a.
The mutual gravitational force between a sphere of mass M and an external sphere
of mass m (say, the earth and an orange) is radial, attractive, and has magnitude
GMm/r2, where r is the distance between the centers of the two spheres
and G = 6.67 × 10-8 cm3/g sec2.
The radial mass distribution within each sphere is arbitrary: for instance, M could be
the mass of a spherical shell.
b.
On the other hand, a spherical mass distribution (such as a spherical mass shell) exerts no
gravitational force on a body located inside it.
c.
The radial equation of motion m dr2/dt2 =
- GMm/r2 has the first integral
d.
If the spherical mass M is contained within a radius r such that the escape
velocity at r exceeds the speed of light c, nothing can escape the gravitational
pull of M, which is then called a black hole. Putting v = c in
Eq. (2), we find that the critical radius R for black hole formation (Schwarzschild
radius) is
a.
Observation shows that the visible universe is expanding. On the average, the relation between
distance r and relative speed v for any two galaxies is v = Hr,
where H is Hubble's constant. The inverse of H has the following meaning: if the
same velocities (not the same H) prevailed in the past, then all galaxies were together
at time t = l/H before the present. The observed value of (1/H) is about
18 billion years.
b.
The equation of motion for an exploding homogeneous universe is obtained as follows. Draw an
arbitrary sphere of radius r with an arbitrarily chosen center, and consider a
"marker" galaxy on the surface of this sphere. It is attracted to the center of the sphere by
the mass M that is inside the sphere. (The outside mass gives zero force, see lb.)
The equation of motion, as in 1c, is
c.
At present, the dominant contribution to M comes from nucleons and electrons, which are
indestructible. Thus, M is constant. The first integral of eq. (4) is then given by
eq. (1). The future of the universe (continued expansion or recontraction) depends on
the sign of E, as discussed in lc. Using v = Hr and M =
4prr3/3, where r is
the mass density, we see that E is positive if H2 >
8prG/3. The "visible" mass density
(r = 2·10 -31 g/cm3) is such that
3H 2/8Gpr = 10, approximately, and therefore
E > 0; but the mass density of "invisible" matter (dust, spent stars) could be enough
to reverse the sign of E. We take E = 0 for simplicity.
d.
We are then led to the "escape velocity" condition, eq. (2), which can be written as
dr/dt = (2GM/r)1/2 and integrated to give
e.
Hubble's law, v = Hr, can be rewritten as H = v/r =
dln r /dt. Using eq. (6), H = (2/3)t. The present value of
the "Hubble time", 1/H0 = 18 billion years, corresponds then to a "true age"
t0 = (2/3)H0 = 12 billion years.
a.
If the universe is filled by a homogeneous one-component fluid, no entropy is generated by the
expansion, which is then reversible. New entropy can be generated by processes analogous to the
nucleation of liquid droplets in an expanding gas, or to the precipitation of compounds from a
solution. Currently, entropy is generated mostly by star formation and by the "cooking" of the
heavy elements in stars out of the "primordial" hydrogen and helium. Starlight carries away the
"latent heat" of star formation and evolution. This heat, however, is not sufficient to
compensate for the adiabatic cooling due to the expansion. It seems obvious then, that the
universe was hotter and more homogeneous in the past.
b.
A simple calculation can be done by neglecting stars altogether. We have then a homogeneous
mixture of H and He (27% He by weight) in thermal equilibrium (why in thermal equilibrium?
Because this is the most likely distribution). The adiabatic expansion of this monatomic gas
obeys TV 2/3 = constant, or Tr2 = constant. Combining this
with eq. (6), we obtain T/T0 = (t0/t)4/3, where
T0 is the present temperature of the gas that has not
been swept up to form stars.
a.
What is the present radiation temperature? We know that the entropy density of radiation is
proportional to T 3 and that the entropy of the radiation contained in an
expanding sphere of radius r has not changed since decoupling occurred. Thus the
radiation temperature T is such that T 3r3 is constant,
or T/T0 = r0/r. We can use eq. (6) to compute the expansion
ratio r0/rd, where rd was the radius at the
time of radiation decoupling (td = 5·105 y). We can do so
because the matter density has been dominant from the time of radiation decoupling to the
present (t0 = 12 Gy). We obtain r0/rd =
(t0/td)2/3 = 1000, and therefore T0 =
10-3·3000 K = 3 K, in agreement with observation.
b.
It turns out, coincidentally, that the decoupling time td is close to the
time when the radiation energy equaled the rest mass-energy of matter. Therefore for
t < td we can treat radiation as the dominant source of
gravitation in the universe. We can still use eq. (2), but now M is given by
c2 times the radiation energy density, times the volume:
Fig. 10.17 Eras in the evolution of the universe. (After Harrison, Annual Review of Astronomy and Astrophysics. 1973.)