NATIONAL RADIO ASTRONOMY OBSERVATORY COMPUTER DIVISION INTERNAL REPORT

Transcription

1 NATIONAL RADIO ASTRONOMY OBSERVATORY COMPUTER DIVISION INTERNAL REPORT COMMENTS ON THE FREQUENCY TO VELOCITY CONVERSION IN THE 413 CHANNEL AUTOCORRELATION RECEIVER PROGRAMS BY PETER STUMPFF Reprt N. 6 Octber 1968

2 SUMMARY: In the 413 channel autcrrelatin receiver prgrams, frequencies are cnverted int velcities by using the equatin fr the relativistic Dppler effect. The high frequency reslu tin seems t justify this very accurate treatment f the data. I dubt that anything is gained by such a refinement,and I think that nly cnfusin is caused. I prpse t cmpute radial velcities in the classical way, with mderate accuracy, but t calculate and print the redshift parameter z with high accuracy.

3 1. INTRODUCTION In ne f the cmputer prgrams*develped at the NRAO fr the new 413 channel autcrrelatin receiver, the apparent radial velcity, u, is calculated with tht equatin 2 2 v - v u = c-f (1) V + V Here, v is the apparent frequency and v is the rest frequency. Equatin (1) is the rigrus slutin f the equatin i -s v = v c (2) 0 vtr^r 2 y "' c the relativistic representatin f the dppler effect as seen by an bserver wh mves with velcity u alng the line f sight. I d nt knw wh riginally made the request t incrprate the relativistic dppler effect in standard data reductin pr grams. But, as a matter f fact, the existing prgrams carry ut the frequency t velcity cnversin in this manner, and at least sme f the users knw abut it and seem t appreciate it. T my knwledge, all radial velcities published until nw were btained frm bserved wave length r frequency shifts by using the classical dppler effe t equatin. The reasns fr this are quite clear: The precisin f measurements was much t small t detect relativistic effects in mst f the applicatins. An exceptin is extragalactic bjects with large redshifts, but here it is the general practice t discuss the redshift parameter z (which fllws in a direct way frm the bservatins, withut making any hypthesis) csmlgically, rather than radial velcity. The new receiver represents a big step frward in terms f frequency reslutin, and this is certainly the reasn fr the relativistic handling f the bserved frequency: Why shuld we nt include a well knwn effect in ur data reductins if ur measurements are precise enugh t detect it? *AC4142; see Cmputer Divisin Internal Reprt N. 5, sectin VI A.

4 -2- In principle, there is nthing wrng with such a decisin. Hwever, ne must bare in his mind that the inclusin f a relativ istic term in radial velcities is a break with traditin. Such breaks are by n means unusual in astrnmy. Fr example, when enugh evidence existed fr the mdern accuracy f star psitins, secnd rder terms had t be intrduced in the nrmal prcedure f reducing an apparent place t a mean place. Hwever, this change was very carefully discussed and did becme effective nly after internatinal agreement. The reasns fr this carefulness are very much clear: One culd fresee that a large amunt f fundamental data wuld be cllected in the future, t be used fr all srts f basic investigatin, and all these data wuld be affected by such a decisin. I dn't knw hw imprtant it is t take standard cmputer prgrams at NRAO very seriusly, in the sense f what I have just tried t describe. But there is n questin that during the cming years many astrnmers will use these prgrams with the feeling that very accurate bserved data are reduced in a very accurate manner. This seems t be enugh justificatin t ask tw questins: (a) Des the imprvement f frequency reslutin, as given by the NRAO 413 channel autcrrelatin receiver, justify the inclusin f relativistic terms in the frequency t velcity cnversin? (b) What are the principle astrnmical prblems which may ccur if ne leaves the traditinal way? It xs clear that if the answer t questin (a) is "N", we can stp these cnsideratins. We shuld then, f curse, eliminate the relativistic features frm ur standard cmputer prgram^. At least we wuld gain sme cmputer time and cmputer memry. Als, we wuld nt betray urselves in terms f accuracy. If the answer is "Yes", hwever, we shuld be aware that we are ging t break with an ld traditin, that we are planning t make much mre accurate bservatins, that we crrespndingly treat these data in a way different frm the previus ne, and that a later interpretatin f the data might lead t new results which never culd have been btained with the ld equipment and with the ld methds. In ther wrds, if the answer t questin (a) is "Yes", we shuld start thinking abut questin (b).

