Doppler shifts in astronomy

Transcription

1 7.4 Doppler shift 253 Diide the transformation (3.4) by as follows: = g 1 bck. (Lorentz transformation) (7.43) Eliminate in the right-hand term with (41) and then inoke (42) to yield = g (1 b cos u). (7.44) Sole for the S-frame frequency by =. (Doppler shift; in terms of u) (7.45) g (1 b cos u Because = 2πn, and = 2πn 0, this epression is identical to our earlier result (37). We will find it useful below to epress the, relation in terms of u, the propagation angle in S. The Table 7.3 transformation (3.8) yields, after similar manipulations (Prob. 42), the frequency obsered in S at angle u as a function of the frequency in S as follows: = g (1 + b cos u ). (Doppler shift; in terms of u ) (7.46) It is identical in form to (45) ecept for the sign preceding b. This arises because, in S, the frame S moes down the negatie -ais, whereas, in S, frame S moes down the positie -ais. The factor b is, we again state, taken to be a positie quantity. Doppler shifts in astronomy The frequencies of spectral lines from celestial sources are often shifted owing to the motions of the emitting object: gaseous clouds, stars, or galaies. The Doppler shifts may be due, for eample, to the orbital motions of a star in a binary system, to the motions of stars in the Galay relatie to the sun, to the motions of stars in other galaies, and to galay recession in an epanding unierse, though the latter is more properly iewed as a cosmological redshift. In most cases, the elocities are sufficiently low that the classical Doppler shift is adequate. Howeer, relatiistic particles hae been discoered in a host of objects such as protostars, binary systems, supernoa remnants, and etragalactic jets. Also, etragalactic objects in the epanding unierse, such as quasars, can be sufficiently distant to hae relatiistic recession speeds. Astronomical sign conention In the classical Doppler shift, the obsered shift of frequency reflects only the component of the elocity along the line of sight, the radial component r. The astronomical conention is that r be positie if it is directed outward and negatie if it is directed inward. Let us further define b r r /c. Here, both r and b r carry sign, unlike b (see our preious discussion). The classical Doppler shift (34) thus takes the form n n 0 n 0 = r c, or n n 0 = 1 b r, ( b r 1; b r > 0 for recession) (7.47)

2 254 Special theory of relatiity in astronomy where n and n 0 are again the obsered and emitted frequencies, respectiely, and where r is the radial component of the elocity. Optical astronomers often work in waelength units rather than frequency units. For a acuum, the relation l = c/n yields l/ = n 0 /n, giing, for b r 1, l = 1 + b r. (b r 1) (7.48) The relatiistic ersion of this is, for strictly radial motion, modified from (38) for the astronomical sign conention (b r > 0 for recession), l 1 + 1/2 br. (Waelength ratio) (7.49) 1 b r The obsered waelength l in (49) is greater than the emitted waelength for a receding emitter. As b r 1, the ratio l/ can grow indefinitely (i.e., as recessional speed r approaches c). In this case, the radiation is greatly reddened and the frequency decreases toward zero, as do the photon energies hn. Redshift parameter The optical spectra of distant luminous objects in the unierse called quasars hae spectral lines shifted by large amounts to lower frequencies. If these redshifts are interpreted as Doppler shifts, they indicate recession elocities approaching the speed of light. These elocities are due to the epansion of the unierse; the epansion is such that the more distant the object, the faster it recedes (AM, Chapter 9). Astronomers define the redshift parameter z as z l = l 1, (Definition of redshift z) (7.50) where again is the emitted waelength. Substitute the ratio of waelengths (49) into this, 1 + 1/2 br z + 1 =, (Redshift-speed relation) (7.51) 1 b r to obtain a relation between z and recessional speed for our Doppler-shift interpretation of the redshift. As the recession speed approaches c, b r 1 and z increases indefinitely. The most distant quasars known are at redshifts z 6. At this redshift, l/ = 7; the obsered waelength is seen times the rest waelength in the quasar frame! An ultraiolet emission line at = nm (Lyman a) would be shifted almost into the near infrared at 850 nm. In this case, the relation (51) yields a speed factor b r = The quasar is receding at 96% the speed of light. We remind the reader that special relatiity is not really appropriate to our unierse with its changing rate of epansion. In general relatiity, redshifts are not iewed as Doppler shifts. Instead they are intrinsic to the epansion of the unierse. The proper iew is that light waes (and hence their waelengths) are stretched as they trael through the epanding intergalactic space en route to earth from the quasar. The relation (51) would apply only to a freely epanding unierse with no matter content.

