Sp.-V/AQuan/1999/10/08:06:45 Page 7

Chapter 2

General Constants and Units

Arthur N. Cox

2.1 Mathematical Constants ...... 7 2.2 Physical Constants ...... 8 2.3 General Astronomical Constants ...... 12 2.4 Astronomical Constants Involving Time ...... 13 2.5 Units ...... 17 2.6 Electric and Magnetic Unit Relations ...... 22

2.1 MATHEMATICAL CONSTANTS [1–3]

Constant Number Log π 3.141 592 653 6 0.497 149 872 7 2π 6.283 185 307 2 0.798 179 868 4 4π 12.566 370 614 4 1.099 209 864 0 2 √π 9.869 604 401 1 0.994 299 745 4 π 1.772 453 850 9 0.248 574 936 3 eore 2.718 281 828 5 0.434 294 481 9

mod = M = log e 0.434 294 481 9 0.637 784 311 3 − 1 1/M = ln 10 2.302 585 093 0 0.362 215 688 7 √2 2.000 000 000 0 0.301 029 995 7 √2 1.414 213 562 4 0.150 514 997 8 √ 3 1.732 050 807 6 0.238 560 627 4 10 3.162 277 660 2 0.500 000 000 0

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Constant Number Log ln π 1.144 729 885 8 0.058 703 021 2 eπ 23.140 692 632 8 1.364 376 353 8 Euler constant γ 0.577 215 664 9 0.761 338 108 8 − 1 1 radian rad = 57.◦295 779 513 1 1.758 122 632 4 = 3 437.746 770 78 3.536 273 882 8 = 206 264.806 25 5.314 425 133 2 1◦ = 0.rad017 453 292 5 0.241 877 367 6 − 2 1 = 0.rad000 290 888 2 0.463 726 117 2 − 4 1 = 0.rad000 004 848 1 0.685 574 866 8 − 6

Square degrees on a sphere = 129 600/π = 41 252.961 25. Square degrees in a steradian = 32 400/π 2 = 32 82.806 35. 1 x2 for Gaussian distribution √ exp − . σ 2π 2σ 2 Probable error/Standard error = r/σ = 0.674 489 750 2. Probable error/Average error = r/η = 0.845 347 539 4. σ/η =√1.253 314 137. ρ = (r/σ )/ 2 = 0.476 936 276 2.

2.2 PHYSICAL CONSTANTS [4, 5]

These fundamental physical constants, mostly in SI units from [5], are the latest available. A revision by Cohen and Taylor is expected by the end of 1998. For many values, the standard error of the last digits follows in parentheses. In the formulations the electron charge e is in esu and e in emu = e/c.

Fundamental constants Speed of (exact) c = 2.997 924 58 × 108 ms−1 c2 = 8.987 551 79 × 1016 m2 s−2 Gravitation constant G = 6.672 59(85) × 10−11 m3 kg−1 s−2 −1 Standard acceleration of gravity (exact) gn = 9.806 65 m s Planck constant 2π = h = 6.626 075 5(40) × 10−34 Js  = 1.054 572 66(63) × 10−34 Js Planck mass (c/G)1/2 = 2.176 71(14) × 10−8 kg Planck length (G/c3)1/2 = 1.616 05(10) × 10−35 m Planck time (G/c5)1/2 = 5.390 56(34) × 10−44 s Elementary charge e = 4.803 206 8(15) × 10−19 C e = 1.602 177 33(49) × 10−20 emu e2 = 23.070 796 × 10−20 in esu e4 = 5.322 616 1 × 10−38 in esu −31 Mass of electron me = 9.109 389 7(54) × 10 kg = 5.485 799 03(13) × 10−4 u Sp.-V/AQuan/1999/10/08:06:45 Page 9