5 -3-2. RELATIVISTIC VELOCITY TERMS AND THE FREQUENCY RESOLUTION OF THE 413 CHANNEL RECEIVER Xf the surce f, the v - radiatin mves alng the line which cnnects surce and bserver, the rigrus relatin between v, v and u is given by equatin (2). Let us first see what the classical equivalent is. In ptical astrnmy, the s-called redshift parameter z is defined by: K - X v - v *. _ % A v Radi astrnmers seem t prefer a slightly different definitin: v - v * z = -2 (4) R v In a relativistic first rder thery, the tw definitins are cmpletely ec^iivalent. This is clearly seen if we cmpute z and z by using the relativistic equatin (2): z %(V + M-) 3 + (5a) _ U G C C C i -. c u,,u. 2,,,u, 3._.. z R =l - = = --%<-) +%(-) - (Bb) By cnversin f these pwer series we btainr 2 u = cz +^cz - (6a) M * R = cz - 3gcz2 (6b) *In classical physics, these tw definitins crrespnd t the tw cases: (a) Observer rests, surce mves. (b) Surce rests, bserver mves- In relativity, this distinctin becmes meaninglesi

6 Which ne f these tw we are using is nt at all imprtant. Let us use the first ne (6a). Then: C2 R = u <a*««i«l term " 2 ]^cz.. B "relativistic term" R In the fllwing cnsideratins, we assume that the velcities invlved are small enugh t let rel :u 2 R (7) be a reasnable apprximatin f the relativistic term. This is the term which, essentially, des nw exist in the cmputer prgrams. We have t cmpare it t the velcity reslutin f the 413 channel receiver. The receiver gives the fllwing, T c _ Bandwidth "channel spacing" 6f = Number f channels A table f all pssible values f Sf can be fund in NRAO Electrnics Divisin Reprt 75, p. 5. Fr the frequency reslutin we may then write: "frequency reslutin" ~ 6v = a.fif (8) where tha numerical factr a is f the rder f 1. What the actual frequency reslutin is in the case f an bservatin shuld be determined frm the btained line prfile. Let us keep, therefre, the 6v itself as the basic argument. Whereas fiv will be apprximately cnstant ver the entire receiver band, the velcity reslutin (Su) will vary with the bserved frequency (v) : 6v 6u s= c "velcity reslutin" (9) with equatins (7) and (9), we can nw calculate the rati u v n rel 2., T, % ** 6u 2. 6v R K W which cmpares relativistic term and velcity reslutin as lng as z_ is nt t large; if z becmes t large, equatin (7) R, JR.

7 5- is n lnger a reasnable representatin f the relativistic effect This restrictin allws us then, f curse, t ignre the third rder term in the last equatin: I Q = 2-6v -,2 *R (10) We are nw prepared t answer questin (a) in the intrductin The relativistic term becmes detectable if Q ^ 1. This, n the ther hand, will ccur when \z I is larger than a certain limit, r if u ^ u. I mm where u. is a characteristic functin f v and Sv. mm Frm equatin (10), we btain easily: u. = 424 mm 6v km/sec (11) 6v = actual frequency reslutin in KHz, v = rest frequency in GHz. T illustrate this relatin, u. is pltted in the diagram mm f Figure 1. The three curves (actually, straight lines in a lg-lg plt) belng t v 0 = 5000 MHz, 2600 MHz and 1420 MHz. Abscissa is ur actual frequency reslutin, rdinate is ^in (km/sec). The interpretatin f the diagram is simply that, fr a given frequency reslutin, the velcities belw the curves cause a dppler effect s small that the relativistic term culd nt be detected by the receiver. The relativistic term is large enugh fr velcities abve the carve. The scale n the right edge f the diagram shws the km/sec equivalent f the relativistic term. The scale alng the curves gives sme even values f the main term f the ttal dppler effect in MHz; they are, f curse, n different psitins alng the varius curves and are cnnected by the smaller straight lines. An example may explain the use f this figure. We assume that we bserve at v 0 = 1420 MHz (21 cm). The receiver cnfigu ratin may be such that 192 channels are distributed ver an 156 KHz band. The table 1 in the Electrnics Divisin Reprt 75 shws that the channel spacing is 6f = 0.81 KHz. If ur actual

