22 From Newton to Hubble
(iii)훺 0 >1, the mass density is overcritical and the Universe is closed. As the scale
factor푎(푡)increases, it reaches a maximum value푎midwhen the expression in
Equation (1.37) vanishes, and where the rate of increase,푎̇mid,alsovanishes.
But the condition (1.37) must stay true, and therefore the expansion must turn
into contraction at푎mid. The solid line in Figure 1.2 describes this case for the
choice훺 0 = 1 .5, whence푎mid=3. For later times the Universe retraces the solid
curve, ultimately reaching scale푎=1 again.This is as far as we can go combining Newtonian mechanics with Hubble’s law.
We have seen that problems appear when the recession velocities exceed the speed
of light, conflicting with special relativity. Another problem is that Newton’s law of
gravitation knows no delays: the gravitational potential is felt instantaneously over all
distances. A third problem with Newtonian mechanics is that the Copernican world,
which is assumed to be homogeneous and isotropic, extends up to a finite distance푟 0 ,
but outside that boundary there is nothing. Then the boundary region is characterized
by violent inhomogeneity and anisotropy, which are not taken into account. To cope
with these problems we must begin to construct a fully relativistic cosmology.
Problems
- How many revolutions has the Galaxy made since the formation of the Solar
System if we take the solar velocity around the galactic center to be 365kms−^1? - Use Equation (1.4) to estimate the mean free path퓁of photons. What fraction
of all photons emitted by stars up to the maximum observed redshift푧=7 arrive
at Earth? - If Hubble had been right that the expansion is given by
퐻 0 =550 km s−^1 Mpc−^1 ,
how old would the Universe be then [see Equation (1.13)]?- What is the present ratio퐾 0 =^235 U∕^238 U on a star 10Gyr old?
- Prove Newton’s theorem that the gravitational force at a radial distance푅from
the center of a spherical distribution of matter acts as if all the mass inside푅
were concentrated at a single point at the center. Show also that if the spherical
distribution of matter extends beyond푅, the force due to the mass outside푅
vanishes. - Estimate the escape velocity from the Galaxy.
References
[1] Ramella, M., Geller, M. J., Pisani, A. and da Costa, L. N. 2002Astron. J. 123 , 2976.
[2] Fang Li Zhi and Li Shu Xian 1989Creation of the Universe. World Scientific, Singapore.
[3] Peebles, P. J. E. 1993Principles of physical cosmology. Princeton University Press,
Princeton, NJ.
[4] Harrison, E. 1987Darkness at night. Harvard University Press, Cambridge, MA.