Steps to the Hubble Constant

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

Why care so much about finding distances in astronomy? If you know the distance to a star, you can determine its luminosity and mass. You 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 you 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. Astronomers have then 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 you to find their mass, luminosity, and star formation history among other things. You are 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, you are 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 keep 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 distance measurements are so important.

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. Each rung depends on the rung below and it is less certain than the previous one.

Rung 1: The Astronomical Unit

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

Rung 2: Geometric Methods

On the next rung of the distance scale ladder, you can 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. Distances to nearby clusters like the Hyades or the Pleiades are found via trigonometric parallax or the moving clusters method (another geometric method). The cluster's main sequence is calibrated 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 the spectral type of star is determined from its spectral lines and the apparent brightness of the star is measured. 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. You first plot the cluster's main-sequence on a color-magnitude diagram with apparent magnitudes, not absolute magnitude. You find how far the unknown main sequence needs to be shifted vertically along the magnitude axis to match the calibrated main sequence. The amount of the shift depends on the distance.

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. You 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 you find Cepheids and/or RR-Lyrae in stars clusters with a distance known through main sequence fitting. Or you 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 you 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

The periods and apparent brightnesses of Cepheids in other nearby galaxies are measured to get their distances. Then the galactic flux and the inverse square law of brightness are used to get the galactic luminosity. You can find the geometric sizes of H-II regions in spiral and irregular galaxies. From this you can calibrate the possible H-II region size--galactic luminosity relation. Or you can calibrate the correlation between the width of the 21-cm line (neutral hydrogen 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 Tully-Fisher 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 = 60 to 70 km/sec/Mpc.

Rung 5b: Luminosity or Size of Bright Feature

Cepheids are found in other nearby galaxies to get their distance. Then the luminosity of several things are calibrated: (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 to 150 megaparsecs depending on the particular method.

Rung 6: Galaxy Luminosity and Inverse Square Law

The Hubble Law is calibrated 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, you can find their distance. You need to take care of the effect on the measured velocities caused by the Milky Way falling into the Virgo Cluster. You can also calibrate the galaxy cluster luminosity function.

The Hubble law relates a galaxy's recession (expansion) speed with its distance: speed = Ho × distance. Measuring the speed from the redshift is easy, but measuring the distance is not. You 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. You use the Hubble Law for all far away galaxies. You 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). You will see in the next chapter that the geometry of the universe determines the fate of the universe.

Rung 4 is a critical one for the distance scale ladder. With the Hubble Space Telescope, astronomers were able to use the Cepheid period-luminosity relation out to distances ten times further than what could be done on the ground. Prevous ground measurements of the Hubble constant were 50 to 100 km/sec/Mpc. Using the Hubbble Space Telescope, astronomers constrained its value to between 64 and 80 km/sec/Mpc with a best value of 72 km/sec/Mpc. The value of 1/Ho is a rough upper limit on the age of the universe (assuming constant recession speeds!), so the new measurements imply an universe age of about 14 billion years. The favorite model for how the recession speeds have changed over the history of the universe gives an age of about 13.7 billion years with this value for the Hubble constant. This agrees with the ages derived for the oldest stars (found in globular clusters) of about 12 to 13 billion years.

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 Hubble Law? What two things does it relate? Why is it important?

A Final Word

The Sombrero Galaxy is one of the most photographed galaxies and this exquisitely beautiful picture from the Hubble Space Telescope shows why. There are number of beautiful objects that draw people to take up astronomy as a profession or a life-long hobby and the Sombrero Galaxy is one of them. Seeking to understand what these objects are made of, how they behave, and how they formed gives us a greater appreciation for the art that surrounds us. Visible in even small telescopes at the southern edge of the Virgo cluster of galaxies, the Sombrero Galaxy is a spiral galaxy more massive than the Milky Way seen nearly edge-on from a distance of about 28 million light years away.

This image also provides a nice illustration of the parts of a spiral galaxy and its history. The oldest stars are in the spherical bulge & stellar halo retaining their randomly-oriented eccentric orbits of the original gas cloud from which the galaxy formed about 13 billion years ago. The stellar halo also sports almost 2000 globular clusters (the Milky Way has only about 150 globular clusters). The more massive stars in the spherical component added heavier elements (dust) to the newly-formed disk in which younger generations of stars are still forming. These younger stars have added to the dust layer that now outlines the disk and spiral arms. The fortunate slight tilt of the disk to our line of sight allows us to easily distinguish the near side of the disk from the far side. The disk star orbits are closely aligned to each other to make the very thin disk characteristic of spiral galaxies. A billion solar-mass black hole lies at the heart of the bright core.

Select the image to go to the webpage in the Hubble Heritage Team's website from which this image came. Larger versions of this image are available from the Hubble Site for the general public.

Sombrero Galaxy -- a beautiful nearly edge-on spiral galaxy with a large bulge

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last updated: June 5, 2004

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Author of original content: Nick Strobel