8 rrt> O O O $ ^5 v^ ^ p S3 ^>i "^ ^ «9 e^ ^ *0 b? g ^ t-n ^r- c5 d d d 1? O ^J'.r-l-tTT ib^btetejiimg ^jm^j^i^^^lw^^lm III N in = M-3 u ui X j j }- u u " > 12. < 3 * - ^C, ->_. w ^» JS^ ^^ O O d

9 -6- terequency reslutin is the same, namely 6\i = 0.81 KHz, we btain frm the diagram: %i n ^ 340 km/sec. This means that fr relative velcities smaller than that we need nt include the ^relativistic term which here is^0.1 km/sec as shwn at the right scale. Only fr velcities larger than 340 km/sec, r fr dqppler effects larger than 1 MHz (see the scale alng the curve),, becmes the relativistic effect significant enugh. In ur example, the cnclusin wuld be that In all types f galactic wrk the relativis tic effect is t small t be detected. Fr extragalactic wrk where we deal with much larger vel cities, the relativistic term is significant in practically all pssible receiver cnfiguratins, prvided that the actual fre quency reslutin is f the same rder as the channel spacing. In general, it seems s as if the answer t questin (a) indeed shuld be "Yes", at least fr certain types f applica tins - Sme additinal cmments shuld be made which d nt have anything t d with the receiver. First, all bserved velcities are reduced t the sun, taking int accunt the annual mtin f the earth. In all nrmal cmputer prgrams, this effect is cmputed with an errr f abut tan/sec, because the peridic perturbatins are nt included. They culd be included, f curse, and the errr wuld becme much smaller, but fr a standard data reductin prgram this wuld be much t elabrate. A relativistic term f 0.02 km/sec wuld ccur fr radial velcities f 100 tan/sec. It is very much questinable, therefre, whether it is meaningful t calculate a relativistic term <0.02 fen/pec if, in anther part f the data reductin, an errr f O.4D2 km/sec is intrduced. I have heard peple saying that the standard reductins made at different bservatries differ by even larger amunts, up t 0.3 km/sec. This wuld indicate that a relativistic term becmes meaningful nly fr velcities -^400 km/sec, i.e., definitely nly fr extragalactic wrk. Furthermre, in extragalactic wrk, as far as ne wants t reduce the data t the LSS, the basic slar mtin has t be cmputed. The mtin f the sun relative t the LSR is certainly nt mre precise than abut 5 km/sec, which means that t include relativistic terms fr velcities belw 200D tan/sec is meaningless.

10 -7- Ignring these difficulties which might be slved eventually in ne way r the ther, we have seen that the new receiver is capable, under certain circumstances, t detect the relativistic term f the dppler effect. This cnclusin brings me then t sectin 3 where I will try t discuss sme f the prblems which are invlved in the relativistic dppler effect, and which are usually verlked if ne des nt really try t understand the definitins f special relativity.