2.2 PHYSICAL CONSTANTS /9

Mass of unit atomic weight u = 1.660 540 2(10) × 10−27 kg (12C = 12 scale) Boltzmann constant k = 1.380 658(12) × 10−23 JK−1 = 8.617 385(73) × 10−5 eV K−1 k1/2 = 1.175 014 × 10−8 erg1/2 K−1/2 Gas constant (12C scale) R = 8.314 510(70) J K−1 mol−1 = 1.987 216 cal K−1 mol−1 = 82.057 83(70) cm3 atm K−1 mol−1 Joule equivalent (chemical, exact) [4] = 4.184 J cal−1 23 −1 Avogadro constant NA = 6.022 136 7(36) × 10 mol 25 −3 Loschmidt constant n0 = 2.686 763(23) × 10 m −3 Volume of -molecule at STP NA/n0 = V0 = 22.414 10(19) × 10 (T = 273.15 K, P = 101 325 Pa) m3 mol−1 −2 Standard atmosphere pressure (exact) P0 = 1 013 250 dyn cm = 760 mmHg Ice point 0◦ C = 273.150 K Triple point (H2O) = 273.160 K −1 Faraday NAe/c = 96 485.309(29) C mol

Atomic constants 1 −1 Rydberg constant for H RH = 10 967 758.306(13) m 1/RH = 911.763 345 0 Aû − Rydberg constant for infinite nuclear mass R∞ = 10 973 731.534(13) m 1 2 4 3 2π mee /ch 1/R∞ = 911.267 053 4 Aû − cR∞ = 3.289 841 950 × 1015 s 1 Fine structure constant α = 7.297 353 08(33) × 10−3 2πe2/hc 1/α = 137.035 989 5(61) α2 = 5.325 136 20 × 10−5 −10 Radius for first Bohr orbit a0 = 0.529 177 249(24) × 10 m 2 2 2 (infinite nuclear mass) h /4π mee −1 −17 Time for (2π) revolutions in first τ0 = 2.418 884 4 × 10 s 1/2 3/2 −1 3 3 4 Bohr orbit me a e = h /8π mee Frequency of first Bohr orbit = 6.579 683 7 × 1015 s−1 π 2 = . × −21 2 Area of first Bohr orbit a0 8 797 356 70 10 m τ −1 = . × 6 −1 Electron speed in first Bohr orbit a0 0 2 187 691 4 10 ms Atomic unit of energy = 4.359 748 2(26) × 10−18 J 2 (Hartree = 2 Rydbergs) e /a0 = 2chR∞ = 27.211 396 1(81) eV Energy of Rydberg ryd = 2.179 874 1(13) × 10−18 J (often adopted as atomic unit) = 13.605 698 1(40) eV Atomic unit of angular momentum h/2π  = 1.054 572 66(63) × 10−34 kg m2s−1 2 2 −15 Classical electron radius e /m0c l = 2.817 940 92(38) × 10 m 2 −2 27 −1 −2 Schrodinger¬ constant for fixed nucleus 8π meh = 1.638 197 48 × 10 cm Schrodinger¬ constant for 1H atom = 1.637 305 78 × 1027 erg−1 cm−2 Sp.-V/AQuan/1999/10/08:06:45 Page 10

