Steps to the Hubble Constant

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This last section is a short essay that summarizes the ways we find distances to various objects in the universe.

Why do we care so much about finding distances in astronomy? If we know the distance to a star, we can determine its luminosity and mass. We can then discover a correlation between luminosity, mass, and temperature for main sequence stars that our physical theories must account for. Finding distances to stellar explosions like planetary nebulae and supernovae enables us to find the power needed to make the gaseous shells visible and how much was needed to eject them at the measured speeds. Stellar distances and distances to other gaseous nebulae are necessary for determining the mass distribution of our galaxy. We then have been able to discover that most of the mass in our Galaxy is not producing light of any kind and is in a dark halo around the visible parts of the Galaxy.

Finding distances to other galaxies enables us to find their mass, luminosity, and star formation history among other things. We're better able to hone in on what is going on in some very active galactic cores and also how much dark matter is distributed among and between galaxy cluster members. From galaxy distances, we're also able to answer some cosmological questions like the large-scale geometry of space, the density of the universe needed to stop the expansion (called W [``Omega'']), age of the universe, and whether or not the universe will be expanding. The cosmological questions will be discussed fully in the next chapter on cosmology. This is only a quick overview of the reasons for distance measurements and is by no means an exhaustive list of reasons why we care about distance measurements.

Now let's take a look at the distance scale ladder. The bottom foundational rung of the ladder is the most accurate and the most certain of all the distance determination methods. As we climb upward, each rung depends on the previous rung and is less certain than the previous one.

Rung 1: The Astronomical Unit

The Earth and Distance to the Sun. We use radar reflections from Venus and its angular separation from the Sun to calculate the numerical value of the Astronomical Unit (AU). We can use radar to measure distances out to 50 AU.

Rung 2: Geometric Methods

On the next rung of the distance scale ladder, we convert trigonometric parallax measurements into distances to the nearby stars using their angular shift throughout the year and the numerical value of the Astronomical Unit. We find distances to nearby clusters like the Hyades or the Pleiades via trigonometric parallax or the moving clusters method (another geometric method). We calibrate the cluster's main sequence in terms of absolute magnitude (luminosity). Geometric methods are used to find distances out to about 100 parsecs (or several hundred parsecs with Hipparcos' data).

Rung 3: Main Sequence Fitting and Spectroscopic Parallax

On the next rung outward we determine the spectral type of star from its spectral lines and measure the apparent brightness of the star. The calibrated color-magnitude diagram is used to get its luminosity and then its distance from the inverse square law of light brightness.

The entire main sequence of a cluster is used in the same way to find the distance to the cluster. We first plot the cluster's main-sequence on a color-magnitude diagram with apparent magnitudes, not absolute magnitude. We find how far the unknown main sequence needs to be shifted vertically along the magnitude axis to match the calibrated main sequence.

The age of the cluster affects the main sequence. An older cluster has only fainter stars left on the main sequence. Also, stars on the main sequence brighten slightly at a constant temperature as they age so they move slightly vertically on the main sequence. We must model the main sequence evolution to get back to the Zero-Age Main Sequence. This method assumes that all Zero-Age main sequence stars of a given temperature (and, hence, mass) start at the same luminosity. These methods can be used to find distances out to 50 kiloparsecs.

Rung 4: Period-Luminosity Relation for Variable Stars

Continuing outward we find Cepheids and/or RR-Lyrae in stars clusters with a distance known through main sequence fitting. Or we can employ the more direct ``Baade-Wesselink method'' that uses the observed expansion speed of the variable star along the line of sight from the doppler shifts in conjunction with the observed angular expansion rate perpendicular to the line of sight. Since the linear expansion rate depends on the angular expansion rate and the distance of the star, the measurement of the linear expansion rate and angular expansion rate will give us the distance of the variable star.

RR-Lyrae have the same time-averaged luminosity (about 49 solar luminosities or an absolute magnitude MV = +0.6). They pulsate with periods < 1 day. Cepheids pulsate with periods > 1 day. The longer the pulsation, the more luminous they are. There are two types of Cepheids: classical (brighter, type I) and W Virginis (fainter, type II). They have different light curve shapes. The period-luminosity relation enables us to find distances out to 4 megaparsecs (40 megaparsecs with the Hubble Space Telescope).