11 -8-3. RELATIVISTIC DOPPLER EFFECT AND ABERRATION,, AMP THE USUAL ASTROHOMIC&L DEFINITION OF RADIAL VELOCITIES Since sectin 2 f this reprt has shwn that relativistic effects start becming significant, we will discuss these effects a little bit mre generally. We are talking, f curse, abut little effects in mst practical cases? but, as I said in the intrductins Either the effects are s little that we dn't like t talk abut them, in which case we simply shuld nt include than in ur reductinsi r, we feel they have t be included in the reductins althugh they are s little. Then, whether we like that r nt, we must at least understand what they mean. In rder t d s, we assume nw the psitin that we -are capable f measuring everything with "unlimited accuracy". Figure 2 shws what happens. Since the bserver, T, is very far away frm the surce, S, the electrmagnetic wave can be apprximated by a plane wave when it arrives at T. If the bserver sslt-!e wuld be in rest relative t B, he wuld find S t be the prpagatin vectr f the wave, and wuld describe the electric vectr f the field by 1=2. exp 2TTi J- 3C. csa + Y csb + z cse -4 If T, hwever, mves alng the x-axis with velcity u, the bserver wuld find s t be the prpagatin vectr, and wuld, crrespndingly, define the field by E t- exp L 2TTiv v XcsA + YcsB + ZcsC -}]. We have chsen an arbitrary relative mtin, u, and then have defined the crdinate systems such that: ; This is n limitatin t generality.* The relatin between v, v, A, A, B, B, C, C is fund by intrducing the Lrentz transfrmatin int the first f the tw equatins:

12 z ^b t^x. Fig. 2 S = Surce, resting in system x, y, z T = Observer, resting in system x, y, z s = Prpagatxn vectr at T as seen by an bserver resting m x, y, z ~ Prpagatin vectr at T as seen by an bserver resting in x, y, z u = Mtin f x, y, z relative t x, y, z The angles 180~A, 180-B, 180-C describe the "apparent" psitin f the surce. The angles 180-A, 180-B, 180-C describe the I, true"psitin f the surce. The difference between the tw psitins is the relativistic aberratin.

13 -9- where x + ut t : y; z_ «z; t = c AfT^^p ' Vi - P c (12) (13) This gives us the relativistic descriptin f dppler effect and aberratin; V = V csa «= csb = csc = 1 - P csa / 2 * A/1 - P^ csa - p 1 - P csa 0 csb 'yl - p 2 H 1 - P csa csc A/I - P 2 ' (14) (15) (16) (17) One shuld nte that the angle which enters (14) is nt the "bserved" directin but the directin in the system f rest Frm equatin (15) we can derive: csa csa + P 1 + P csa (15a) If we substitute this in equatin (14), we btain the equivalent frmula: V = (14a} Frm (14) and (14a) we get Z R = V - V V P csa Vi ^""l (18) = 1 (18a)

14 -10- Fr further discussins it will be helpful t develp sme f these expressins in pwer series: csa = csa - p sin A - p csa sin A - p cs A sin A } i (19) VJ w I csa = csa 4 Psin A - p csa sin A 4 p cs A sin A-*- } 2 3 iz = P csa - %p 4 ^csa * p (20) < R t z^ = P csa - %p (2cs 2 A - 1) 4 %p 3 csa (2cs 2 A - 1) (20a) i R The next step, in ur cnsideratin, must be t find a rela tin between the term "Radial Velcity" (as used until nw in astrnmy) and the relativistic dppler effect. In rder t gain a clear picture f this relatin, we shuld ignre the varius standard crrectins which nrmally are applied t bservatins such as diurnal crrectin, annual crrectin and crrectin due t basic slar mtin. In ther wrds we assume that the bserver is identical with the lcal standard f rest. The mean psitin , fr instance, describes the directin t the surce (in the 1950 crdinate system). Such a"mean psitin" is, actually, an "apparent" psitin because all ur psitin catalgues cntain still the individual aberratin f the surces (caused by the mtin f the surce relative t the lcal standard f rest). Even if we wanted, we culd nt crrect fr thip individual aberratin because we d nt knw the directin f the spatial mtin. The term "Radial Velcity" is defined as the prjectin f space velcity n the line f sight. In ur ntp.tin, this means that Radial Velcity V = u-csa. (21) With the new abbreviatin it fllws that f =- (22) M C P = ^seca. (23)

