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v intervals of several years. The convergence point is immediately apparent; all the stars seem to be heading toward one position on the celestial sphere at a ~ 6h and 8 ~ +9°. The motion, d9/dt (radians per second), is a very small quantity, of magnitude 0.075"/yr for a typical star in the Hyades cluster.

Recall that a vector direction in three-dimensional space may be represented as a point on the infinitely distant celestial sphere. Thus the convergence point represents a vector direction in space. This direction is the direction of the velocity vector v of our sample star or any other star in the cluster. (Remember that in our model all the cluster stars have the same velocity v.) The situation is illustrated with the three sketches of Fig. 3b wherein a star at a distance r = r1 from us has velocity vector v parallel to the dashed arrow. As the star recedes, the perceived angle between the star and the dashed line gradually decreases: 91 > 92 > 93. In the limit of very long times, the angle approaches zero. Thus the star's position converges to the point on the celestial sphere which represents the direction of the velocity vector, as stated.

The utility of the convergence-point determination is that the angle between the current star position and its velocity vector (convergence point) is a measurable quantity. This angle 9 is shown in both figures (Figs. 3a,b) for star A at time t1 .Itisa large angle, of order 20°. In our limited lifetime, star A in the Hyades cluster moves hardly at all, only about 4" in 50 years, or ~10-4 the distance to the convergence point.

Distance to a cluster star

The velocity vector v has two components, a radial component vr and a tangential component in the plane of the sky, v (Fig. 3b, time t1). The radial velocity vr is readily obtained from spectroscopic measures of line frequencies. The frequency shift due to the Doppler effect yields directly vr according to (16).

The tangential component v 9 is related to the measurable proper motion d9/dt, d v9 = r— (m/s; transverse velocity) (9.21)

dt and to the observable 9 (Fig. 3b), tan 9 = — (9.22)

These two equations include two unknowns, v and r. Thus one can solve for the distance r, vr tan

/ moving cluster method)

Thus, with knowledge of the measured quantities, 9 (rad), vr (m s-1) and d9/dt (rad s-1) the distance r (m) to the star is obtained.

Distance to the cluster

The distances of about 40 stars in the Hyades cluster may be obtained in this manner. These distances, together with the celestial (angular) positions, locate all the measured stars in three-dimensional space. Thus the distance to the cluster is obtained. The distance to the Hyades cluster is found from this method quite precisely, r = 144 ± 3 LY. Such distances are the underpinning of other distance scales because both cepheid variables and main-sequence stars found in clusters serve as standard candles (see below). The cluster distance serves to calibrate their luminosities. Other clusters sufficiently nearby (so the required measurements can be made) include the Ursa Major group, the Scorpio-Centaurus group, the Praesepe cluster, and the double cluster h and x Persei. The latter is at quite a large distance, 7600 LY.

Secular and statistical parallaxes

The use of parallax motions can be extended to larger distances by making use of the sun's motion relative to the barycenter of the stars in the nearby solar region of the Galaxy. These stars define the local standard of rest (LSR), the frame of reference in which their peculiar motions (vector velocities in three-dimensional space) average to zero. The solar motion in this frame of reference amounts to 4.1 AU/yr, as presented previously in Section 4.3.

Secular parallax

In traditional parallax determinations of distances, the baseline motion of the earth is limited to ±1 AU. The annual oscillatory motion of the earth leads to an oscillatory motion of the apparent track of the star under study. It is thus easy to identify unambiguously the apparent stellar motion (parallax) that is due to the earth's orbital motion.

The sun carries the earth over much greater distances than 1 AU over the years, for example 20 x 4.1 = 82 AU in 20 years relative to the LSR. One would hope to use this larger motion to make parallax measurements of more distant stars than is possible with traditional parallax. Unfortunately, though, in this case, there is no telltale oscillatory motion of the star track that identifies the part of the apparent motion due to the sun's motion.

The sun moves steadily along a more or less straight line at constant speed since the time between stellar encounters is very long. Similarly, another star whose distance we hope to learn, will have its own peculiar motion relative to the LSR, and this is also approximately straight line motion at constant speed. The apparent motion of the star against the more distant "background" stars according to a hypothetical observer at the solar system barycenter is thus a straight line due to the motions of both the sun and the star. Since the motion of the star is generally not independently known, one can not trivially extract the portion of the apparent motion due solely to the sun's (actual) motion, which is needed to arrive at a parallax distance. However, it is possible to extract the apparent motion due to the sun's motion alone with the appropriate statistical average over a large number of stars.

