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6.1. Radial-Velocity Corrections for Earth Motion
6. COMPUTER PROGRAMS FOR RADIO ASTRONOMY
6.1. Radial-Velocity Corrections for Earth Motion* 6.1.1. Introduction
In 1842, Christian Doppler published a paper quantitatively describing the relationship between line-of-sight velocities and wave frequencies. At the time, the wave theory of light introduced by Huygens was becoming accepted over Newton’s corpuscular theory because of its ability to explain interference phenomena. Doppler was investigating implications of the wave nature of light. Using the analogy of wave trains in the ocean moving toward (or against) a moving ship, Doppler derived formulas relating the wave frequency perceived by an observer on a moving ship to that seen by a stationary observer. He correctly noted that the shift in frequency depends not only upon the relative velocity between the wave source and the ship, but also upon whether the ship moves with respect to the wave source, or the source with respect to the ship. For if the ship moves with a velocity v, toward the wave source, and if the waves move through the ocean with a velocity v, then the effective velocity of the wave train with respect to an observer aboard ship would be v, + u, and the observed frequency f of the wave train is
f = (0, + 41no = f o U + vslv),
(6.1.1)
where fo and l o are the actual frequency and wavelength, respectively, of waves emitted by the source. Thus, the observer would perceive a higher wave frequency owing to the ship’s motion. If the wave source moves toward the stationary ship, the physical separation between wave crests would decrease to l o ( l - v,/v), and the observer would also perceive a higher wave frequency
f = V I A =fo/(l
- v,lv),
(6.1.2)
but different than that given by Eq. (6.1.1). Such frequency changes due to relative velocities between source and observer are known categorically as Doppler efsecrs. It is important to note that the effect here is quantitatively ~~~
* Chapter 6.1 is by
M. A. Gordon 277
278
6.
COMPUTER PROGRAMS FOR RADIO ASTRONOMY
different depending upon whether the source moves relative to the observer, or the observer moves relative to the source. Doppler believed the equations describing water waves also described electromagnetic waves propagating through the ether, and, in his paper, did not hesitate to apply these equations to astronomy. He suggested that all stars had the same intrinsic color as the sun, and that the observed variations in the color of stars reflected not differences in their temperature and chemical composition, but differences in their radial velocities with respect to the earth. This suggestion created bitter opposition to his ideas, which persisted long after Doppler’s death, until careful spectroscopic observations showed that there were indeed differences in the temperature and composition of stars, and that differences in apparent color were not due wholly to velocity effects. (Fizeau had privately pointed out that, owing to the continuous nature of stellar radiation, Doppler effects probably could not cause significant changes in apparent colors of stars.) 6.1.2. Special Relativity
Doppler’s equations assume fundamental concepts regarding the vector interaction of the velocities of light, source, observer, and medium (ether). In his 1849 experiment to measure the velocity of light, Fizeau investigated, but failed to find, any effect of the earth’s motion upon the velocity of light. Yet, Doppler’s ideas required variations in the effective propagation velocity. Such early experiments cast doubt as to whether light waves propagated in ether the same way as ocean waves in water and acoustic waves in gas. In 1887, Michelson and Morley performed their sophisticated experiment which indicated that the velocity of light was a constant, unaffected by vector addition or subtraction of the earth’s velocity. These results led Lorentz (and, independently, FitzGerald) in 1895 to propose that one leg of the MichelsonMorley apparatus changed length as a function of velocity such that, if the velocity of light indeed changed because of the earth’s motion, the arm of the measuring device would change length correspondingly, thus preventing the investigators from detecting the changes in the velocity of light. This suggestion led Lorentz to consider further the nature of velocity and electromagnetic phenomena, which in 1904 resulted in a paper which paralleled the more complete exposition by Einstein. In 1905, Einstein published his now famous paper in which he postulated that the velocity of light was independent of the motion of its source. In his discussion, he derived the “ Doppler effect ” of special relativity to be f=fo
1 - (u/c) cos 2
[1 - ( 0 I c
2
)I
f#J
112’
(6.1.3)