15 -11- Frm equatin (18a) we btain then the rigrus relatin between frequency shift and radial velcity: R ^2 2^ -JT- 1-6 sec A 1 4 r (24) Or, frm equatin (20a): z R = - hf(2 - sec 2 A) 4 ^3(2 Inversin f this series gives 2 2 t" = z fi 4 %z (2 - sec A) " * * 2. sec A) (25) (26) r Radial Velcity V = V +AV c re js V = c*z^ (classical term) c R 2 2 4^V. = %c(2 - sec A) z (relativistic term) re i R UJR. *-. fc-jm^ M (27) In sectin 2 we saw that in the existing cmputer prgrams, the relativistic term instead was calculated by u rel H C 25' i.e., in the naive applicatin f special relativity, the classical radial velcity is crrected by a term which invlves the assump tin that the surce has n tangential mtin. The difference between the naive and the crrect relativistic terms is determined by the functin 2 - sec A which is given in the next table fr sme values f A: A sec A 0 e, * 10, b ; , i , B0? ? ,

16 -12- If the space velcity is inclined by^s** : 2 - sec^a = 0, i.e., in this case disappears the relativistic secnd rder term. These cnsideratins fail, f curse, if A is in the neighbrhd f 90 Q. If the true spatial mtin wuld be perpendicular t the "true" line f sight, it. i.e.,.e.. if A^ A - = = 90, then we btain frm equatin (18): w =3 1 - Z R 1 1 -yr - P2' (28) Ithich is the "transversal dppler effect" (blue shift). (15) shws bhat under this cnditin "Equatin csa «-P, (29) i.e., the individual aberratin f the surce reaches its maximum value. Fr small space velcities, csa wuld still be small and, -therefre, sec A wuld be large. Cnvergence f the series in equatin (26) exists nly if bth yand seca are nt t large, i.e., ifysmall andlcsal^l. The latter cnditin, hwever, means that 1 csa^l^l, t, and this excludes the special case A - 90 which was just cnsidered. These cnsideratins shw that it is nt pssible t apply a relativistic crrectin withut knwing the space directin f Ehe relative mtin, r withut measuring the "individual aberra tin" Applicatin f equatin (1), r inclusin f the term %cz 2, implies the assumptin A = 0 r 180. Perhaps this is a gd assumptin in a statistical sense fr a large sample f surces, but fr a statistics we can ignre the small relativistic crrectin anyway. Sme remarks shuld be made abut a cnnected prblem althugh this has certainly n practical meaning at the present. Let us assume theit ur z * s were very accurate, even, in the lw velcity R range (u 100 km/sec). Let us further assume that we wuld knw the directin f u. In this case, we culd apply the relativistic crrectin prperly. Hwever, we wuld then f curse als have t pay attentin t the fact that fr the annual velcity crrectin and fr the basic slar mtin crrectin the Einstein therem f velcity additin must be applied rather than the classical

17 -13 vectr additin. Otherwise we wuld again intrduce an errr in the relativistic terms. Let us hpe that relativistic crrectins in the lw velcity range never becme imprtant! Fr high velcities, f curse, they are imprtant, but this is a different thing. Csmlgists discuss z r z rather than radial velcities. Therefre, it might be a gd idea ir the standard data reductin prgrams wuld print accurate values f z and/r z - quantities which can be derived withut hyptheses directly frm bservatins and which will directly reflect the accuracy f the bservatins. The accuracy f z is given by til 6v -6 I AzJ " R< = v.10 where 6v = actual frequency reslutin in KKz, and v = rest fre quency in GHz. The highest pssible accuracy fr z, in the case f the 413 channel receiver, wuld be btained with 6v = 0.1 KHz, v =1.42 GHz : \Az j = 7.10~ f R 1 If the data reductin prgrams wuld cmpute classical velcities with mderate accuracy, but in additin give values f z and/r z j with 7 decimal places, they wuld present useful and clearly defined data fr further discussins. j The parameter z is preferable because this is the ne which has been in use since Hubble's time. The nly difference between z and z R is that if P varies frm -1 t ±?, z ges frm -1 t 4CD whereas z ges frm - t f1. R