The average peculiar motion of a large group of stars is zero relative to the LSR, by definition. Thus, if the apparent motions of many stars in our local region are measured relative to the distant background stars, their average apparent motion must be due solely to the sun's motion. The average apparent motion would be a drift in the backward direction, opposite to the sun's motion in the LSR. This is the parallax motion that can be used to obtain distances. This method is called secular parallax. In the following discussion, we illustrate how this is done while considering the observer to be at the barycenter of the solar system. In practice, one can subtract out the wobble in the observed tracks due to the earth's orbital motion.

Consider a large sample of stars that are known to be all of approximately the same luminosity because they all are of the same "spectral type". If, in addition, we select from these only those that exhibit the same apparent magnitude or flux, the sample will then consist of stars that are all at about the same distance from the sun. Further, among these "constant-distance" stars, consider, for simplicity, only those that lie in a direction roughly normal to the solar velocity vector; that is, 9V ~ 90° (Fig. 4a). In the secular-parallax method, one measures the component of the proper motion that lies parallel to the sun's velocity vector, i.e., along the great circle passing through the star and the apex of the solar motion (Fig. 4a). This is called the upsilon component (d9/dt)v of the proper motion.

To obtain a long baseline of solar motion, one measures the changing positions of all the stars in the selected sample relative to the background stars on plates taken over a period of, say, ~20 years. The average of all the upsilon motions is presumed to be due to the sun's motion relative to the LSR because the peculiar motions of the sample should average to zero relative to the LSR.

For normal parallax, we used the relation (19), r ~ a/9pm, where a is the semimajor axis of the earth's orbit. In our case, for plates taken 20 years apart, the sun would have moved the above quoted a' = 82 AU, and the stars in our sample at 9V ~ 90° would have moved, on average, some angular distance 0par' toward the antapex; see stars in Fig. 4b. These two values, a' and 0par' can be used in place of a and 0par to calculate the distance: r = a'/9pJ of the stars in our constant-distance sample.

Equivalently one can use the motions that occur in a 1-s time interval, averaging over the 20 years. Replace a' with the distance of travel of the sun per second, %, and replace 0par' with the average upsilon motion in unit time (radians per

Figure 9.4. (a) The three measurable components of the motion of a star relative to the sun, shown for a star on the celestial sphere: the radial component of linear velocity vr measurable through spectroscopy, and the two transverse components of angular velocity in the plane of the sky measurable through imaging. The upsilon component, (d6 /dt )v , is the motion of the star parallel to the great circle connecting the star and the apex of the solar motion relative to the local standard of rest. The tau component (d6 /dt)T lies perpendicular to the great circle. (b) Observer's view of the sky. The closer stars (large dots) exhibit a general backward drift due to the sun's motion toward the apex as well as their own random motions. Caution: the upsilon and velocity symbols are similar.

Figure 9.4. (a) The three measurable components of the motion of a star relative to the sun, shown for a star on the celestial sphere: the radial component of linear velocity vr measurable through spectroscopy, and the two transverse components of angular velocity in the plane of the sky measurable through imaging. The upsilon component, (d6 /dt )v , is the motion of the star parallel to the great circle connecting the star and the apex of the solar motion relative to the local standard of rest. The tau component (d6 /dt)T lies perpendicular to the great circle. (b) Observer's view of the sky. The closer stars (large dots) exhibit a general backward drift due to the sun's motion toward the apex as well as their own random motions. Caution: the upsilon and velocity symbols are similar.

+ r = --— (m; secular parallax; distance to the (9.24)

' av sample of stars normal to solar apex)

This is the desired distance to our sample of constant-distance stars. The numerator is the speed of sun in the LSR, 19.5 x 103 m/s = 4.1 AU/year. This illustrates the principle of the method. In practice, one would make use of stars at other angles 9V and would take into account a modest spread of types or luminosities, i.e., distances, in the sample.