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1 6 −1 Hyperfine structure splitting of H νH = 1 420.405 751 768 × 10 s ground state Doublet separation in 1H atom = 0.365 866 231 cm−1 2 2 2 10 −1 (1/16)RHα [1 + α/π + (5/8 − 5.946/π )α ] = 1.096 839 36×10 s 1 −31 Reduced mass of electron in H atom me(mp/mH) = 9.104 431 3 × 10 kg Mass of 1H atom = 1.673 534 4 × 10−27 kg = 1.007 825 050(12) u Mass of proton = 1.672 623 1(10) × 10−27 kg = 1.007 276 470(12) u Mass of neutron = 1.674 928 6(10) × 10−27 kg = 1.008 664 904(14) u Mass of deuteron = 3.343 586 0(20) × 10−27 kg = 2.013 553 214(24) u Mass energy of unit atomic mass uc2 = 1.492 419 1 × 10−10 J = 931.494 2(28) MeV 2 −14 Rest mass energy of electron mec = 8.187 111 1 × 10 J = 0.510 999 06(15) MeV Mass ratio proton/electron = 1 836.152 701(37) 7 −1 Specific electron charge e/me = 1.758 819 62 × 10 emu g 17 −1 e/me = 5.272 808 6 × 10 esu g Quantum of magnetic flux h/e = 1.379 510 77 × 10−17 erg s esu−1 hc/e = 4.135 669 2 × 10−7 cm2 −1 Quantum of circulation h/me = 7.273 896 2 erg s g −12 Compton wavelength h/mec = 2.426 310 58(22) × 10 m −13 h/2πmec = 3.861 593 23(35) × 10 m Band spectrum constant h/8π 2c = 27.992 774 × 10−40 gcm (moment of inertia/wave number) Atomic specific heat constant = 4.799 216 × 10−11 sK c2/c = h/k −21 Magnetic moment of 1 Bohr magneton µB = 9.274 015 4(31) × 10 µ 1 α 1/2 5/2τ −1 = / π −1 B= 2 me a0 0 he 4 mec erg gauss Electron magnetic moment µe = 1.001 159 652 193(10)µB −3 Proton magnetic moment µp = 1.521 032 202(15) × 10 µB 4 Gyromagnetic ratio of proton γp = 2.675 221 28(81) × 10 −1 −1 corrected for diamagnetism of H2O rad s gauss −24 Magnetic moment of 1 nuclear magneton µn = 5.050 786 6(17) × 10 −1 he/4πmpc erg gauss Atomic unit of magnetic moment = 2.541 747 8 × 10−18 erg gauss−1 2µB/α Magnetic moment per mole of 1 Bohr = 5 584.938 8 erg gauss−1 mol−1 magneton per molecule Zeeman displacement = 4.668 643 7(14) × 10−5 −1 −1 3/4πmec (e in emu) cm gauss in frequency = 1.399 624 18(42) × 106 s−1 gauss−1 Sp.-V/AQuan/1999/10/08:06:45 Page 11

2.2 PHYSICAL CONSTANTS /11

The electron–volt and photons [5] −10 Wavelength associated with 1 eV λ0 = 12 398.428 2 × 10 m −1 Wave number associated with 1 eV s0 = 8 065.538 51 cm = 8.065 538 51 kiloÐkayser 14 −1 Frequency associated with 1 eV ν0 = 2.417 988 36(72) × 10 s −19 Energy of 1 eV E0 = 1.602 177 33(49) × 10 J = 0.073 498 617 6 ryd Photon energy associated with unit hc = 1.986 448 0 × 10−23 J wave number Photon energy associated with = 1.986 448 0 × 10−8/λ erg (λ in A)û wavelength λ Speed of 1 eV electron = 5.930 968 92 × 105 ms−1 8 1/2 (2 × 10 (e/mec)) Speed2 = 3.517 639 23 × 1011 m2 s−2 Wavelength of electron of energy V in eV = V −1/2(12.264 263 × 10−8) cm −1/2 −1/2 h(2me E0) V Temperature associated with 1 eV = 11 604.45 K E0/k Temperature associated with 1 eV = 5039.75 K in common logs = (E0/k) log e Temperature associated with 1 kilo-kayser = 624.849 3 K in common logs = 103(hc/k) log e Energy of 1 eV per molecule = 23 060.054 2 cal mol−1

Radiation constants Radiation density constant a = 7.565 91(25) × 10−15 8π 5k4/15c3h3 erg cm−3 K−4 StefanÐBoltzmann constant = ac/4 σ = 5.670 51(19) × 10−5 erg cm−2 K−4 s−1 −5 First radiation constant c1 = 3.741 774 9(22) × 10 (emittance) = 2πhc2 erg cm2 s−1 First radiation constant (radiation density) 8πhc = 4.992 487 0 × 10−15 erg cm radiation constant = hc/kc2 = 1.438 769(12) cm K Wien displacement law constant = 0.289 775 5 c2/4.965 114 23

Some general constants [1, 5] Density of mercury (0◦ C, 760 mmHg) = 13.395080 g cm−3 Ratio, grating to Siegbahn scale of X-ray wavelengths [5] λg/λs = 1.002 077 89(70) [λs (Cu Kα1) = 1.537 400 kXu] Lattice spacing of Si (in vacuum, 22.5◦ C) = 0.543 101 96(11) × 10−9 m Molar volume of Si = 12.058 817 9(89) cm3 mol−1 Maximum density of water = 0.999 972 g cm−3 Cesium resonance frequency = 9 192 631 770 Hz (defining the SI second) [6] Sp.-V/AQuan/1999/10/08:06:45 Page 12