3 7.5 Aberration 255 On the other hand, the redshift parameter z (50) is a widely used obserational parameter that is independent of any theory of the epansion. Cosmologists construct theories to match the measured redshifts of distant galaies. In the following discussions, we will continue to use the self-consistent Lorentz relations wherein the speed factor b is a positie definite quantity insofar as S moes down the positie -ais of S. 7.5 Aberration The apparent direction of radiation will differ according to obserers in two frames of reference that are moing with respect to each other. This phenomenon is known as aberration. This is not the transerse bunching of electric field lines (Fig. 7.3a,b). Rather, aberration refers to the propagation directions of electromagnetic waes. Aberration results in displaced positions on the celestial sphere (stellar aberration) and in the beaming of radiation from astronomical jets. One can eplain stellar aberration with a simple classical argument (AM, Chapter 4), but now we hae the tools to calculate the aberration properly in the contet of special relatiity. Transformation of k direction The task is simply to compare the angle of the propagation ectors measured in S and S; that is, u and u, respectiely (Fig. 7.4). The comparison of angles in the two frames is obtained through the Lorentz transformations for k, (Table 7.3). The dispersion relation for radiation in a acuum (41) is alid in any inertial frame, and so we can write the equality k = k = c, (Dispersion relation) (7.52) where k = (k 2 + k y 2 ) 1/2, k = (k 2 + k y 2 ) 1/2, and k z = k z = 0. The quantities of interest are cos u = k k ; cos u = k k. (7.53) Inoke the transformation for k (3.5) of Table 7.3 and diide by k to yield k k = g + b. (7.54) k k c k Apply (52) to the first and last terms of (54) as follows: k k = g + b. (7.55) k k Eliminate / with the Doppler relation (46) and epress the ratios k /k and k /k as the cosine functions (53) cos u = cos u + b. (Transformation of k ector directions) (7.56) 1 + b cos u

4 256 Special theory of relatiity in astronomy This is the desired epression that relates the directions of k and k. This relation does not depend on the location or frame of the emitter. It simply compares the directions of any bit of propagating radiation according to obserers in two frames, S and S. The angle in S can be calculated from the angle in S (emitter frame) and the speed parameter b. This result could also hae been obtained directly from the two Doppler relations (45) and (46) (Prob. 51). The general effect of (56) is to rotate a propagation ector of radiation emanating from a source toward the forward direction of the source motion, and so u < u or cos u > cos u (Prob. 53). The following limiting cases of aberration are easily etracted from (56): b = 0 cos u = cos u (7.57a) u = 90, 270 cos u = b (7.57b) b 1 cos u 1. (7.57c) In the first case, at b = 0, the two obserers are effectiely in the same inertial frame, and thus the angles u and u do not differ. The second case will be applied to stellar aberration as an illustratie eample. The third reeals intense beaming, or the headlight effect, which is an etreme case of aberration. We discuss these latter two cases, respectiely, in the net section immediately below and in Section 6 (under Beaming ). Stellar aberration The earth s motion in its orbit about the sun leads to changes in the propagation direction of starlight. This causes star positions to appear slightly displaced ( 20 ) from their actual positions in the sky. The magnitude and direction of this displacement for a gien star change throughout the year and hence are easily detectable. The effect is small because the earth s orbital speed is only 29.8 km/s, which is much less than the speed of light. Earth as stationary frame To be in accord with our preious eamples, we first place the emitting star in the moing frame S and the earth obserer in the stationary frame S (Fig. 7.5a). Consider the limiting case (57b) in which the radiation is emitted at eactly 90 or 270 in S and the angle of k (in S) is specified by cos u = b, a positie alue. Thus, in S, the k ector is rotated forward of 90, toward positie, as shown. In our case, = km/s, and b = Because the angle u 90 is small, we write cos u = sin (u 90 ) (u 90 ). Taking care to use radians, we hae from (57b) (u (π/2)) b, or u = π 2 b = π 2 ( ) rad = (Stellar aberration at u = 90 ) (7.58) The propagation ector in S is rotated from the ertical by 20.5 in the direction of the source motion. To receie such radiation, a telescope on S (Fig. 7.5a) would hae to be pointed opposite to the propagation ector and tilted to the left, as shown. This is the stellar aberration effect for a star that lies 90 from the star elocity direction. The effect at other angles could be obtained here, but let us first reframe the discussion.

5 7.5 Aberration 257 (a) Obserer in S y S (b) Obserer in S y S k Star k measured by S obserers y S k = 90 k measured by S obserers y S * Star k k = Fig. 7.5: Stellar aberration. (a) Obserer at rest in stationary S frame and emitting star in moing S frame. The k ector that lies normal to the direction of motion of S is rotated in S by a small amount toward the direction of the S motion. The rotation is 20.5 if S moes at the speed of the earth in its orbit about the sun; b earth = The telescope in S is tilted toward the left to intercept the ray (i.e., in the direction S moes relatie to S ). (b) Deity iew with star at rest in stationary S frame at eleation c. The earth and astronomer are at rest in a (moing) S frame with the star at eleation c. Again, the telescope must be tilted in the direction of the (earth) motion relatie to the star frame (S) in this case to the right to the smaller angle c < c. Stars as stationary frame The stellar aberration problem is usually described from the point of iew of an obserer at rest with respect to the stars. In this iew, the astronomer is on a moing earth that is rushing through starlight from a stationary source. Also, conentionally, the angles are defined to be the directions to the star, or more precisely, the negatie of the propagation directions k and k. Let us then use our standard frames S and S, placing the star in the stationary S and the astronomer in the moing S and further defining the angles c and c to the star (Fig. 7.5b). Because the S frame moes down the positie -ais of S, the transformation of the k directions in (56) remains alid. Again, we desire to find the astronomer s angle in terms of the angle in the emitter frame. Thus, we need to modify (56) to gie cos u in terms of cos u. One can sole (56) for cos u, or equialently, interchange primes and nonprimes while reersing the signs preceding each b. The directions u and c in Fig. 7.5b are directly opposed, and so c = 180 u and c = 180 u, giing cos u = cos c and cos u = cos c. These ariable changes applied to the epression for cos u just described yield (Prob. 52) cos c = cos c + b, (Transformation of direction to a star) (7.59) 1 + b cos c