displaystyle M_{<}}$ ${\displaystyle r_{0}}$

With the initial determination that the mass element is at rest at the beginning, follows through one-time integration

${\displaystyle {\frac {\mathrm {d} r}{\mathrm {d} t}}=-{\sqrt {{\frac {8\pi }{3}}G\rho r_{0}^{2}\left({\frac {r_{0}}{r}}-1\right)}}}$

and through twice

${\displaystyle t_{\mathrm {ff} }={\sqrt {{\frac {3}{8\pi }}{\frac {1}{G\rho r_{0}^{2}}}}}\int _{0}^{r_{0}}\mathrm {d} r\,{\sqrt {\frac {r}{r_{0}-r}}}={\sqrt {{\frac {3\pi }{32}}{\frac {1}{G\rho }}}}}$

In particular, the free fall time of each mass element only depends on the original density of the gas cloud. If the mass distribution of the cloud is homogeneous, the free fall time of a mass element is independent of its original position: All atoms of a gas cloud arrive in the center at the same time.

Taking into account the fact that a gas cloud does not completely collapse to a point before the stellar nuclear fusion ignites, the lower limit in the last integral can be set from zero to a final, finite radius . It is still approximately valid if is. ${\displaystyle r_{f}}$ ${\displaystyle r_{f}\ll r_{0}}$

literature

• Rudolf Kippenhahn and Alfred Weigert: Stellar Structure and Evolution . 1st edition. Springer, 1990, ISBN 978-3-642-61523-8 , pp. 256 f . (English).