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2.3 GENERAL ASTRONOMICAL CONSTANTS by Alan D. Fiala

Astronomical unit of distance = mean SunÐEarth distance = semimajor axis of Earth orbit [2, 6]. AU = 1.495 978 706 6 × 1011 m. Parsec (= 206 264.806 AU) pc = 3.085 677 6 × 1016 m. = 3.261 563 8 light (Julian) year. Light (Julian) year = 9.460 730 472 × 1015 m. Light time for 1 AU [6] = 499.004 783 70 s = 0.005 775 518 33 d. Solar mass M = 1.9891 × 1030 kg. Solar radius R = 6.955 08 ± 0.00026 × 108 m. − Solar radiation L = 3.845(8) × 1033 erg s 1. Earth mass M⊕ = 5.974 2 × 1024 kg. − Earth mean density ρ¯⊕ = 5.515gcm 3. Earth equatorial radius [6] = 6378.136 km.

◦ ◦ Galactic pole (J2000.0) α3 = 192. 859 481 23 δ3 =+27. 128 251 20 ◦   12h51m26.s2755 +27 7 41. 704 ◦ Direction of galactic center (J2000.0) α1 = 266.404 996 25 δ1 =−28. 936 172 42 ◦   17h45m37.s1991 −28 56 10. 221 Solar motion toward galactic center [7] U = 10.00 ± 0.36 km s−1 toward direction of galactic rotation V = 5.23 ± 0.62 km s−1 vertically up in north direction W = 7.17 ± 0.38 km s−1 Galactic rotation [8] R0 = 7.66 ± 0.32 kpc −1 Vcirc = 237 ± 12 km s Sun’s equatorial horizontal parallax [6] = 8.794 144(3) = 4.263 521 × 10−5 rad Moon’s equatorial horizontal parallax = 3422.608 at mean distance Constant of nutation [6] = 9.2025 Constant of aberration [6] = 20.495 52 2π × 206 265 × AU ct(1 − e2)1/2 t = sidereal year, e = Earth orbital eccentricity Gaussian gravitational constant k in n2a3 = k2(1 + m), where m = mass of planet in solar units, n = mean daily in AU k = 0.0172 020 989 5 rad motion, and a = semimajor axis (a defining constant) = 3548.187 607 = 0.◦985 607 668 6 k/86400 = 2π/(sidereal year in sec) k = 1.990 983 675 × 10−7 rad, for use with of time Heliocentric gravitational constant = AU3(k)2 = 1.327 124 40 × 1026 cm2 s−1 Semimajor axis of Earth orbit in terms of AU = 1.000 001 057 266 65 AU Sp.-V/AQuan/1999/10/08:06:45 Page 13

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Mass ratios [6, 9] M⊕/M“ = 81.300 59 M /M⊕ = 332 946.05 M /(M⊕ + M“) = 328 900 56(2) Obliquity of ecliptic (fixed ecliptic of J2000.0)  = 23◦26 21.4119