Statistical parallax

The method of statistical parallaxes makes use of the other component of the proper motion in the plane of the sky, the tau component (d9/dt)T (Fig. 4). This motion is perpendicular to the solar motion and is unaffected by it. Only the peculiar motion of the star affects the tau component, and this, by definition, should average to zero in the LSR and also in the frame of the sun, for a large sample of nearby stars. The tau component, like the upsilon component, is an angular velocity.

Again consider a sample of stars at a common distance and in a direction normal to the solar velocity. For stars in these particular (normal) directions, the radial velocity vr (component along the line of sight) is also unaffected by the solar motion; hence it too is due only to peculiar motions. Furthermore, one expects the absolute value of the speeds in the radial and tau directions, averaged over the stars in the sample, to be equal because all directions of the motions are assumed to be equally probable.

The radial velocity vr may be obtained from the spectroscopic Doppler shifts (16) of the spectral lines, and the average of the absolute values |vr |av can be calculated. The tau components of the stars' angular motions can also be measured and the absolute values averaged, |(d9/dt)T |av. It is necessary to take the absolute values before averaging; a straight average would yield zero velocity. Alternatively, one could calculate the root-mean-square values of the t components.

The relation between linear azimuthal velocity in the t direction vt and the angular velocity in the same direction (d9/dt)t for a single object at distance r is simply vt = r (d9/dt)t , from the definition of the radian. The same relation will hold for the averages of the linear and angular t motions for a large number of stars. If the averages were based on a single object, the relation would surely hold. Or you can prove this by carrying the relation through the averaging process. The relation with the average values could be solved for the distance r.

In our case, we do know the values of (d9/dt)t and hence also the average |(d9/dt)t |av but we do not know the linear velocities vt . Nevertheless, as noted, we presume these to be equal, on average, to the radial velocities we did measure. Thus our two measured (average) values can yield the desired distance r to our sample objects,

* r =- (m; statistical parallax; distance to (9.25)

' a group of stars

In practice, one must again take into account the effect of stars in the sample having a spread of distances as well as projection effects for stars not normal to the solar apex. This technique has yielded distances of stars out to ~3000 LY.

Standard candles

Subsequent steps in the distance ladder involve the identification of standard candles. The calibration of the standard candles is a step-by-step process. First the distance to a nearby object is obtained, say by trigonometric parallax. This distance, together with the measured flux, yields the luminosity L according to (3). If the object is a recognizable member of a class for which all members have the same luminosity, the members of the class can serve as standard candles out to greater distances. The luminosity must be sufficiently high so that it can be detected at these larger distances, and it must have spectral or temporal characteristics that make it recognizable. Whenever another such object is found, its distance follows directly from its measured flux and the known luminosity of the class.

If one of these objects is located in a cluster of objects on the sky, the distance of the cluster thus also becomes known. One of the other cluster objects may be a more luminous object of another type which may well be suitable as a new standard candle. With knowledge of its distance and measured flux, its luminosity is known. With its greater luminosity, it can be used out to even greater distances. Of course, it must also exhibit easily recognizable spectral or temporal characteristics so other members of the class can be identified. This second standard candle would be yet another step in the distance ladder.

Luminous normal stars and cepheid variables (see below) are important standard candles. Also, planetary nebulae, supernovae, and galaxies themselves are used as standard candles. The determination of the distances to distant galaxies plays a major part in the effort to determine the past and future history of the universe.

Spectroscopic classification

All normal stars can be classified according to their spectral types. Each type is characterized by a particular luminosity and a unique combination of spectral features that follow from the mass of the star and its stage of evolution. Among the stars burning hydrogen in their core (like our sun), the low-mass stars are generally cool (color red) with luminosities as low as 10-3L Q whereas high-mass stars are white hot in color and have luminosities up to about 5 x 105 L Q. Stars of the various types can therefore be used as a whole series of standard candles. Since these standard candles are based on their spectral type and since they yield distances, the process is sometimes called spectroscopic parallax.

The luminosities of individual stars obtained with the parallax method may be used to calibrate this system. However, the stars sufficiently close to the earth to be measured directly via parallax are mostly low-mass stars that are not particularly luminous. Thus the distance to which they can be used as standard candles is limited. The rarer more luminous stars must be calibrated indirectly.

