Hubble's Law

Hubble's law is the statement in astronomy that galaxies move away from each other, and that the velocity with which they recede is proportional to their distance. It leads to the picture of an expanding universe and, by extrapolating back in time, to the Big Bang theory.

The law was first formulated by Edwin Hubble in 1929. Hubble compared the distances to nearby galaxies to their redshift, found a linear relationship, and interpreted the redshift as caused by the receding velocity. His estimate of the proportionality constant, now known as Hubble's constant, was however off by a factor of about 10. Furthermore, if one takes Hubble's original observations and then use the most accurate distances and velocities currently known, one ends up with a random scatter plot with no discernable relationship between redshift and velocity. Nevertheless the relationship was confirmed by observations after Hubble.

The law can be stated as follows:

v = H₀ D

where v is the receding velocity of a galaxy due to the expansion of the universe (typically measured in km/sec), H₀ is Hubble's constant, and D is the current distance to the galaxy (measured in mega parsec Mpc).

One can derive Hubble's law mathematically if one assumes that the universe expands (or shrinks) and that the universe is homogeneous, meaning that all points within it are equal.

For most of the second half of the 20th century the value of H₀ was estimated to be between 50 and 90 km/sec/Mpc. The value of the Hubble constant was the topic of a long and rather bitter controversy between Gérard de Vaucouleurs who claimed the value was 100 and Alan Sandage who claimed the value was 50. The Hubble Key Project significantly improved the determination of the value and in May 2001 published its final estimate of 72+/-8  km/sec/Mpc. In 2003 the satellite WMAP further improved that determination to 71+/-4, using a completely independent method, based in the measurement of anysotropies in the cosmic microwave background radiation.

Hubble's constant is "constant" in the sense that it is believed to work for all velocities and distances right now. The value of H (usually called Hubble parameter to distiguish it from its value now, the Hubble constant) decreases over time however. If one assumes that all galaxies retain their speed relative to us and do not accelerate or deccelerate, then we have D = vt and it follows that H = 1/t, where t is the time since the Big Bang. This allows to estimate the age of the universe from H.

Based on recent observations, it is now believed that galaxies accelerate away from us, which means that H > 1/t (but still decreases over time) and the estimate 1/H₀ (between 11 and 20 billion years) for the age of the universe is too low.

There are several additional notes to be made:

• The distance D to nearby galaxies can be estimated for example by comparing their apparent luminosity to their postulated absolute luminosity.

If the galaxies are far away, one has to take as D the distance to the galaxy right now, not when the light from it was emitted. This distance is extremely hard to determine.

• The velocity v is defined to be the time rate of change of D.

For relatively nearby galaxies, the velocity v can be determined from the galaxy's redshift z using the formula v ≈ zc where c is the speed of light. However, only the velocity due to the expansion of the universe should be used: all galaxies move relative to each other independent of the expansion of the universe, and these relative velocities, called peculiar velocities are not accounted for by Hubble's law. For far away galaxies, v cannot easily be determined from the redshift z and can be larger than c.

• Systems that are gravitationally bound, such as galaxies or our planetary system, are not subject to Hubble's law and do not expand.

External links and references:

• The Hubble Key Project: http://www.ipac.caltech.edu/H0kp/H0KeyProj.html
• Freedman et. al.: Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant. The Astrophysical Journal, Volume 553, Issue 1, pp. 47-72. Online at http://adsabs.harvard.edu/cgi-bin/nph-bib_query?2001ApJ...553...47F