Hubble's Law and the Distance Scale

The greatest leaps in our understanding of our place in the universe come from new methods of determining distances.  Examples

• Copernicus and Kepler -- scale and dynamics of the solar system
• Bessel -- parallax of stars, and placement of stars at vast distances
• Hubble -- use of Cepheid variables to determine distance to galaxies
• Hubble's Law

• Henrietta Leavitt discovered a relationship in apparent magnitude and period for Cepheids in the Small Magellenic Cloud (SMC).  Since they are all at about the same distance, she recognized that this was a period-luminosity relation.
• Harlow Shapley set about calibrating the scale and applied it to Cepheids in the Milky Way, determining the size and scale of our galaxy.  (Unfortunately, he calibrated using Pop II Cepheids in globular clusters, and so made an error.)
• Edwin Hubble determined the distance to the Andromeda galaxy in 1924, thus proving that these "spiral nebulae" were external galaxies.


Immediately, the scale of the universe was clear, since spiral nebulae can be seen down to limits of detectability.

In 1912, even before Hubble showed that galaxies are external island universes, Slipher observed the red-shifted spectral lines from galaxies.  From this it had already been argued that spiral nebulae were external galaxies based on their unusual velocities.

In 1929, Hubble published his paper announcing what is now called Hubble's Law.  Recall that for v << c, redshift is given by

(Dl/l_o) = v/c

In astrophysics, we use z = (Dl/l_o) as the redshift, so velocities are related
to redshift simply by
 
v = cz.


Hubble found a linear relationship between distance and redshift
 

v = cz = H_od


where H_o is a constant of proportionality, now called the Hubble Constant.  Hubble's finding implies that the more distant a galaxy, the larger the recession velocity.  This leads directly to the expansion of the universe, which we will discuss in some detail.

Before this discovery, the prevailing view was that the universe was static (in a steady state).  This led Einstein, just a few years before (1915) to choose a value for an integration constant in his General Theory of Relativity that gave a static universe.  He called this the biggest blunder of his life after Hubble's announcement, but his second biggest blunder was to set it to zero, thus giving a uniform expansion.  We now suspect that it has some other value, such that the universe is accelerating its expansion!

Note that since z = v/c, one might expect that the greatest value for the redshift is unity (which would occur at v = c).  However, this expression is only valid for v << c.  Recall that the relativistic version of the Doppler Shift formula is
 

l = l_o [(1 + v_r /c)/ (1 - v_r /c)]^1/2


and the corresponding relativistic form of Hubble's Law is
 

d ~ (c/H_o) [(z + 1)² - 1]/[(z + 1)² + 1].

Hubble could measure the constant H, but it was not very accurate because it was based on those galaxies within reach of the Cepheid distance-scale.  One of the key projects of the Hubble Space Telescope was to pin down the value of  with much better than the factor of two accuracy it has had for so long.  The final results of this key project are in, and the announced value is

H_o ~ 72 +/- 8 km s^-1 Mpc^-1.

Here is the abstract of the journal article:

The Astrophysical Journal, 553:47-72, 2001 May 20
© 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

Final Results from theHubbleSpace Telescope KeyProject to Measure the Hubble Constant

Wendy L. Freedman, Barry F. Madore, Brad K. Gibson, Laura Ferrarese, Daniel D. Kelson, Shoko Sakai, Jeremy R. Mould, Robert C. Kennicutt, Jr., Holland C. Ford,  John A. Graham,  John P. Huchra, Shaun M. G. Hughes, Garth D. Illingworth, Lucas M. Macri, and Peter B. Stetson

