Positional Astronomy

The position of a star can be measured either with respect to some reference stars (relative astrometry) or with respect to a fixed coordinate frame (absolute astrometry).

Fig. 2.22. Astronomers discussing observations with the transit circle of Helsinki Observatory in 1904



Absolute coordinates are usually determined using a meridian circle, which is a telescope that can be turned only in the meridional plane (Fig. 2.22). It has only one axis, which is aligned exactly in the east-west direction. Since all stars cross the meridian in the course of a day, they all come to the field of the meridian circle at some time or other. When a star culminates, its altitude and the time of the transit are recorded. If the time is determined with a sidereal clock, the sidereal time immediately gives the right ascension of the star, since the hour angle is h = 0 h. The other coordinate, the declination 8, is obtained from the altitude:

8 = a - (90°-$), where a is the observed altitude and $ is the geographic latitude of the observatory.

Relative coordinates are measured on photographic plates (Fig. 2.23) or CCD images containing some known reference stars. The scale of the plate as well as the orientation of the coordinate frame can be determined from the reference stars. After this has been done, the right ascension and declination of any object in the image can be calculated if its coordinates in the image are measured.

All stars in a small field are almost equally affected by the dominant perturbations, precession, nutation, and aberration. The much smaller effect of parallax, on the other hand, changes the relative positions of the stars.

The shift in the direction of a star with respect to distant background stars due to the annual motion of the Earth is called the trigonometric parallax of the star. It gives the distance of the star: the smaller the parallax, the farther away the star is. Trigonometric parallax is, in fact, the only direct method we currently have of measuring distances to stars. Later we shall be introduced to some other, indirect methods, which require

^ Fig. 2.23. (a) A plate photographed for the Carte du Ciel project in Helsinki on November 21, 1902. The centre of the field is at a = 18 h 40 min, S = 46°, and the area is 2° x 2°. Distance between coordinate lines (exposed separately on the plate) is 5 minutes of arc. (b) The framed region on the same plate. (c) The same area on a plate taken on November 7, 1948. The bright star in the lower right corner (SAO 47747) has moved about 12 seconds of arc. The brighter, slightly drop-shaped star to the left is a binary star (SAO 47767); the separation between its components is 8"

Fig. 2.24. The trigonometric parallax n of a star S is the angle subtended by the radius of the orbit of the Earth, or one astronomical unit, as seen from the star

certain assumptions on the motions or structure of stars. The same method of triangulation is employed to measure distances of earthly objects. To measure distances to stars, we have to use the longest baseline available, the diameter of the orbit of the Earth.

During the course of one year, a star will appear to describe a circle if it is at the pole of the ecliptic, a segment of line if it is in the ecliptic, or an ellipse otherwise. The semimajor axis of this ellipse is called the parallax of the star. It is usually denoted by n .It equals the angle subtended by the radius of the Earth's orbit as seen from the star (Fig. 2.24).

The unit of distance used in astronomy is parsec (pc). At a distance of one parsec, one astronomical unit subtends an angle of one arc second. Since one radian is about 206,265", 1pc equals 206,265 AU. Furthermore, because 1 AU = 1.496 x 1011 m, 1 pc ~ 3.086 x 1016 m. If the parallax is given in arc seconds, the distance is simply r = 1/n, [r ] = pc, [n ]=" . (2.35)

In popular astronomical texts, distances are usually given in light-years, one light-year being the distance light travels in one year, or 9.5 x 1015m. Thus one parsec is about 3.26 light-years.

The first parallax measurement was accomplished by Friedrich Wilhelm Bessel (1784-1846) in 1838. He found the parallax of 61 Cygni to be 0.3". The nearest star Proxima Centauri has a parallax of 0.762" and thus a distance of 1.31 pc.

Fig. 2.25a-c. Proper motions of stars slowly change the appearance of constellations. (a) The Big Dipper during the last ice age 30,000 years ago, (b) nowadays, and (c) after 30,000 years

Fig. 2.25a-c. Proper motions of stars slowly change the appearance of constellations. (a) The Big Dipper during the last ice age 30,000 years ago, (b) nowadays, and (c) after 30,000 years

In addition to the motion due to the annual parallax, many stars seem to move slowly in a direction that does not change with time. This effect is caused by the relative motion of the Sun and the stars through space; it is called the proper motion. The appearance of the sky and the shapes of the constellations are constantly, although extremely slowly, changed by the proper motions of the stars (Fig. 2.25).

The velocity of a star with respect to the Sun can be divided into two components (Fig. 2.26), one of which is directed along the line of sight (the radial component or the radial velocity), and the other perpendicular to it (the tangential component). The tangential velocity results in the proper motion, which can be measured by taking plates at intervals of several years or decades. The proper motion p has two components, one giving the change in declination and the other, in right ascension, pa cos 8. The coefficient cos 8 is used to correct the scale of right ascension: hour circles (the great circles with a = constant) approach each other towards the poles, so the coordinate difference must be multiplied by cos 8 to obtain the true angular separation. The total proper motion is

Fig. 2.26. The radial and tangential components, vr and vt of the velocity v of a star. The latter component is observed as proper motion

The greatest known proper motion belongs to Barnard's Star, which moves across the sky at the enormous speed of 10.3 arc seconds per year. It needs less than 200 years to travel the diameter of a full moon.

In order to measure proper motions, we must observe stars for decades. The radial component, on the other hand, is readily obtained from a single observation, thanks to the Doppler effect. By the Doppler effect we mean the change in frequency and wavelength of radiation due to the radial velocity of the radiation source. The same effect can be observed, for example, in the sound of an ambulance, the pitch being higher when the ambulance is approaching and lower when it is receding.

The formula for the Doppler effect for small velocities can be derived as in Fig. 2.27. The source of radiation transmits electromagnetic waves, the period of one cycle being T. In time T, the radiation approaches the observer by a distance s = cT, where c is the speed of propagation. During the same time, the source moves with respect to the observer a distance s' = vT, where v is the speed of the source, positive for a receding source and negative for an approaching one. We find that the length of one cycle, the wavelength X, equals

Fig. 2.26. The radial and tangential components, vr and vt of the velocity v of a star. The latter component is observed as proper motion

Fig. 2.27. The wavelength of radiation increases if the source is receding

If the source were at rest, the wavelength of its radiation would be X0 = cT. The motion of the source changes the wavelength by an amount

AX = X - X0 = cT + vT - cT = vT , and the relative change AX of the wavelength is

AX v

X0 c

This is valid only when v ^ c. For very high velocities, we must use the relativistic formula

If x is given in arc seconds per year and r in parsecs we have to make the following unit transformations to get vt in km/s:

Hence vt = 4-74 ¡xr , [vt] = km/s , M = 7a, [r] = pc •

The total velocity v of the star is then v = V vr2 + v2 •

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