To find the value of the critical density (^) cwe begin the same way we would to find
the escape velocity from the earth. The gravitational potential energy Uof a spacecraft
of mass mon the surface of the earth, whose mass is Mand radius is R, is U
GmMR. (A negative potential energy corresponds to an attractive force.) To escape
permanently from the earth, the spacecraft must have a minimum kinetic energy ^12 m 2
such that its total energy Eis 0:
E KEU m 2
0 (13.9)
This gives 2 GMR 11.2 km/s for the escape velocity.
Now we consider a spherical volume of the universe of radius Rwhose center is the
earth. Only the mass inside this volume affects the motion of a galaxy on the surface
of the sphere provided the distribution of matter in the universe is uniform, which it
seems to be on a large enough scale. If the density of matter inside this volume is ,
the volume contains a total mass of M ^43 R^3. According to Hubble’s law (Sec. 1.3),
the outward velocity of a galaxy Rfrom the earth due to the expansion of the uni-
verse is proportional to R. Hence HR, where His Hubble’s parameter.Calling the
galaxy’s mass m, if it has just enough speed never to return, we have from Eq. (13.9)
m 2
m(HR)^2 R^3 c
Critical density (^) c (13.10)
3 H^2
8 G
4
3
Gm
R
1
2
GmM
R
1
2
GmM
R
1
2
502 Chapter Thirteen
Open, < (^) c
Flat, = (^) c
Closed, > (^) c
Figure 13.14Two-dimensional analogies of the geometry of space in open, flat, and closed universes.
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