6. I .
RADIAL-VELOCITY CORRECTIONS FOR EARTH MOTION
279
where 4 is the angle between the velocity and the line of sight between source and observer, v is the velocity of the source in the observer’s reference frame, and c is the speed of light. Unlike Doppler’s original formulation, this equation always predicts a frequency shift even if the light source (or observer) moves at right angles (cos (p = 0) to the line of sight joining the two objects. If the source and observer move toward each other, cos (p = 1 , and Eq. (6.1.3) becomes
(6.1.4) which is quite different from Doppler’s original formulation given in Eqs. (6.1.1) and (6.1.2). This difference results from the fact that light waves and water waves do not propagate in the same way. 6.1.3. Conventional Tabulation o f Redshifts
Astronomical spectroscopy began considerably before Einstein’s paper appeared, and thus wavelength (frequency) shifts of spectral lines are always catalogued as velocity effects calculated by Doppler’s original formulations, with the assumption that sources move relative to the earth. Optical astronomers calculate radial velocities from their most basic measurements : the dispersion of wavelengths on a spectroscopic plate. Hence, = c[(A - AO)/AOI>
uop1
(6.1.5)
where uopl is positive for a receding source if A is the observed wavelength. The astronomical redshift parameter z is defined to be
z
= (A - A())/&.
(6.1.6)
The system generally used by radio astronomers differs from that used by optical astronomers. Presumably, the difference stems from the fact that, unlike the case for the optical spectrometer, the radio spectrometer sorts information by frequency. Hence, the convention arose that uradio
= c[(fO
-f)lfol.
(6.1.7)
To date, neither optical nor radio astronomers conventionally use special relativity [Eq. (6.1.4)] to calculate radial velocities from observations. Because of the differences between radial velocities quoted by radio and optical astronomers, errors’ are apt to result when one uses optically determined redshifts of high velocity objects like quasars to position narrow-band radio spectrometers. And, occasionally, uopt or z are calculated with a comC . Heiles and G . K. Miley, Asfrophys. J . 160, L83 (1970).
280
6.
COMPUTER PROGRAMS FOR RADIO ASTRONOMY
bination of rest wavelengths measured in a vacuum (Ao) and observed wavelengths measured in the air (A) of the astronomical spectrography. Finally, with low-dispersion optical spectrographs, the “ line ” spectra are in fact often blends of several lines. Thus, radial velocities calculated from blends erroneously presumed to be single transitions often contain substantial errors. This problem is a well-known one, and a variety of techniques have been used to minimize resulting errors. 6.1.4. The Aberration of Light
Early astronomers noted that the stars seemed to change position with season. This effect arises because the motion of the earth changes the aspect angle of an observer with respect to the direction of arrival of rays from stars. If 8‘ is the apparent angle between the direction of the earth’s velocity u and the star, and 0 is the actual angle, simple trigonometry shows that tan 8’ = sin O/[cos 8
+ (u/c)].
(6.1.8)
where c is the velocity of light. For the most adverse case, position errors as large as 41 arc sec can occur. 6.1.5. Velocity Reference Frames
In considering the effects of relative velocities of source and observer, we must choose some standard reference velocity. The observer moves on the surface of the earth, which moves around the moon-earth barycenter, which moves around the solar system barycenter, which moves with respect to local stars, which move around the galactic center, etc. For standardization, the observer usually references his velocity measurements to one of two conventional reference velocities. 6.1.5.1. Heliocentric System. Perhaps the most obvious velocity-reference system is that of the sun. In the late nineteenth century, when stellar spectroscopy began, the orbital elements of the earth’s motion about the sun were known to accuracies considerably greater than could be measured from stellar absorption spectra. Consequently, radial velocities measured from the earth could be easily referenced to the sun without loss of accuracy. As mentioned in Section 6.1.3, one of the largest sources of error in these measurements is due to the possibility of unresolved blends of spectral lines. The relative intensity of blended components is probably the same amongst stars of the same spectral type, and here the relative differences in radial velocities can be accurately measured. For example, the radial velocities of F and B stars can be measured relative to the sun with considerable accuracy. Stars of other types may have blends of different relative intensities to those of the sun, and serious measurement errors can arise because of the differences in line shapes.