2.4 ASTRONOMICAL CONSTANTS INVOLVING TIME [6] by Alan D. Fiala

The basic unit of time is the Systeme` International (SI) second which is defined to be the duration of 9 192 631 770 cycles of one of the hyperfine transitions of the ground state of 133Cs. Based on this defined unit, International Atomic Time (TAI) is formed from statistical analysis of individual frequency standards and time scales based on atomic clocks in many countries. It was introduced in January 1972, and is a coordinate time scale. Universal Time (UT) is the measure of time used for all civil time keeping, and conforms closely to the mean diurnal motion of the Sun. It is directly related to sidereal time by means of an adopted numerical formula. It does not refer to the motion of the Earth and is not precisely related to the hour angle of the Sun. UT0 is the uncorrected observed rotational time scale derived from observation of sidereal time at a particular station. When this time scale is corrected for the shift in longitudes caused by polar motion, it is designated UT1. This still contains the variable rotation of the Earth and is generally implied when the symbol “UT” is used without qualification. Coordinated Universal Time (UTC) is the time scale distributed by radio signals, satellites, communication media, as the basis for civil time keeping around the world. UTC is maintained within 0.9 second of UT1 by the introduction of leap seconds. UTC differs from TAI by an integer number of seconds, which difference changes when leap seconds are introduced. Dynamical time represents the independent variable of the equations of motion of the bodies in the Solar System. It depends on the theory of relativity being used, as does the transformation between barycentric and geocentric time scales. In the transformation, the constants can be chosen so that the timescales have only periodic variations with respect to each other. The dynamical time scale for apparent geocentric ephemerides was chosen to be unique and independent of the theories; the barycentric timescales are theory dependent. Terrestrial Dynamical Time (TDT), or Terrestrial Time (TT), is the idealized time on the geoid of the Earth and is approximated as being equal to TAI + 32.184 seconds. Terrestrial Time is a continuation of Ephemeris Time (ET), beginning 1977 Jan. 1.0 TAI. The relationship between UT and TT changes according to the variations in the rotation of the Earth. Barycentric Dynamical Time (TDB) is the relativistically transformed time for referring equations of motion to the barycenter of the Solar System. It is defined to contain only periodic variations with respect to TDT. The time scales Geocentric Coordinate Time (TCG) and Barycentric Coordinate Time (TCB) are the time-like arguments appropriate for coordinate systems defined with respect to the geocenter of the Earth and the barycenter of the Solar System, respectively, including all relativistic transformations from terrestrial time. Up to 1984, the tropical year was used as the basis of time for reference systems and the Besselian year was used as the epoch for such reference frames, thus designated, for example, as B1950.0. Since 1984, the Julian Century has been used as the time unit for reference frames and the standard epoch is then designated as, for example, J2000.0. Sp.-V/AQuan/1999/10/08:06:45 Page 14

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Sidereal time is defined by the hour angle of the equinox. The relationship between Greenwich Mean Sidereal Time (GMST) and UT1 is specified by an adopted equation, which is often considered to be the definition of UT1. At 0 hours UT1: = . + .s + .s 2 − . × −6 3 GMST 24110 54841 8640184 812866Tu 0 093104Tu 6 2 10 Tu

seconds of time, where Tu = du/36525, du is the number of days of Universal Time elapsed since JD 2451545.0 UT1 (2000 January 1, 12 hrs UT1), taking on values ±0.5, ±1.5, etc. The ratio of mean sidereal time to UT1 is

 − − r = 1.002 737 909 350 795 + 5.900 6 × 10 11T − 5.9 × 10 15T 2,

where T is the number of Julian centuries elapsed since JD 2 451 545.0. The ratio of UT1 to mean sidereal time is

 − − 1/r = 0.997 269 566 329 084 − 5.868 4 × 10 11T + 5.9 × 10 15T 2.

The relationships between time scales in seconds of time are:

TT = TDT = ET = TAI + 32.184, T = ET − UT = TDT − UT = TT − UT, AT = TAI − UTC, DUT ∼ UT = UT1 − UTC, TDB = TDT + P, TCG − TT = 6.969 290 4 × 10−10(JD− 2 443 144.5) × 86 400, −8 −2 TCB − TCG = 1.480 813 × 10 (JD− 2 443 144.5) × 86 400 + ve · (x − xe)c + P, TCB − TDB = 1.550 506 × 10−8(JD− 2 443 144.5) × 86 400, P = 0.001 656 8 sin(35 999.37T + 357.5) + 0.000 022 4 sin(32 964.5T + 246) + 0.000 013 8 sin(71 998.7T + 355) + 0.000 004 8 sin(3 034.9T + 25) + 0.000 004 7 sin(34 777.3T + 230),

where T is the elapsed time from J2000.0 measured in Julian centuries and the coefficients are rounded at their last digits [6, 10, 11]. Arguments are in degrees. Here xe and ve denote the barycentric position and velocity of the Earth’s center of mass, the difference (x − xe) is the vector distance of the observer from this center of mass, and c is the speed of light.

