2.3.10. Direct Astrophysical Distance Measurements

The previous, cumbersome and somewhat tedious discussion should have convinced the reader that the determination of a precise value for H₀ requires a distance measuring technique which completely bypasses the distance scale ladder. There are many steps on the ladder where systematic error comes into play and the entire technique is always open for criticism on the basis of bias associated with sample selection. Direct astrophysical distance indicators do not rely on distance estimates to intermediate galaxies and hence have the potential of providing definitive distances. Furthermore, these techniques can be applied over large distances where any perturbation from expansion velocity becomes negligible. However, these direct indicators are extremely model-dependent, as they rely on the astrophysics of some process to provide the distance and that astrophysics must be modeled. Still, this is a major improvement as various protagonists can now argue over the physics of the situation, rather than sample selection techniques. To date, the most promising kinds of direct astrophysical distance measurements are the following:

The Sunyaev-Zeldovich (SZ) effect

The CMB radiation is nearly a perfect blackbody with temperature of 2.74 K. When this radiation passes through a cluster of galaxies that has a hot Intracluster Medium (ICM - see Chapter 3), then hot electrons in that plasma scatter the CMB photons to higher energies and thus distort the original blackbody spectrum. The overall effect is to increase the CMB photon energies preferentially at the shorter wavelengths. At longer wavelengths, this effect causes a net reduction in the temperature of the background radiation that passes through the cluster plasma. The measured decrease in temperature, T / T, depends on the total amount of scattering which in turn depends on the electron temperature (T_e), the electron density (n_e) and path length through the plasma (dl). The temperature decrement can then be formulated as

(15)

It is convenient to replace the integral in equation 15 with average mean values for n_e and T_e, thus

(16)

where L is now the physical path length. To estimate the mean values of n_e and T_e we can make use of the observed X-ray flux, S_x

(17)

where V = the volume of the cluster, n_p = ion density and D = distance from the observer to the cluster. In thermal equilibrium n_p n_e. For a spherical cluster V L³. More generally, V L(_{x y}) (D²) where _xy is the angular size of the cluster in projected coordinates x and y. Using equation 15 to solve for n_e² and inserting that into equation 17 yields

(18)

which becomes

(19)

Everything on the left hand side of equation 19 is a measurable quantity from which the physical path length can be measured. Once the physical size of the cluster is known, its angular size on the sky can be used to determine D and hence H₀. This method has been applied to a few clusters and the derived values of H₀ are generally in the range 25 -75. The most recent determination is based on observations of the Coma cluster where Herbig et al. (1995) derive H₀ = 74 ± 29. The large error bar is a reflection of the difficulty of this method as there are several complications:

Any small scale structure in the X-ray gas is fatal as the method uses smooth, average values of n_e and T_e. Small scale clumping of the gas in which there are differences in n_e and T_e will produce spurious results.

T_e must be determined and in general, it is quite difficult to determine X-ray temperatures from the extant data due to limited spectral coverage of most X-ray detectors.

The measurements of the microwave background decrement itself are very hard. The Herbig et al. data are made with the best system currently available for measuring this and hence they probably have the most credible result.

The method really works best for perfectly spherical clusters (where _xy can just be replaced by L²). Irregularly shaped clusters can cause spurious errors.

Since the SZ effect itself is one of a frequency shift it can be simulated if the cluster itself is moving. In this case, T v / c. For the case of a cluster moving with respect to the CMB reference frame at 1000 km s^-1, then T will have the same value as it would in the case of a non-moving plasma with T_e 1 kev.

Because of all these complicating effects, distances derived from the SZ effect are highly suspect and model dependent. The most serious of these complications involves fine-scale structure in the X-ray emitting plasma. Recent data from new X-ray satellites such as ROSAT which is capable of detecting this fine-scale structure, and ASCA which is capable of measuring temperatures directly (provided the signal strength is high) should offer improved SZ distances to some clusters in the near future.

Gravitational Lensing

Gravitational lensing occurs whenever the light from a distant point source passes very near by a massive object. The space around that object is distorted and the light path can take on a number of different trajectories which can reach the observer. Hence, the observer sees not only multiple objects but in some cases amplified objects, depending upon the nature of the mass concentration. The details of this were first laid out by Refsdel (1964). For the case of the light from a distant point source encountering an isothermal sphere the critical radius for lensing and amplification is given by

(20)

where _m is the velocity dispersion of the lensing mass. To a reasonable degree of approximation (see Chapter 3) clusters of galaxies have a potential like that of an isothermal sphere and hence, in principle, can be gravitational lenses. D_LS is the angular diameter distance between the lens and the source and D_OS is the angular diameter distance between the observer and the source. In more descriptive terms, r_crit is related to differences in the potential that the multiple light paths take. These potential differences give rise to delays in arrival times of the light from these multiple components that reaches an observer. D_LS can be approximately determined if the redshifts of both the lensing source and the lensed object are unknown (in general the redshift of the lens is not known) and _m can, in principle be measured as well.

Figure 2-23: CCD image showing the double quasar Q 0957+561. Obtained by Dr. Rudolph Schild, Smithsonian Astrophysical Observatory and reproduced with permission.

To date, one system has been discovered that lends itself to this kind of analysis. The distant quasar Q 0957+561 (see Figure 2-23) is lensed by a foreground cluster of galaxies of known redshift. The two brightest images of the quasar are known as A and B. For a particular model of the lensing geometry and the mass distribution of the cluster it can be shown (see Falco et al. 1991; Surpi et al. 1996) that

(21)

where _AB^-1 is the measured time delay. The principle advantage of using lensed QSOs is that, in general, QSOs are variable in their luminosity output at all wavelengths. Continuous monitoring of this system at both radio and optical wavelengths can then determine _AB^-1. Radio observations are less susceptible to sampling gaps caused by poor weather.