6. I .
RADIAL-VELOCITY CORRECTIONS FOR EARTH MOTION
28 I
The intrinsic absolute accuracy of optical radial velocities measured from stellar absorption spectra is therefore limited to perhaps something less than 1 km/sec, even for measurements taken by the best spectrographs by the most experienced observers. (Measurement errors of 0.2 km/sec are often quoted, but such numbers refer to precision rather than accuracy.) Such accuracy does not warrant distinction between the sun and the solar system barycenter. Thus, the term heliocentric velocity assumes the sun to be the focus of the earth's orbital motion. (Motion with respect to the solar-system barycenter is discussed in Section 5.6.2.) The reader may wish to refer to work by Campbell and Moore2 and by Petrie3 for more detailed information about stellar velocity measurements, and to work by Herrick4 for a convenient table to reduce radial velocities to the sun. 6.1.5.2. Local Standard of Rest. Analysis of velocities determined by stellar spectroscopy shows that the sun has a systematic motion with respect to neighboring stars. The situation is not simple, because this systematic motion is also a function of the spectral type of stars used for the radial velocity measurements. If we define a kinematic reference frame moving with the average velocity of these neighboring stars, then this local standard of rest can be said to be a function of stellar type. Clearly, some convention is necessary. Standard solar motion is defined to be the average velocity of spectral types A through G as found in general catalogs of radial velocity, without regard to luminosity class. Here, the motion of the sun is 19.5 km/sec toward lSh right ascension (RA) and 30" declination (6) for the epoch 1900.0, which corresponds to galactic coordinates ( I , 6) of (56", 23"). Basic solar motion' is the most probable velocity of stars in the solar neighborhood, thereby being heavily weighted by the radial velocities of the most common spectral types A , gK,and dM in the vicinity of the sun. In this system, the sun moves at 15.4 km/sec toward the direction (1, 6 ) of (51", 23"). The conventional reference frame used for galactic studies is essentially that of standard solar motion. The convention of Local Standard of Rest (LSR) assumes the sun to move at the rounded velocity of 20.0 km/sec toward 18h RA and 30" 6 (1900.0). Since the stars used in the determination include the earlier spectral types of the distribution, their ages are comparatively younger, and, presumably, their velocities are closer to that of the interstellar gas. For some purposes, it is necessary to know the motion of the Local Stan-
* W. W. Campbell and J.
H. Moore, Publ. Lick Obseru. 16 (1928). R. M. Petrie, in "Basic Astronomical Data" (K. A. Strand, ed.), p. 64. Univ. of Chicago Press, Chicago, Illinois, 1963. S. Herrick, Jr., Lick Obs. Bull. 470, 17, 35 (1935). A. N. Vyssotsky and E. N. Janssen, Asrron. J . 56, 58 (1951).
282
6.
COMPUTER PROGRAMS FOR RADIO ASTRONOMY
dard of Rest about the galactic center. Observations6 of globular clusters show the sun to move at 167 & 30 km/sec toward an (l, b) of (90", 0"). (With these uncertainties, the motions of the sun and the Local Standard of Rest are indistinguishable.) Similar observations' of galaxies in the Local Group show the sun to move at 292 If: 32 km/sec toward (106", -6"). The reason for the difference between the two velocities is unclear, but based on such observations, the Local Standard of Rest is usually assumed to move at 250 km/sec toward (90", 00) about the Galactic Standard of Rest. 6.1.6. Calculation of Radial Velocities
The calculation of the radial velocity between a position on the earth's surface and a distant astronomical object involves a number of velocity terms. Just how many terms need to be included is determined by the accuracy required. Table I lists a number of these terms and their approximate maxiTABLE I. Velocities Involved in the Radial-Velocity Computation
Component Source with respect to Local Standard of Rest (LSR) LSR with respect to solar system barycenter Solar system barycenter with respect to earth-moon barycenter Earth-moon barycenter with respect to earth center Earth center with respect to telescope Planetary perturbations upon earth's orbit
Approximate maximum velocity (km/sec) -
20
30 0.1 0.5
0.01 3
mum size. In practice, of course, the contributions of these velocity components will be less according to direction cosines. Ball' has written a program to calculate the velocity of a telescope with respect to the LSR. This program includes terms described in Table I except those due to planetary perturbations of the earth's orbit, and the program omits the motion of the sun around its barycenter. Appendix A lists this T. D. Kinman, Mon. Nor. Roy. Aslron. SOC.119, 559 (1959).
' M. L. Humason and H. D. Wahlquist, Astron. J. 60, 254 (1955).
* J. A. Ball, M.I.T. Lincoln Lab. Tech. Note T N 1969-42, Lexington, Massachusetts (16 July 1969).
