The ABC's of Distances

It is almost impossible to tell the distances of objects we see in the sky. Almost, but not quite, and astronomers have developed a large variety of techniques. Here I will describe 26 of them. I will ignore the work that went into determining the astronomical unit: the scale factor for the Solar System, and just consider distances outside of the Solar System.

A. TRIGONOMETRIC PARALLAX

This methods rates an A because it is the gold standard for astronomical distances. It is based on measuring two angles and the included side of a triangle formed by 1) the star, 2) the Earth on one side of its orbit, and 3) the Earth six months later on the other side of its orbit.

The parallax of a star is one-half the angle at the star in the diagram above. Thus the parallax is the angle at the star in an Earth-Sun-star triangle. Since this angle is always very small, the sine and tangent of the parallax are very well approximated by the parallax angle measured in radians. Therefore the distance to a star is

D[in cm] = [Earth-Sun distance in cm]/[parallax in radians]

The first stellar parallax (of the star 61 Cygni) was measured by Friedrich Wilhelm Bessel (1784-1846) in 1838. Bessel is also known for the Bessel functions in mathematical physics.

B. Moving Clusters

Not many stars are close enough to have useful trigonometric parallaxes. But when stars are in a stable star cluster whose physical size is not changing, like the Pleiades, then the apparent motions of the stars within the cluster can be used to determine the distance to the cluster.

D[in cm] = VT[in cm/sec]/[d(theta)/dt]

D[in pc] = (VR/4.74 km/sec)*tan(theta)/{d(theta)/dt[in "/yr]}

C. Secular Parallax

Another method can be used to measure the average distance to a set of stars, chosen to be all about the same distance from the Earth.

D[in cm] = V(sun)[in cm/sec]/(mu [in radians/sec])

D[in pc] = 4.16/(mu [in "/yr])

D. Statistical Parallax

When the stars have measured radial velocities, then the scatter in their proper motions can be used to determine the mean distance. It is

                  (scatter in VR)[in cm/sec]
D[in cm] = ----------------------------------------
           (scatter in d(theta)/dt)[in radians/sec]

E. Kinematic Distance

The pattern of differential rotation in our galaxy can be used to determine the distance of a source when its radial velocity is known.

F. Expansion Parallax

The distance to an expanding object like a supernova remnant like a supernova remnant such as Tycho can be determined by measuring:

1. the angular expansion rate d(theta)/dt using pictures taken many years apart, and
2. the radial velocity of expansion, VR, using the Doppler shift of lines emitted from the front and back of the expanding shell. When a spectrograph is pointed at the center of the remnant a double line is seen, with the red shifted emission coming from the back of the shell while the blue shifted emission comes from the front.

The distance is then calculated using

D = VR/d(theta)/dt          with theta in radians

G. Light Echo Distance

The center elliptical ring around SN1987A in the LMC appears to be due to an inclined circular ring around the progenitor. When the pulse of ultraviolet light from the supernova hit the ring, it lit up in ultraviolet emission lines which were observed by the International Ultraviolet Explorer (IUE). The first detection of these lines at time, t1, and also the time when the lines from the last part of the ring to be illuminated, t2, were both clearly evident in the IUE light curve of the UV lines. If t0 is the time that we first saw the supernova, then the extra light travel times to the front and back of the ring are:

t1 - t0 = R(1 - sin(i))/c
t2 - t0 = R(1 + sin(i))/c

R = c(t1-t0 + t2-t0)/2

D = R/theta           with theta in radians

H. Spectroscopic Visual Binaries

If a binary orbit is observed both visually and spectroscopically, then both the angular size and the physical size of the orbit are known. The ratio gives the distance.

I. Baade-Wesselink Method

The Baade-Wesselink method is applied to pulsating stars. Using the color and flux light curves, one finds the ratio of the radii at different times:

              sqrt[Flux(t2)/SB(Color(t2)]
R(t2)/R(t1) = ---------------------------
              sqrt[Flux(t1)/SB(Color(t1)]

J. Spectroscopic Eclipsing Binaries

In a double-lined spectroscopic binary, the projected size of the orbit a*sin(i) is found from the radial velocity amplitude and the period. In an eclipsing binary, the relative radii of the stars R1/a and R2/a and the inclination of the orbit i are found by analyzing the shapes of the eclipse light curves. Using the observed fluxes and colors to get surface brightnesses, the angular radii of the stars can be estimated. R1 is found from i, a*sin(i) and R1/a; and with theta1 the distance can be found.

K. Expanding Photosphere Method

The Baade-Wesselink method can be applied to an expanding star: the variations in radius do not have to be periodic. It has been applied to Type II supernovae, which are massive stars with a hydrogen rich envelope that explode when their cores collapse to from neutron stars. It can also be applied to Type Ia supernovae, but these objects have no hydrogen lines in their spectra. Since the surface brightness vs color law is calibrated using normal, hydrogen-rich stars, the EPM is normally used on hydrogen-rich supernovae, which are Type II. The Type II SN1987A in the Large Magellanic Cloud has been used to calibrate this distance indicator.