Received 2000 July 30; accepted 2000 December 19

ABSTRACT

We present here the final results of the Hubble Space Telescope (HST) Key Project to measure the Hubble constant. We summarize our method, the results, and the uncertainties, tabulate our revised distances, and give the implications of these results for cosmology. Our results are based on a Cepheid calibration ofseveral secondary distance methods applied over the rangeof about 60400 Mpc.The analysis presented here benefits from a numberof recent improvements and refinements, including (1) a larger LMC Cepheid sample to define the fiducialperiod-luminosity (PL) relations, (2) a more recent HSTWide Field and PlanetaryCamera 2 (WFPC2) photometriccalibration, (3) a correction for Cepheid metallicity, and (4) a correction for incompleteness bias in theobserved Cepheid PL samples. We adopt a distancemodulus to the LMC (relative to which the more distant galaxies are measured) of m₀(LMC)= 18.50 ± 0.10 mag, or 50 kpc. New, revised distances are given for the 18 spiral galaxies for which Cepheids have been discovered as part of theKey Project, as well as for 13 additionalgalaxies with published Cepheid data. The new calibrationresults in a Cepheiddistance to NGC 4258 in better agreement with the maser distance to this galaxy. Based on these revised Cepheid distances, we find values (in km s^-1 Mpc^-1) ofH₀ = 71 ± 2 (random) ± 6 (systematic) (Type Ia supernovae), H₀ = 71 ± 3 ± 7 (Tully-Fisher relation), H₀ = 70 ± 5 ± 6 (surface brightness fluctuations), H₀ = 72 ± 9 ± 7 (Type II supernovae), and H₀ = 82 ± 6 ± 9 (fundamental plane). Wecombine these results forthe different methods withthree different weighting schemes,and find good agreementand consistency with H₀= 72 ± 8 km s^-1 Mpc^-1. Finally,we compare these results with other, global methodsfor measuring H₀.

You can see that there is still some uncertainty, since the different techniques do not agree exactly, but the disagreement is much smaller than previously.  The uncertainty is due to the gaps in the distance scale,

• between distances determined by trigonometric parallax and Cepheids,
• and between Cepheid distances and the Redshift scale.

These gaps have been largely closed by Hipparchos (refining trigonometric parallaxes, including a sample of Cepheids), and by the Hubble Space Telescope (HST) (extending the distance that Cepheids can be observed).  Note the units, relating recession velocity in km/s with distance in Mpc.  Many astronomers use a parametrized form of equations with H_o as

H_o = 100 h km s^-1 Mpc^-1,

with h being 0.72 to give the Hubble result.

The observed velocities of galaxies are due to both peculiar velocities, and to the recession velocity due to expansion of space (recessional motion is called the Hubble flow).  Since the latter grows with distance, the motions of nearby galaxies are dominated by their peculiar velocities.

Example: The quasar PC 1247+3406 is moving away from us at over 94% of the speed of light.  What is its redshift?  What is its distance (parametrized with h)?

z = (Dl/l_o) = (l-l_o)/l_o = {[(1 + v/c)/ (1 - v /c)]^1/2 - 1} = 2.48

d  = (c/100 h) [(z + 1)² - 1]/[(z + 1)² + 1]
    = (3 x 10⁵/100 h) (0.847) = 2.54/h Gpc

The units of H_o, km s^-1 Mpc^-1, scale as L/T / L = 1/T.  What time is represented by H_o^-1?  It is the time taken for the galaxies to reach their current distance from us at their current recession velocity--that is, the Age of the Universe  or the Hubble Time.

• For H_o = 50 km s^-1 Mpc^-1 , H_o^-1= 1 Mpc / 50 km/s

                                                    = 10⁶ (3 x 10¹³ km) / 50 km/s
                                                    = 6 x 10¹⁷ s = 20 billion years
• For H_o = 100 km s^-1 Mpc^-1 , H_o^-1 = 10 billion years

so an accurate value for H_o is rather important!  The Hubble value for H_o gives an age of

H_o^-1 = 13.9 billion years.                                                  (Age of universe)

There is an alternative way to determine the age of the universe, which is to date globular clusters by looking at their H-R diagrams.  The age of globular clusters tends to be just over 14 by, in good agreement with the established value of H_o.  An accurate value for H_o, as we have seen, gives us confidence that we know the age and size of the universe.  The size of the observable universe, by definition, is the size given by assuming we can see to infinite redshift (where the recession velocity reaches the speed of light).  At infinite redshift, the factor involving z becomes unity and we have:

d  = (c/100 h) [(z + 1)² - 1]/[(z + 1)² + 1]
    = (3 x 10⁵/100 h) = 3.00/h Gpc = 4.17 Gpc.                    (Size of visible universe)

Note that if the expansion universe were to slow down with time, new galaxies outside our currently visible universe would become visible.  If it were to speed up, galaxies currently in our visible universe would leave it!  We will discuss such cosmological questions later.