6. I .
RADIAL-VELOCITY CORRECTIONS FOR EARTH MOTION
283
subroutine, which can be used to compute the radial-velocity correction with sufficient accuracy for most spectroscopy of astronomical objects. The absolute accuracy is approximately 0.02 km/sec. For times less than a year or so, the relative accuracy is 0.005 km/sec. For more detailed discussion of the astronomical considerations involved in this computation problem, the reader is referred to standard book^.^-'^ A tabulation of radial velocity components has been prepared by McRae and W e ~ t e r h o u t . ’ ~ The computer program called DOP, listed in Appendix A, may be used to calculate the velocity component of the observer with respect to the Local Standard of Rest as projected onto a line specified by the right ascension and declination (RAHRS,RAMIN,RASEC ;DDEG,DMIN,DSEC) for the epoch of date with time specified as follows: NYR last two digits of the year (for 19XX A.D.); NDAY day number (time UT); NHUT, NMUT, NSUT time (hours, minutes, seconds UT). The location of the observer is specified by the latitude (ALAT), the geodetic longitude in degrees (OLONG), and the elevation in meters above sea level (ELEV). The program gives as output the local mean sidereal time in days (XLST), the velocity component in kilometers per second of the sun’s motion with respect to the Local Standard of Rest as projected onto the line of sight to the source (VSUN), and the total velocity component in kilometers per second (Vl). Positive velocity corresponds to increasing distance between source and observer. The program DOP in Appendix A is written as a subroutine in a standard v e r ~ i o n of ’ ~ the Fortran language with the following exception : certain variables are calculated in double precision as declared by nonstandard statements as noted by comments in the program. ACKNOWLEDGMENTS I thank M. S. Roberts and B. G . Clark for their vital comments. R. J . Trumpler and H. F. Weaver, “Statistical Astronomy,” p. 251. Univ. of California Press, Berkeley, California, 1953. l o D. Mihalas and P. M . Routly, “Galactic Astronomy.” Freeman, San Francisco, California, 1968. *‘W. M . Smart, “Text-Book on Spherical Astronomy.” Cambridge Univ. Press, London and New York, 1962. l 2 ’‘ Explanatory Supplement to the Astronomical Ephemeris.” H.M. Stationery Office, London, 1968. l 3 D. A. McRae and G . Westerhout, Table for the Reduction of Velocities to the Local Standard of Rest. The Observatory, Lund, Sweden (1956). l 4 Amer. Std. Ass., Sect. Committee X3, Comniun. Ass. Comput. Machinery 7,590 (1964).
6.1. Radial-Velocity Corrections for Earth Motion* 6.1.1. Introduction
In 1842, Christian Doppler published a paper quantitatively describing the relationship between line-of-sight velocities and wave frequencies. At the time, the wave theory of light introduced by Huygens was becoming accepted over Newton’s corpuscular theory because of its ability to explain interference phenomena. Doppler was investigating implications of the wave nature of light. Using the analogy of wave trains in the ocean moving toward (or against) a moving ship, Doppler derived formulas relating the wave frequency perceived by an observer on a moving ship to that seen by a stationary observer. He correctly noted that the shift in frequency depends not only upon the relative velocity between the wave source and the ship, but also upon whether the ship moves with respect to the wave source, or the source with respect to the ship. For if the ship moves with a velocity v, toward the wave source, and if the waves move through the ocean with a velocity v, then the effective velocity of the wave train with respect to an observer aboard ship would be v, + u, and the observed frequency f of the wave train is
f = (0, + 41no = f o U + vslv),
(6.1.1)
where fo and l o are the actual frequency and wavelength, respectively, of waves emitted by the source. Thus, the observer would perceive a higher wave frequency owing to the ship’s motion. If the wave source moves toward the stationary ship, the physical separation between wave crests would decrease to l o ( l - v,/v), and the observer would also perceive a higher wave frequency
f = V I A =fo/(l
- v,lv),
(6.1.2)
but different than that given by Eq. (6.1.1). Such frequency changes due to relative velocities between source and observer are known categorically as Doppler efsecrs. It is important to note that the effect here is quantitatively ~~~
* Chapter 6.1 is by
M. A. Gordon 277
278
6.