Figure 2-24: Spectacular image of arclets and rings representing gravitational lensing associated with the cluster Abell 2218. CCD image taken with the Hubble Space Telescope.

Only one source (Q 0957+561) has been discovered to date that is suitable and deriving H₀ on the basis of the statistics of one event is dubious.

The time delay is actually quite difficult to measure because it is so long. For Q 0957+561, _AB^-1 is of order 1.5 years. In the optical, the components are sufficiently faint that they can not be measured in moonlight which results in significant gaps and irregular sampling in the timing data. Reconstruction of the intrinsic time delay from irregularly sampled data is difficult. Because of this the radio data provides the best means for estimating the time delay. In fact for Q 0957+561 there has been somewhat of a controversy regarding the value for _AB^-1 as values of either 415 or 535 days can fit the timing data. Very recently, Turner et al. (1996) present convincing data in favor of the 415 day period.

The intrinsic density distribution of the lens itself must be known. For clusters of galaxies (Chapter 3) this is almost impossible to know. Detailed maps of the x-ray distribution can help in this regard. In addition to Q 0957+561, faint background galaxies are also lensed. As these galaxies are not point sources, their multiple images are not point sources; instead they are thing arcs or arclets (Figure 2-24). In principle, the size of these arcs is related to the mass distribution in the lens. Fischer et al. (1996) have used observations of lensed galaxies to determine the mass distribution of the lensing cluster towards Q 0957+561. Interestingly, their results do not agree very well with the mass distribution which has been inferred from the X-ray map of the cluster.

Because of the difficulty in determining the mass distribution of the lens, derivations of H₀ from this method are not yet credible. For the Q 0957+561 system, the strongest statement which can be made from the observed time delay and reasonable modeling of the mass distribution is that H₀ 90 ± 30 km s^-1 (see Kochanek 1991).

Superluminal Motion in Radio Sources

It has been known for 20 years that some radio sources have small-scale components which seem to be separating from one another at a velocity that exceeds the speed of light. This is now understood to be an illusion which is caused by the relativistic acceleration of a plasma down a beam pipe which is pointed at the observer. The knots which are seen in the radio jets are shocks in this relativistic flow. These knots often exhibit cm-waveband variability which is due to the passage of these shocks through the optically thick surface of the flow. During this passage the flux increases, followed by adiabatic energy loss and a decline in the flux. The increase in the percentage polarization that accompanies such activity is associated with the shock compression of an initially tangled magnetic field, establishing a `preferred direction', and causing a significant percentage polarization for observers viewing radiation emitted in the plane of the compression . A stationary observer on the `1 / -cone' of the flow sees both maximum possible superluminal motion, and high percentage polarization, because aberration causes radiation emitted in the plane of a shock traveling along the flow to be swung into the line of sight.

Novel work by Phillip Hughes and his collaborators at the University of Michigan have used this relationship between increased polarization and maximum superluminal motion to construct a geometrical shock model which allows the distance to the source to be determined. In this model, description of the flow dynamics makes use of the analytic jump conditions for shocks in a relativistic gas. Predicted polarized flux light curves are obtained by performing radiation transfer calculations through the plasma at many epochs. Although the models contain a large number of free parameters, they can be well-constrained because of the wealth of information contained in high time resolution, multi-frequency, flux and polarization data. In particular, the shape of the total flux profile is strongly influenced by time delay effects, and thus by viewing angle, while the degree of polarization is sensitively dependent on relativistic aberration and hence on flow speed and viewing angle. Models of two well-defined outbursts in the source BL Lac that occurred in the early 1980s have an optimal fit to both the light curve profile and percentage polarization for an angle of view of almost 40°. With angle of view and flow speed known, the apparent speed of structures can be calculated, and thus the angular separation rate of source components can be predicted as a function of cosmological distance. Comparison with the actual rate, determined by VLBI, allows the distance to be determined, and when combined with the known redshift, H₀ may be estimated. An initial application of this technique yielded a value somewhat in excess of 100 km s^-1 Mpc^-1. Although more refined models admit a somewhat smaller viewing angle, and smaller value of H₀, an important point is that at these large angles of view (which have received strong support from independent modeling of VLBI data), no source speed is compatible with values of H₀ close to 50 km s^-1 Mpc^-1, because the apparent component speed is too small, and superluminal motion in BL Lac would simply not be observed (see Hughes et al. 1991).

Virial Masses:

This is a straightforward procedure but it is unlikely to be applicable to any real astrophysical source due to the presence of dark matter. The dynamical mass of a rotating cloud of gas is

(22)

where R is some characteristic physical scale. R is related to distance via D where is the angular size of the gas cloud. If we imagine that this cloud has no dark matter in it and no stars such that the gas comprises 100% of the dynamical mass then the distance follows directly as we can use the observed flux (F₀) of emission from that gas (assume that it is neutral hydrogen). In this case

(23)

and only one value of D satisfies the observational constraints provided by F₀ and V_c. This technique has been applied to one gas-rich system to date by Staveley-Smith et al. 1990 resulting in an upper limit on H₀ of 70 ± 7. If other gas-rich systems can be detected and if they have reasonable dynamics, then this method may provide a statistically interesting measure of H₀.

In sum, consideration of the possible forms of direct astrophysical distance measures has yielded some promising candidates. The principle limitations are 1) the availability of real astrophysical sources that are ideal and 2) resulting values of H₀ remain model dependent. However, the mere fact that these astrophysical distance measures are returning values of H₀ that do lie in the range 50 - 100 is quite encouraging and hence continued pursuit in this direction is a viable and appealing alternative to having to derive H₀ via the cumbersome distance scale ladder.