D = sqrt[L/(4*pi*F)]

L. Main Sequence Fitting

When distances to nearby stars were found using trigonometric parallaxes in the late 19th and early 20th century, it became possible to study the luminosities of stars. Einar Hertzsprung and Henry Norris Russell both plotted stars on a chart of luminosity and temperature. Most stars fall on a single track, known as the Main Sequence, in this diagram, which is now known as the H-R diagram after Hertzsprung and Russell. Often the absolute magnitude is used instead of the luminosity, and the spectral type or color is used instead of the temperature.

When looking at a cluster of stars, the apparent magnitudes and colors of the stars form a track that is parallel to the Main Sequence, and by correctly choosing the distance, the apparent magnitudes convert to absolute magnitudes that fall on the standard Main Sequence.

M. Spectroscopic Parallax

When the spectrum of a star is observed carefully, it is possible to determine two parameters of the star as well as the chemical abundances in the star's atmosphere. The first of these two parameters is the surface temperature of the star, which determines the spectral type in the range OBAFGKM from hottest to coolest. The hot O stars show ionized helium lines, the B stars show neutral helium lines, the A stars have strong hydrogen lines, the F and G stars have various metal lines, and the coolest K and M stars have molecular bands. The spectral classes are further subdivided with a digit, so the Sun is a G2 star.

The second parameter that can be determined is the surface gravity of the star. The higher the surface gravity, the higher the pressure in the atmosphere, and high pressure leads to line broadening and also reduces the amount of ionization in the atmosphere. The surface gravity is denoted by the luminosity class denoted by a Roman numeral from I to V with I being the lowest gravity and V being the highest (except for class VI which is sometimes seen and for white dwarfs which separately classified.) Stars with high surface gravity (class V) are called dwarfs while stars with medium gravity (class III) are called giants and stars with low gravity (class I) are called supergiants. The use of surface gravity to determine the luminosity of a star depends on three relations:

L = 4*pi*sigma*T4*R2
L = A*Mb                      Mass-luminosity law with b = 3-4
g = G*M/R2

One warning about this method: it only works for normal stars, and any given single object might not be normal. Main sequence fitting in a cluster is much more reliable since with a large number of stars it is easy to find the normal ones.

N. RR Lyrae Distance

O. Cepheid Distance

Cepheid variable stars are pulsating stars, named after the brightest member of the class, Delta Cephei. These stars pulsate because the hydrogen and helium ionization zones are close to the surface of the star. This more or less fixes the temperature of the variable star, and produces an instability strip in the H-R diagram.

P. Planetary Nebula Luminosity Function

Q. Brightest Stars

R. Largest H II Region Diameters

T. Type Ia Supernovae

U. Tully-Fisher Relation

L = Const * V(rot)4

V. Faber-Jackson Relation

L = Const * sigma(v)4

W. Brightest Cluster Galaxies

X. Gravitational Lens Time Delay

When a quasar is viewed through a gravitational lens, multiple images are seen, as shown in diagram below.

Y. Sunyaev-Zeldovich Effect

Hot gas in clusters of galaxies distorts the spectrum of the cosmic microwave background observed through the cluster. The diagram below shows a sketch of this process. The hot electrons in the cluster of galaxies scatter a small fraction of the cosmic microwave background photons and replace them with slightly higher energy photons.

The difference between the CMB seen through the cluster and the unmodified CMB seen elsewhere on the sky can be measured. Actually only about 1% of the photons passing through the cluster are scattered by the electrons in the hot ionized gas in the cluster, and these photons have their energies increased by an average of about 2%. This leads to a shortage of low energy photons of about 0.01*0.02 = 0.0002 or 0.02% which is gives a decrease in the brightness temperature of about 500 microK when looking at the cluster. At high frequencies (higher than about 218 GHz) the cluster appears brighter than the background. This effect is proportional to (1) the number density of electrons, (2) the thickness of the cluster along our line of sight, and (3) the electron temperature. The parameter that combines these factors is called the Kompaneets y parameter, with y = tau*(kT/mc²). Tau is the optical depth or the fraction of photons scattered, while (kT/mc²) is the electron temperature in units of the rest mass of the electron.

The X-ray emission, I_X, from the hot gas in the cluster is proportional to (1) the square of the number density of electrons, (2) the thickness of the cluster along our line of sight, and (3) depends on the electron temperature and X-ray frequency. As a result, the ratio

y2/IX = CONST * (Thickness along LOS) * f(T)

If we assume that the thickness along the LOS is the same as the diameter of the cluster, we can use the observed angular diameter to find the distance.

This technique is very difficult, and years of hard work by pioneers like Mark Birkinshaw yielded only a few distances, and values of H_o that tended to be on the low side. Recent work with close packed radio interferometers operating at 30 GHz has given precise measurements of the radio brightness decrement for 18 clusters, but only 3 of these have adequate X-ray data.

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And finally:

Z. The Hubble Law

The Doppler shift gives the redshift of a distant object which is our best indicator of its distance, but we need to know the Hubble constant.

But wait, there's MORE! Pulsar dispersion measures and interstellar extinction increase with distance along a given line of sight and can be used to determine distances. The globular cluster luminosity function can be used to estimate the distance to a galaxy from the observed brightness of its globular clusters.

© 1997-1999 Edward L. Wright. Last modified 15-Jun-1999