COMPUTER PROGRAMS FOR RADIO ASTRONOMY
different depending upon whether the source moves relative to the observer, or the observer moves relative to the source. Doppler believed the equations describing water waves also described electromagnetic waves propagating through the ether, and, in his paper, did not hesitate to apply these equations to astronomy. He suggested that all stars had the same intrinsic color as the sun, and that the observed variations in the color of stars reflected not differences in their temperature and chemical composition, but differences in their radial velocities with respect to the earth. This suggestion created bitter opposition to his ideas, which persisted long after Doppler’s death, until careful spectroscopic observations showed that there were indeed differences in the temperature and composition of stars, and that differences in apparent color were not due wholly to velocity effects. (Fizeau had privately pointed out that, owing to the continuous nature of stellar radiation, Doppler effects probably could not cause significant changes in apparent colors of stars.) 6.1.2. Special Relativity
Doppler’s equations assume fundamental concepts regarding the vector interaction of the velocities of light, source, observer, and medium (ether). In his 1849 experiment to measure the velocity of light, Fizeau investigated, but failed to find, any effect of the earth’s motion upon the velocity of light. Yet, Doppler’s ideas required variations in the effective propagation velocity. Such early experiments cast doubt as to whether light waves propagated in ether the same way as ocean waves in water and acoustic waves in gas. In 1887, Michelson and Morley performed their sophisticated experiment which indicated that the velocity of light was a constant, unaffected by vector addition or subtraction of the earth’s velocity. These results led Lorentz (and, independently, FitzGerald) in 1895 to propose that one leg of the MichelsonMorley apparatus changed length as a function of velocity such that, if the velocity of light indeed changed because of the earth’s motion, the arm of the measuring device would change length correspondingly, thus preventing the investigators from detecting the changes in the velocity of light. This suggestion led Lorentz to consider further the nature of velocity and electromagnetic phenomena, which in 1904 resulted in a paper which paralleled the more complete exposition by Einstein. In 1905, Einstein published his now famous paper in which he postulated that the velocity of light was independent of the motion of its source. In his discussion, he derived the “ Doppler effect ” of special relativity to be f=fo
1 - (u/c) cos 2
[1 - ( 0 I c
2
)I
f#J
112’
(6.1.3)
6. I .
RADIAL-VELOCITY CORRECTIONS FOR EARTH MOTION
279
where 4 is the angle between the velocity and the line of sight between source and observer, v is the velocity of the source in the observer’s reference frame, and c is the speed of light. Unlike Doppler’s original formulation, this equation always predicts a frequency shift even if the light source (or observer) moves at right angles (cos (p = 0) to the line of sight joining the two objects. If the source and observer move toward each other, cos (p = 1 , and Eq. (6.1.3) becomes
(6.1.4) which is quite different from Doppler’s original formulation given in Eqs. (6.1.1) and (6.1.2). This difference results from the fact that light waves and water waves do not propagate in the same way. 6.1.3. Conventional Tabulation o f Redshifts
Astronomical spectroscopy began considerably before Einstein’s paper appeared, and thus wavelength (frequency) shifts of spectral lines are always catalogued as velocity effects calculated by Doppler’s original formulations, with the assumption that sources move relative to the earth. Optical astronomers calculate radial velocities from their most basic measurements : the dispersion of wavelengths on a spectroscopic plate. Hence, = c[(A - AO)/AOI>
uop1
(6.1.5)
where uopl is positive for a receding source if A is the observed wavelength. The astronomical redshift parameter z is defined to be
z
= (A - A())/&.
(6.1.6)
The system generally used by radio astronomers differs from that used by optical astronomers. Presumably, the difference stems from the fact that, unlike the case for the optical spectrometer, the radio spectrometer sorts information by frequency. Hence, the convention arose that uradio
= c[(fO
-f)lfol.
(6.1.7)
To date, neither optical nor radio astronomers conventionally use special relativity [Eq. (6.1.4)] to calculate radial velocities from observations. Because of the differences between radial velocities quoted by radio and optical astronomers, errors’ are apt to result when one uses optically determined redshifts of high velocity objects like quasars to position narrow-band radio spectrometers. And, occasionally, uopt or z are calculated with a comC . Heiles and G . K. Miley, Asfrophys. J . 160, L83 (1970).
280
6.
COMPUTER PROGRAMS FOR RADIO ASTRONOMY
bination of rest wavelengths measured in a vacuum (Ao) and observed wavelengths measured in the air (A) of the astronomical spectrography. Finally, with low-dispersion optical spectrographs, the “ line ” spectra are in fact often blends of several lines. Thus, radial velocities calculated from blends erroneously presumed to be single transitions often contain substantial errors. This problem is a well-known one, and a variety of techniques have been used to minimize resulting errors. 6.1.4. The Aberration of Light
Early astronomers noted that the stars seemed to change position with season. This effect arises because the motion of the earth changes the aspect angle of an observer with respect to the direction of arrival of rays from stars. If 8‘ is the apparent angle between the direction of the earth’s velocity u and the star, and 0 is the actual angle, simple trigonometry shows that tan 8’ = sin O/[cos 8
+ (u/c)].