2.4.1 Reduction of Time Scales

The variations in the Earth’s rotation rate have resulted in differences between time based on it and that based on planetary orbits. The differences between the ephemeris and the (generally slower) universal time are given in Tables 2.1 and 2.2 for the last 130 years. For dates back to 1620, see the Astronomical Almanacs [12]. Even earlier to the year 1500 B.C., one can find a table in the Canon of Lunar Eclipses [13]. Before 1884, T = ET − UT, after 1984, T = TDT − UT, and after 1989, the differences are for exactly 1 Jan. 0h UTC. Sp.-V/AQuan/1999/10/08:06:45 Page 15

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Table 2.1. Reduction of time scales from 1870 to 1974. Year T Year T Year T Year T Year T

1870 +1.61 1895 −6.47 1920 +21.16 1945 +26.77 1970 +40.18 1875 −3.24 1900 −2.72 1925 +23.62 1950 +29.15 1971 +41.17 1880 −5.40 1905 +3.86 1930 +24.02 1955 +31.07 1972 +42.23 1885 −5.79 1910 +10.46 1935 +23.93 1960 +33.15 1973 +43.37 1890 −5.87 1915 +17.20 1940 +24.33 1965 +35.73 1974 +44.49

Table 2.2. UTC leap seconds since 1971 and starting at the given date. Year T Year T Year T Year T

1972, Jan. 1 10 1977, Jan. 1 16 1983, July 1 22 1992, July 1 27 1972, July 1 11 1978, Jan. 1 17 1985, July 1 23 1993, July 1 28 1973, Jan. 1 12 1979, Jan. 1 18 1988, Jan. 1 24 1994, July 1 29 1974, Jan. 1 13 1980, Jan. 1 19 1990, Jan. 1 25 1996, Jan. 1 30 1975, Jan. 1 14 1981, July 1 20 1991, Jan. 1 26 1997, July 1 31 1976, Jan. 1 15 1982, July 1 21

Day 1 day = 24 hours = 1440 minutes = 86400 SI seconds Period of rotation of Earth (referred to fixed stars) In mean sidereal time = 86 164.090 54 SI seconds. In mean solar time = 23h56m04.s090 549. 1 day of mean sidereal time = 0.997 269 566 33 of mean solar time. 1 day of mean solar time = 1.002 737 909 35 of mean sidereal time. Rate of rotation = 15.041 067 178 669 10 s−1. = 7.292 115 10 × 10−5 rad s−1.

Year 1 Julian year = 365.25 days = 8766 hours = 525 960 minutes = 31 557 600 SI seconds ddhms Tropical (equinox to equinox) 365.242 189 7 365 05 48 45.19 Sidereal (fixed star to fixed star) 365.256 36 365 06 09 10 Anomalistic (perihelion to perihelion) 365.259 64 365 06 13 53 Eclipse (Moon’s node to Moon’s node) 346.620 05 346 14 52 52 Gaussian (Kepler’slawfora = 1) 365.256 90 365 06 09 56 Julian (based on Julian calendar) 365.25 365 06 Gregorian (based on Gregorian calendar) 365.242 5 365 05 49 12 Sp.-V/AQuan/1999/10/08:06:45 Page 16

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Calendar Julian Dates (see Chapter 27 also) 1900 January 0.5 = JD 2 415 020.0, 1925 January 0.5 = JD 2 424 151.0, 1950 January 0.5 = JD 2 433 282.0, 2000 January 0.5 = JD 2 451 544.0, 2050 January 0.5 = JD 2 469 807.0, 2100 January 0.5 = JD 2 488 069.0.

