## Methods of Distance Measurement

In order to study the structure of the Milky Way, one needs to know how various kinds of objects, such as stars, star clusters and interstellar matter, are distributed in space. The most important ways of measuring the distances will first be considered.

Trigonometric Parallaxes. The method of trigonometric parallaxes is based on the apparent yearly back-and-forth movement of stars in the sky, caused by the orbital motion of the Earth. From Earth-based observations the trigonometric parallaxes can be reliably measured out to a distance of about 30 pc; beyond 100 pc this method is no longer useful. The situation is, however, changing. The limit has already been pushed to a few hundred parsecs by the Hipparcos satellite, and Gaia will mean another major leap in the accuracy.

Fig. 17.4. Because of the motion of the Sun towards the apex, the average radial velocity of the nearby stars appears largest in the apex and antapex directions

The Motion of the Sun with Respect to the Neighbouring Stars. The Local Standard of Rest. The motion of the Sun with respect to the neighbouring stars is reflected in their proper motions and radial velocities (Fig. 17.4). The point towards which the Sun's motion among the stars seems to be directed is called the apex. The opposite point is the antapex. The stars near the apex appear to be approaching; their (negative) radial velocities are smallest, on the average. In the direction of the antapex the largest (positive) radial velocities are observed. On the great circle perpendicular to the apex-antapex direction, the radial velocities are zero on the average, but the proper motions are large. The average proper motions decrease towards the apex and the antapex, but always point from the apex towards the antapex.

In order to study the true motions of the stars, one has to define a coordinate system with respect to which the motions are to be defined. The most practical frame of reference is defined so that the stars in the solar neighbourhood are at rest, on the average. More precisely, this local standard of rest (LSR) is defined as follows:

Let us suppose the velocities of the stars being considered are distributed at random. Their velocities with respect to the Sun, i. e. their radial velocities, proper motions and distances, are assumed to be known. The local standard of rest is then defined so that the mean value of the velocity vectors is opposite to the velocity of the Sun with respect to the LSR. Clearly the mean velocity

of the relevant stars with respect to the LSR will then be zero. The motion of the Sun with respect to the LSR is found to be:

 Apex coordinates a = 18 h 00 min = 270° 1 = 56° S = +30° b = +23° Solar velocity V0 = 19.7 kms-1

The apex is located in the constellation of Hercules. When the sample of stars used to determine the LSR is restricted to a subset of all the stars in the solar neighbourhood, e. g. to stars of a given spectral class, the sample will usually have slightly different kinematic properties, and the coordinates of the solar apex will change correspondingly.

The velocity of an individual star with respect to the local standard of rest is called the peculiar motion of the star. The peculiar velocity of a star is obtained by adding the velocity of the Sun with respect to the LSR to the measured velocity. Naturally the velocities should be treated as vectors.

The local standard of rest is at rest only with respect to a close neighbourhood of the Sun. The Sun and the nearby stars, and thus also the LSR, are moving round the centre of the Milky Way at a speed that is ten times greater than the typical peculiar velocities of stars in the solar neighbourhood (Fig. 17.5).

Statistical Parallaxes. The velocity of the Sun with respect to neighbouring stars is about 20kms-1. This means that in one year, the Sun moves about 4 AU with respect to the stars.

Let us consider a star S (Fig. 17.6), whose angular distance from the apex is & and which is at a distance r from the Sun. In a time interval t the star will move away from the apex at the angular velocity u/t = because of the solar motion. In the same time interval, the Sun will move the distance s. The sine theorem for triangles yields

sin u u because the distance remains nearly unchanged and the angle u is very small. In addition to the component due to solar motion, the observed proper motion has a component due to the peculiar velocity of the star. This can be removed by taking an average of (17.1) for

Fig. 17.5. The local standard of rest, defined by the stars in the solar neighbourhood, moves with respect to the galactic centre. However, the average value of the stellar peculiar velocities with respect to the LSR is zero
Fig. 17.6. When the Sun has moved the distance s towards the apex, the direction to the star S appears to have changed by the angle u

a sample of stars, since the peculiar velocities of the stars in the solar neighbourhood can be assumed to be randomly distributed. By observing the average proper motion of objects known to be at the same distance one thus obtains their actual distance. A similar statistical method can be applied to radial velocities.

Objects that are at the same distance can be found as follows. We know that the distance modulus m — M and the distance r are related according to:

where A is the interstellar extinction. Thus objects that have the same apparent and the same absolute magnitude will be at the same distance. It should be noted that we need not know the absolute magnitude as long as it is the same for all stars in the sample. Suitable classes of stars are main sequence A4 stars, RRLyrae variables and classical cepheids with some given period. The stars in a cluster are also all at the same distance. This method has been used, for example, to determine the distance to the Hyades as explained in Sect. 16.2.

Parallaxes based on the peculiar or apex motion of the Sun are called statistical or secular parallaxes.

Main Sequence Fitting. If the distance of a cluster is known, it is possible to plot its HR diagram with the absolute magnitude as the vertical coordinate. Another cluster, whose distance is to be determined, can then be plotted in the same diagram using the apparent magnitudes as the vertical coordinate. Now the vertical distance of the main sequences tells how much the apparent magnitudes differ from the absolute ones. Thus the distance modulus m — M can be measured. This method, known as the main sequence fitting, works for clusters whose stars are roughly at the same distance; if the distances vary too much, a clear main sequence cannot be distinguished.

Photometric Parallaxes. The determination of the distance directly from (17.2) is called the photometric method of distance determination and the corresponding parallax, the photometric parallax. The most difficult task when using this method usually involves finding the absolute magnitude; there are many ways of doing this. For example, the two-dimensional MKK spectral classification allows one to determine the absolute magnitude from the spectrum. The absolute magnitudes of cepheids can be obtained from their periods. A specially useful method for star clusters is the procedure of main sequence fitting. A condition for the photometric method is that the absolute magnitude scale first be calibrated by some other method.

Trigonometric parallaxes do not reach very far. For example, even with the Hipparcos satellite, only a few cepheid distances have been accurately measured by this method. The method of statistical parallaxes is indispensable for calibrating the absolute magnitudes of bright objects. When this has been done, the photometric method can be used to obtain distances of objects even further away.

Other examples of indicators of brightness, luminosity criteria, are characteristic spectral lines or the

periods of cepheids. Again, their use requires that they first be calibrated by means of some other method. It is a characteristic feature of astronomical distance determinations that the measurement of large distances is based on knowledge of the distances to nearer objects.

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