(6.1.8)
where c is the velocity of light. For the most adverse case, position errors as large as 41 arc sec can occur. 6.1.5. Velocity Reference Frames
In considering the effects of relative velocities of source and observer, we must choose some standard reference velocity. The observer moves on the surface of the earth, which moves around the moon-earth barycenter, which moves around the solar system barycenter, which moves with respect to local stars, which move around the galactic center, etc. For standardization, the observer usually references his velocity measurements to one of two conventional reference velocities. 6.1.5.1. Heliocentric System. Perhaps the most obvious velocity-reference system is that of the sun. In the late nineteenth century, when stellar spectroscopy began, the orbital elements of the earth’s motion about the sun were known to accuracies considerably greater than could be measured from stellar absorption spectra. Consequently, radial velocities measured from the earth could be easily referenced to the sun without loss of accuracy. As mentioned in Section 6.1.3, one of the largest sources of error in these measurements is due to the possibility of unresolved blends of spectral lines. The relative intensity of blended components is probably the same amongst stars of the same spectral type, and here the relative differences in radial velocities can be accurately measured. For example, the radial velocities of F and B stars can be measured relative to the sun with considerable accuracy. Stars of other types may have blends of different relative intensities to those of the sun, and serious measurement errors can arise because of the differences in line shapes.
6. I .
RADIAL-VELOCITY CORRECTIONS FOR EARTH MOTION
28 I
The intrinsic absolute accuracy of optical radial velocities measured from stellar absorption spectra is therefore limited to perhaps something less than 1 km/sec, even for measurements taken by the best spectrographs by the most experienced observers. (Measurement errors of 0.2 km/sec are often quoted, but such numbers refer to precision rather than accuracy.) Such accuracy does not warrant distinction between the sun and the solar system barycenter. Thus, the term heliocentric velocity assumes the sun to be the focus of the earth's orbital motion. (Motion with respect to the solar-system barycenter is discussed in Section 5.6.2.) The reader may wish to refer to work by Campbell and Moore2 and by Petrie3 for more detailed information about stellar velocity measurements, and to work by Herrick4 for a convenient table to reduce radial velocities to the sun. 6.1.5.2. Local Standard of Rest. Analysis of velocities determined by stellar spectroscopy shows that the sun has a systematic motion with respect to neighboring stars. The situation is not simple, because this systematic motion is also a function of the spectral type of stars used for the radial velocity measurements. If we define a kinematic reference frame moving with the average velocity of these neighboring stars, then this local standard of rest can be said to be a function of stellar type. Clearly, some convention is necessary. Standard solar motion is defined to be the average velocity of spectral types A through G as found in general catalogs of radial velocity, without regard to luminosity class. Here, the motion of the sun is 19.5 km/sec toward lSh right ascension (RA) and 30" declination (6) for the epoch 1900.0, which corresponds to galactic coordinates ( I , 6) of (56", 23"). Basic solar motion' is the most probable velocity of stars in the solar neighborhood, thereby being heavily weighted by the radial velocities of the most common spectral types A , gK,and dM in the vicinity of the sun. In this system, the sun moves at 15.4 km/sec toward the direction (1, 6 ) of (51", 23"). The conventional reference frame used for galactic studies is essentially that of standard solar motion. The convention of Local Standard of Rest (LSR) assumes the sun to move at the rounded velocity of 20.0 km/sec toward 18h RA and 30" 6 (1900.0). Since the stars used in the determination include the earlier spectral types of the distribution, their ages are comparatively younger, and, presumably, their velocities are closer to that of the interstellar gas. For some purposes, it is necessary to know the motion of the Local Stan-
* W. W. Campbell and J.
H. Moore, Publ. Lick Obseru. 16 (1928). R. M. Petrie, in "Basic Astronomical Data" (K. A. Strand, ed.), p. 64. Univ. of Chicago Press, Chicago, Illinois, 1963. S. Herrick, Jr., Lick Obs. Bull. 470, 17, 35 (1935). A. N. Vyssotsky and E. N. Janssen, Asrron. J . 56, 58 (1951).
282
6.
