104 2x10"

Velocity (km/s)

104 2x10"

Velocity (km/s)

Hubble's law.The distance is plotted on the vertical axis, and the radial velocity (from the redshift) is plotted on the horizontal axis.This is the result of the HST Key Project to study Hubble's law, and data using different distance indicators are shown with different colored symbols. The labelled lines show lines corresponding to different values of H. [John Huchra, CFA]

Suppose the rate of expansion has been constant over the age of the universe. If all objects started very close together at t = 0 (whatever that time means), then the current distance between any two objects would be d= vt0

where t0 is the current age of the universe. Solving for v gives v = (1/to)d

At first it might seem unusual that we are in some special part of the universe, so that all things are moving away from us in a very particular way. However, we interpret Hubble's law as telling us that all galaxies are moving away from each other. This motion represents the overall expansion of the universe. (We will discuss this in more detail in Chapter 20.) To visualize this, we can imagine the galaxies as dots on the surface of a balloon, as shown in Fig. 18.10. As the balloon is blown up, all the dots move away from all the other dots. In Fig. 18.11, we see that the separations between any pair of galaxies increases, and that the larger separations increase faster. This means that we could observe from any of the galaxies, and we would still obtain Hubble's law.

Suppose that in some time At, the balloon expands so that all distances are multiplied by a factor (1 + f). If two objects were initially a distance d apart, their distance at the end of the interval is (1 + f)d. The change in the distance between the two objects is fd, so the average relative velocity of the two objects is fd/At. This is the same form as Hubble's law.

Fig 18.10.

The universe as an expanding balloon.The galaxies are painted on the surface of the balloon.As the balloon expands, each galaxy moves away from every other galaxy.This is a two-dimensional analogy to help us with the visualization.

The universe as an expanding balloon.The galaxies are painted on the surface of the balloon.As the balloon expands, each galaxy moves away from every other galaxy.This is a two-dimensional analogy to help us with the visualization.

Separations at Earlier Time

Separations at Later Time

Change in Separation

The effect of all galaxies moving away from each other.The two frames show the positions of the galaxies at different times, with the bottom frame being later. In the top frame the green arrows show the separations at the earlier time. In the bottom frame, the blue arrows show the separations at the later time.The red arrows show the change in separation between the two times for each pair of galaxies (the difference between the blue and green arrows). You can see that the galaxies that were initially farther apart have the greatest change in separation, and the galaxies that were closest have the least change in separation.

This is the same as Hubble's law, if we make the identification

Then, 1/H0, called the Hubble time, is the age of the universe if the expansion has been constant. Actually, as we will discuss in Chapter 20, the expansion is not constant. If the expansion is slowing down, the actual age of the universe is less than the Hubble time. If the expansion is speeding up, the actual age of the universe is greater than the Hubble time.

The value that Hubble obtained for H0 was 500 km/s/Mpc. Note that the units of the Hubble constant may seem strange, but there is a distance in the numerator (km) and in the denominator (Mpc), so the units really work out to 1/time. We use km/s/Mpc because with it we can express both distance and velocity in convenient units. A value of 500 km/s/Mpc works out to a Hubble time of 2 X 109 yr (2 Gyr; see Problem 18.6). This was a cause for concern, since our understanding of stellar evolution and the HR diagrams of globular clusters in our galaxy tell us that these clusters are about 10 Gyr old. (In fact, radioactive dating places the age of the Solar System at over 4 Gyr.) It is disconcerting to have the universe younger than things in it. However, there is an immediate error in Hubble's value due to confusion between type I and type II Cepheids as distance indicators. Over the years, other refinements have been made. As we will see below, the currently accepted value for the Hubble constant is in the range 50-100 km/s/Mpc.

Apart from telling us something interesting about the universe, Hubble's law is also of great value when determining distances to distant objects. It is important that this only be used for objects that are far enough away that their velocities relative to us are dominated by the expansion of the universe, as shown in Fig. 18.12. We say that objects must be far enough away to be in the Hubble flow. Objects within our own cluster of galaxies are not in the Hubble flow. Their motions are dominated by the dynamics of our cluster. Even nearby clusters have random velocities relative to us that are a significant fraction of their Hubble law velocity.

Example 18.3 Hubble's law and distance

For some cluster, we measure v = 103 km/s. What is the distance? Take H0 = 65 km/s/Mpc.

solution

We find the distance from d = v/H0

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