Stars form by the gravitational collapse of interstellar gas clouds. Use dimensi
ID: 801697 • Letter: S
Question
Stars form by the gravitational collapse of interstellar gas clouds. Use dimensional analysis to find an equation that you would use to estimate the collapse timescale for a cloud with a density rho = 1 M pc^-3. How does the collapse time that you estimated compare to the lifetime of a star like the Sun? Similarly, gravity is trying to pull all of the mass in the Universe closer together. This gravitational pull has been trying to overcome the expansion of the Universe ever since the Big Bang, nearly 14 billion years ago. Use dimensional analysis to estimate the average density of the Universe. About how many hydrogen atoms are there in 1 m^3 of "empty" space? What is the density in units of M/kpc^3? (You may assume that the Universe is made up entirely of hydrogen, and that it has reached its maximum extent.)Explanation / Answer
a. Jeans instablity is the phenomenon which the causes the gas clouds to collapse and hence lead to the formation of stars. It occurs when the internal pressure in the gas clouds is not strong enough to prevent the collapsing of clouds due to the weight of the matter present in the cloud. The spherical cloud must be in hydrostatic equilibrium to remain stable.
dp/dr=-Gp(r)Menc(r)/r2
where p= pressure
Menc= enclosed mass within the cloud
p(r) density of gas at radius r
G= gravitational constant
r= radius of the cloud
The gas cloud will be unstable if the it is very massive at a particular temperature or either it is too cold to at the particular mass so that it would overcome the gas pressure.
The mass above which a cloud will collapse is known as jean mass and this mass depends on temperature and density of the cloud and is about tens to thousands of solar masses.
Equation for jean mass
Mj=(5Kt/GM)3/2*(3/4pi* P)1/2
if Mcloud>Mj then collapse
Mj=1500 Msun for diffuse HI cloud
b. Since the universe is expanding and the gravitational overcomes this expansion and it can collapse eventuallly.Critical density of universe is the density of matter present in the universe which is sufficient enough to halt its expansion.
Critical density given by: P critical=3 H^2/*PiG
where H= hubble constant=65 km/s/Mpc=(65*1000)/3.1*1022
=2.1*10-18/sec
G= gravitational constant=6.67*10-11Nm2/kg2
Now P critical=3*(2.1*10-18)^2/8*3.14*6.67*10-11
=3*4.41*10-36/167.55*10-11
=13.23*10-36/167.55*10-11
=0.0789*10-25
=7.8*10-27kg/m3 approximately equal to 10-26
which is about mass of 5 hydogen atoms in 1 m3 of universe
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