Rung 5a: Galaxy Luminosity vs. Another Bright Feature

We measure the periods and apparent brightnesses of Cepheids in other nearby galaxies to get their distances. Use galactic flux and inverse square law of brightness to get galactic luminosity. We can find the geometric sizes of H-II regions in spiral and irregular galaxies. From this we can calibrate the possible H-II region size--galactic luminosity relation. Or we can calibrate the correlation between the width of the 21-cm line (neutral H emission line) and the spiral galaxy luminosity. The width of the 21-cm line is due to rotation of the galaxy. This correlation is called the Fisher-Tully relation: infrared luminosity = 220 × Vrot4 solar luminosities if Vrot is given in units of km/sec. Elliptical galaxies have a correlation between their luminosity and their velocity dispersion, vdisp, within the inner few kpc called the Faber-Jackson law: vdisp approximately equals 220 × (L/L*)(1/4) km/sec, where L* = 1.0 × 1010 × (Ho/100)-2 solar luminosities in the visual band and the Hubble constant Ho = 50 - 100 km/sec/Mpc.

Rung 5b: Luminosity or Size of Bright Feature

Find Cepheids in other nearby galaxies to get their distance. We can then calibrate the luminosity of several things: (a) the supernova type 1a maximum luminosity in any type of galaxy; (b) the globular cluster luminosity function in elliptical galaxies; (c) the blue or red supergiant stars relation in spirals and irregulars; (d) the maximum luminosity--rate of decline relation of novae in ellipticals and bulges of spirals; and (e) the planetary nebula luminosity function in any type of galaxy.
The Rung 5 methods can be used to measure distances out to 50 - 150 megaparsecs depending on the particular method.

Rung 6: Galaxy Luminosity and Inverse Square Law

We can calibrate the Hubble Law using rung 4 methods for nearby galaxy distances and rung 5 methods for larger galaxy distances. If those rung 5 galaxies are like the nearby ones (or have changed luminosity in a known way), then by measuring their apparent brightness and estimating their luminosity OR by measuring their angular size and estimating their linear size, we can find their distance. We need to take care of the effect on the measured velocities caused by the Milky Way falling into the Virgo Cluster. We can also calibrate the galaxy cluster luminosity function.

The Hubble law relates a galaxy's recession (expansion) speed with its distance: speed = Ho × distance. the redshift is easy, but measuring the distance is not. We can calibrate the Hubble law using galaxies out to 500 megaparsecs.

Rung 7: Hubble Law

This is the final rung in the distance scale ladder. We use the Hubble Law for all far away galaxies. We can make maps of the large-scale structure of the universe. The Hubble Law is also used to determine the overall geometry of the universe (how the gravity of the universe as a whole has warped it). We will see in the next chapter that the geometry of the universe determines the fate of the universe.

Rung 4 is the critical one now for the distance scale ladder. With the fixed Hubble Space Telescope, we are able to use the Cepheid P-L relation out to distances ten times further than what we can do now on the ground. The ground measurements of the Hubble constant are 50--100 km/sec/Mpc (a factor of two in range!). With Cepheid observations at farther away distances, we're able to constrain its value to 75 - 85 km/sec/Mpc. The value of 1/Ho is a rough upper limit on the age of the universe (assuming constant recession speeds!). The new Hubble constant measurements are implying an universe age of only 12 - 13 billion years. This is in conflict with the ages derived for the oldest stars (found in globular clusters) of about 15 - 16 billion years. Right now, there is a lot more confidence in the age determinations for the oldest stars than for the age of the universe. This is because we are still quite uncertain as to the history of the expansion speeds and what all can affect the expansion speed. So the recent Hubble Space Telescope distance measurements have forced astronomers to attack the deficiencies in the theory of the universe expansion.

Recent very accurate parallax measurements from the Hipparcos satellite call for revisions in the calibration of the Cepheid period-luminosity relation and the distances to globular clusters that may slightly lower the derived ages for the globular clusters and slightly increase the derived age of the universe enough so the globular cluster ages may just fit under the universe age boundary. Stay tuned for more late-breaking announcements!

Review Questions

  1. Why is finding accurate extragalactic distances so important?
  2. What are the more accurate or more certain ways to measure distances? What are the less accurate (less certain) ways to measure distances? What assumptions do we make when using the less certain techniques?
  3. What is the {\em Hubble Law}? What two things does it relate? Why is it important?
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last update: 03 December 1997


Nick Strobel -- Email: strobel@lightspeed.net

(661) 395-4526
Bakersfield College
Physical Science Dept.
1801 Panorama Drive
Bakersfield, CA 93305-1219