COMPUTER PROGRAMS FOR RADIO ASTRONOMY
dard of Rest about the galactic center. Observations6 of globular clusters show the sun to move at 167 & 30 km/sec toward an (l, b) of (90", 0"). (With these uncertainties, the motions of the sun and the Local Standard of Rest are indistinguishable.) Similar observations' of galaxies in the Local Group show the sun to move at 292 If: 32 km/sec toward (106", -6"). The reason for the difference between the two velocities is unclear, but based on such observations, the Local Standard of Rest is usually assumed to move at 250 km/sec toward (90", 00) about the Galactic Standard of Rest. 6.1.6. Calculation of Radial Velocities
The calculation of the radial velocity between a position on the earth's surface and a distant astronomical object involves a number of velocity terms. Just how many terms need to be included is determined by the accuracy required. Table I lists a number of these terms and their approximate maxiTABLE I. Velocities Involved in the Radial-Velocity Computation
Component Source with respect to Local Standard of Rest (LSR) LSR with respect to solar system barycenter Solar system barycenter with respect to earth-moon barycenter Earth-moon barycenter with respect to earth center Earth center with respect to telescope Planetary perturbations upon earth's orbit
Approximate maximum velocity (km/sec) -
20
30 0.1 0.5
0.01 3
mum size. In practice, of course, the contributions of these velocity components will be less according to direction cosines. Ball' has written a program to calculate the velocity of a telescope with respect to the LSR. This program includes terms described in Table I except those due to planetary perturbations of the earth's orbit, and the program omits the motion of the sun around its barycenter. Appendix A lists this T. D. Kinman, Mon. Nor. Roy. Aslron. SOC.119, 559 (1959).
' M. L. Humason and H. D. Wahlquist, Astron. J. 60, 254 (1955).
* J. A. Ball, M.I.T. Lincoln Lab. Tech. Note T N 1969-42, Lexington, Massachusetts (16 July 1969).
6. I .
RADIAL-VELOCITY CORRECTIONS FOR EARTH MOTION
283
subroutine, which can be used to compute the radial-velocity correction with sufficient accuracy for most spectroscopy of astronomical objects. The absolute accuracy is approximately 0.02 km/sec. For times less than a year or so, the relative accuracy is 0.005 km/sec. For more detailed discussion of the astronomical considerations involved in this computation problem, the reader is referred to standard book^.^-'^ A tabulation of radial velocity components has been prepared by McRae and W e ~ t e r h o u t . ’ ~ The computer program called DOP, listed in Appendix A, may be used to calculate the velocity component of the observer with respect to the Local Standard of Rest as projected onto a line specified by the right ascension and declination (RAHRS,RAMIN,RASEC ;DDEG,DMIN,DSEC) for the epoch of date with time specified as follows: NYR last two digits of the year (for 19XX A.D.); NDAY day number (time UT); NHUT, NMUT, NSUT time (hours, minutes, seconds UT). The location of the observer is specified by the latitude (ALAT), the geodetic longitude in degrees (OLONG), and the elevation in meters above sea level (ELEV). The program gives as output the local mean sidereal time in days (XLST), the velocity component in kilometers per second of the sun’s motion with respect to the Local Standard of Rest as projected onto the line of sight to the source (VSUN), and the total velocity component in kilometers per second (Vl). Positive velocity corresponds to increasing distance between source and observer. The program DOP in Appendix A is written as a subroutine in a standard v e r ~ i o n of ’ ~ the Fortran language with the following exception : certain variables are calculated in double precision as declared by nonstandard statements as noted by comments in the program. ACKNOWLEDGMENTS I thank M. S. Roberts and B. G . Clark for their vital comments. R. J . Trumpler and H. F. Weaver, “Statistical Astronomy,” p. 251. Univ. of California Press, Berkeley, California, 1953. l o D. Mihalas and P. M . Routly, “Galactic Astronomy.” Freeman, San Francisco, California, 1968. *‘W. M . Smart, “Text-Book on Spherical Astronomy.” Cambridge Univ. Press, London and New York, 1962. l 2 ’‘ Explanatory Supplement to the Astronomical Ephemeris.” H.M. Stationery Office, London, 1968. l 3 D. A. McRae and G . Westerhout, Table for the Reduction of Velocities to the Local Standard of Rest. The Observatory, Lund, Sweden (1956). l 4 Amer. Std. Ass., Sect. Committee X3, Comniun. Ass. Comput. Machinery 7,590 (1964).