The luminosities of the more luminous stars were first calibrated from observations of the stars in the Hyades cluster, the distance of which is known (see above). The Hyades cluster does not contain the most luminous stars because they have used up their nuclear fuel (burned out). However, younger clusters do contain them. The already-calibrated less-luminous stars or cepheid variables (see below) in a younger cluster give its distance. This allows the luminosities of the rare

Figure 9.5. Cluster of H2O masers in radio source Sgr B-2 near the center of the Galaxy. The arrows represent directions and speeds of masers on the sky. Note that the masers are clustered within an area of only 4" extent. The masers are expanding from apoint within the region indicated by the dashed cross, presumably the location of a newly formed star. Modeling of these data and also radial velocity data yields a distance to the galactic center of ~23 000 LY. [From M. Reid et al., ApJ 330,809(1988)]

Figure 9.5. Cluster of H2O masers in radio source Sgr B-2 near the center of the Galaxy. The arrows represent directions and speeds of masers on the sky. Note that the masers are clustered within an area of only 4" extent. The masers are expanding from apoint within the region indicated by the dashed cross, presumably the location of a newly formed star. Modeling of these data and also radial velocity data yields a distance to the galactic center of ~23 000 LY. [From M. Reid et al., ApJ 330,809(1988)]

luminous stars it contains to be determined. In this way they too become standard candles.

Galactic center and Crab nebula distances

The distance to the center of our galaxy has been obtained by radio astronomers with a method that made use of a cluster of H2O masers in a star-forming region in the close vicinity of the galactic center (Fig. 5). The masers are probably condensations of interstellar gas and dust in the vicinity of a luminous newly formed star. They operate similarly to a laser but radiate in the microwave frequency range; they emit at precise frequencies characteristic of a constituent molecule, H2O in this case.

The angular positions of the individual masers are measured repeatedly with very long baseline interferometry to obtain their angular velocities. Radial linear velocities are obtained from Doppler shifts of the spectral lines. The angular motion of the masers on the sky (arrows in Fig. 5) shows them to be moving away, more or less, from a common point on the sky, probably an (unseen) star. If a spherical expansion is assumed (e.g., due to a stellar wind), these data can be modeled to give a distance to the source.

Consider a maser to be part of the close edge of a spherical shell of gas moving outward from the central star. It will be directly approaching the observer, so the measured Doppler shift will yield the velocity magnitude v = vr (16). Consider next another maser on the right edge of the same shell as viewed by the same observer. This maser moves in the plane of the sky normal to the line of sight and will have only a tangential component of velocity ve = r (d9/dt), where d9/dt is the directly measured angular displacement per unit time. In our model of spherical expansion, the two masers would have the same speed. The distance to the star follows from equating the two speeds, dd

The complete analysis will include all the masers associated with this object, the results being appropriately weighted and averaged. The published result from this experiment is 23 ± 5 kLY which is substantially less than the value used for the distance to the galactic center for many years previously, 32 600 LY (= 10 000 pc). In this text, we adopt 25 000 LY as the nominal value to the galactic center. Note that the underlying principle of the analysis is similar to that discussed above for the statistical parallax, namely the simultaneous measure of angular and linear velocities.

The same method has been used by optical astronomers to obtain the distance to the Crab nebula which is the remnant of a supernova explosion several arcmin in size with a bluish synchrotron nebula and an expanding shell of red hydrogen-emitting filaments (Fig. 1.3). The radial and angular velocities of the filaments are measured to yield a distance of ~6000 LY. The expansion is not perfectly spherical which leads to some uncertainty in this distance.

Cepheid and RR Lyrae variables

Cepheid variables are luminous stars that are 300 to 30 000 times the luminosity of the sun (300-30 000 LQ). Their luminosities vary periodically with periods of days (1-100 d); see Fig. 6a. The luminosity variation is due to oscillation in the size (height) and temperature of the stellar atmosphere associated with variation in the degree of ionization of the atmospheric gases. The period of the oscillations is highly correlated with the average luminosity (Fig. 6b). A cepheid variable with a period of 100 days is roughly 40 times more luminous than a cepheid variable with a 1 day period. These stars are sufficiently luminous to be found in galaxies out to ~60 MLY with the Hubble Space telescope, e.g., in the Virgo cluster of galaxies.

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