Length of the month ddhms Synodic (new Moon to new Moon) 29.530 59 29 12 44 03 Tropical (equinox to equinox) 27.321 58 27 07 43 05 Sidereal (fixed star to fixed star) 27.321 66 27 07 43 12 Anomalistic (perigee to perigee) 27.554 55 27 13 18 33 Draconic (node to node) 27.212 22 27 05 05 36

Orbit of the Moon about the Earth Sidereal mean motion of Moon 2.661 699 489 × 10−6 rad s−1 Mean distance of Moon from Earth 3.844 × 10 ∗ 5km 60.27 Earth radii 0.002 570 AU Equatorial horizontal parallax 57 02.608 at mean distance 3422.608 Mean distance of center of Earth from EarthÐMoon barycenter 4.671 × 103 km Mean eccentricity 0.054 90 Mean inclination to ecliptic 5.◦145 396 Mean inclination to lunar equator 6◦ 41 Limits of geocentric declination ±29◦ = = = 1 Saros 223 lunations 19 passages of Sun through node 6585 3 days Period of revolution of node 6798 days Period of revolution of perigee 3232 days Mean orbital speed 1023 m s−1 = 0.000 591 AU day−1 Mean centripetal acceleration 0.002 72 m s−2 = 0.0003 g.

Precession Annual rates of precession (T in centuries from J2000.0) general precession in longitude 50.290 966 + 0.022 222 6T, lunisolar precession in longitude 50.387 784 + 0.004 926 3T, planetary precession −0.018 862 3 − 0.047 612 8T, geodesic precession (relativistic nonperiodic Coriolis effect) 1.92T . Sp.-V/AQuan/1999/10/08:06:45 Page 17

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2.5 UNITS

The seven SI base units are: meter (m), kilogram (kg), second (s), ampere (A), (K), mole (mol), and candela (cd) [14]. All other units are derived from these. Units used with SI are: the time units of minute (min), hour (h), and day (d); the plane angle units of radian (rad), degree (◦), minute (), and (arc)second (); the solid angle unit, steradian (sr); the volume unit liter; (L); the mass unit metric ton (t); and the land area hectare (ha). Other experimentally determined units used with SI are: the special energy unit (eV), and the atomic mass unit (u). Units used in astronomy and astrophysics are often not standard but unique to the special subfield. This procedure is followed for many chapters in this book. They are frequently defined at the beginning of each chapter. Table 2.3 gives the SI unit prefixes.

Table 2.3. The SI prefixes. Factor Prefix Symbol Factor Prefix Symbol − 1024 yotta Y 10 1 deci d − 1021 zetta Z 10 2 centi c − 1018 exa E 10 3 milli m − 1015 peta P 10 6 micro µ − 1012 tera T 10 9 nano n − 109 giga G 10 12 pico p − 106 mega M 10 15 femto f − 103 kilo k 10 18 atto a − 102 hecto h 10 21 zepto z − 101 deka da 10 24 yocto y

Unconventional (nonstandard) units sometimes used in astronomy and astrophysics are listed below.

Length Angstrom unit Aû = 10−8 cm = 10−10 m Micron µ = µm = 10−4 cm=10−6 m Foot ft = 30.4800 cm = 12 in Inch in = 2.540 000 cm Mile = 1.609 344 km = 5280 ft Nautical mile [2] = 1.853 km = 6080 ft Area Square foot ft2 = 929.03 cm Acre = 4046.85 m2 = 43560 ft2 Barn = 10−24 cm2 Sp.-V/AQuan/1999/10/08:06:45 Page 18

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Volume Cubic foot ft3 = 28 316.8 cm3 = 6.229 British gallons = 7.481 US gallons Fluid ounce = 480 minims (British and US) = 28.413 cm3 (British) = 29.574 cm3 (US) Mass Kilogram (SI unit) kg = 1000 g Pound avoirdupois British lb = 453.592 37 g = 7000 grains US lb = 453.592 43 g = 7000 grains Pound troy and apothecary = 373.242 g = 5760 grains Grain (all systems) = 0.064 798 9 g Carat = 0.2000 g Slug = 14.594 kg Ton = tonne = 2240 lb = 1.016 047 × 106 g Metric ton = 106 g Energy Joule (SI unit) J = 107 erg Calorie [4] (exact) cal = 4.184 J = 4.184 × 107 erg Kilowatt-hour = 3600 × 103 J = 8.6042 × 105 cal British thermal unit BTU = 1055 J = 252.0 cal Therm = 100 000 BTU Foot-pound = 1.355 82 × 107 erg Kiloton of TNT = 4.184 × 1019 erg Power Watt (SI unit) = 107 erg/s = Js−1 British horse-power = 745.7 W Force de cheval = 735.5 W Force Newton (SI unit) N = 105 dyn Poundal = 1.3825 × 104 dyn Pound weight = 4.4482 × 105 dyn Slug = 14.594 kg Gram weight = 980.665 dyn Acceleration Standard gravity 1 = 1cms−2 Gravity (equator) g = 978.031 cm s−2 = 32.09 ft s−2 Gravity (pole) g = 983.217 cm s−2 = 32. 26 ft s−2 Speed Mile per hour = 44.704 cm s−1 = 1.4667 ft s−1 Knot = 51.47 cm s−1 Sp.-V/AQuan/1999/10/08:06:45 Page 19

