Physics 321 

Prof. Dale E. Gary
NJIT 
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. ExamplesPeriodLuminosity Relation for Cepheids
 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
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 redshifted 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
Hubble found a linear
relationship between distance and redshift
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
and the corresponding
relativistic form of Hubble's Law is
Hubble could measure the constant H, but it was not very accurate because it was based on those galaxies within reach of the Cepheid distancescale. 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 isH_{o} ~ 72 +/ 8 km s^{1} Mpc^{1}.Here is the abstract of the journal article:The Astrophysical Journal, 553:4772, 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 fiducialperiodluminosity (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 m_{0}(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}) ofH_{0} = 71 ±^{ }2 (random) ± 6^{ }(systematic) (Type Ia supernovae),^{ }H_{0} = 71 ±^{ }3 ± 7 (TullyFisher^{ }relation), H_{0} = 70^{ }± 5 ± 6^{ }(surface brightness fluctuations), H_{0}^{ }= 72 ± 9^{ }± 7 (Type II^{ }supernovae), and H_{0} =^{ }82 ± 6 ±^{ }9 (fundamental plane). Wecombine these results forthe different methods withthree different weighting schemes,and find good agreementand consistency with H_{0}= 72 ± 8^{ }km s^{1} Mpc^{1}. Finally,we compare these results^{ }with other, global methodsfor measuring H_{0}.
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 H_{o} as
 between distances determined by trigonometric parallax and Cepheids,
 and between Cepheid distances and the Redshift scale.
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}) = (ll_{o})/l_{o} = {[(1 + v/c)/ (1  v /c)]^{1/2 } 1} = 2.48d = (c/100 h) [(z + 1)^{2}  1]/[(z + 1)^{2} + 1]
= (3 x 10^{5}/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 velocitythat is, the Age of the Universe or the Hubble Time.so an accurate value for H_{o} is rather important! The Hubble value for H_{o} gives an age of
 For H_{o} = 50 km s^{1} Mpc^{1} , H_{o}^{1}= 1 Mpc / 50 km/s
= 10^{6} (3 x 10^{13} km) / 50 km/s
= 6 x 10^{17} s = 20 billion years For H_{o} = 100 km s^{1} Mpc^{1} , H_{o}^{1} = 10 billion years
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 HR 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]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.
= (3 x 10^{5}/100 h) = 3.00/h Gpc = 4.17 Gpc. (Size of visible universe)