Physics 321
Astrophysics II:  Lecture #21
Prof. Dale E. Gary

Hubble's Law and the Distance Scale

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
Period-Luminosity Relation for Cepheids Hubble's Law
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/lo) = v/c
Redshift, Distance, and the Age of the Universe
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
Ho ~ 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


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 60–400 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,

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 Ho as
Ho = 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/lo) = (l-lo)/lo = {[(1 + v/c)/ (1 - v /c)]1/2 - 1} = 2.48

= (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 Ho, km s-1 Mpc-1, scale as L/T / L = 1/T.  What time is represented by Ho-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.

so an accurate value for Ho is rather important!  The Hubble value for Ho gives an age of
Ho-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 Ho.  An accurate value for Ho, 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:
= (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.