2.5 UNITS /19

Pressure Pascal (SI unit) = 10 dyn cm−2 = 10 µb (occasionally called Bar) µb = 1.000 dyn cm−2 Bar bar = 1.000 × 106 dyn cm−2 = 0.986 923 atm = 1.0197 × 103 g-weight cm−2 Millibar mb = 10−3 bar = 103 µb = 103 dyn cm−2 Atmosphere (standard) atm = 1.013 250 × 106 dyn cm−2 = 760 mmHg = 1013.25 mb Millimeter of mercury ( = 1 Torr) mmHg = 1333.22 dyn cm−2 = 0.001 315 8 atm Inch of mercury = 3.386 38 × 104 dyn cm−2 = 0.033 421 atm Pound per square inch = 6.8947 × 104 dyn cm−2 = 0.068 046 atm Density Kilogram/cubic meter (SI unit) = 1.000 × 10−3 gcm−3 Density of water (4◦ C) = 0.999 972 g cm−3 Density of mercury (0◦ C) = 13.5951 g cm−3 Solar mass/cubic parsec = 6.770 × 10−23 gcm−3 −5 −3 STP gas density = 4.4616 × 10 µ0 gcm where µ0 is molecular weight Temperature Degree scales (Kelvin K, K = deg C = 1.8 deg F Celsius (centigrade) C, Fahrenheit F) Temperature comparisons 0◦ C = 273.150 K = 32◦ F 100◦ C = 373.150 K = 212◦ F Triple point of natural water = 273.160 K = 0.010◦ C Viscosity (dynamic) P = 1gcm−1 s−1 = 0.1Pas SI unit N s m−2 = 10gcm−1 s−1 Viscosity (kinematic) Stokes = 1cm2 s−1 SI unit m2 s−1 = 10000 cm2 s−1 Frequency Hertz Hz = cycle s−1 Kayser (a wave number unit) cm−1 = c Hz  3 × 1010 Hz Rydberg frequency cR∞ = 3.289 84 × 1015 Hz Frequency in first Bohr orbit 2cR∞ = 6.5797 × 1015 Hz Frequency of free electron = 2.7993 × 106 H Hz in magnetic field H (gauss) 3 1/2 Plasma frequency associated = 8.979 × 10 Ne Hz −3 with electron density Ne in cm ) Sp.-V/AQuan/1999/10/08:06:45 Page 20

20 / 2 GENERAL CONSTANTS AND UNITS

Angular velocity (= 2π frequency) Unit of angular velocity = 1 rad s−1 = 2π Hz 1 of arc per tropical year = 1.536 314 7 × 10−13 rad s−1 1 of arc per day = 5.611 269 5 × 10−11 rad s−1 Angular velocity of Earth on its axis = 7.292 115 2 × 10−5 rad s−1 Mean angular velocity of Earth in its orbit = 1.990 986 7 × 10−7 rad s−1 Momentum Linear momentum, SI unit = 105 gcms−1 = 1kgms−1 mc = 2.730 93 × 10−17 gcms−1 Angular momentum SI unit = 107 gcm2 s−1 = 1kgm2 s−1 Electron momentum in first Bohr orbit = 1.993 × 10−19 gcms−1 Quantum unit  = 1.0546 × 10−27 erg s Homogeneous sphere angular momentum = (2/5)R2Mω (R = radius, M = mass, ω = angular velocity) Angular momentum of solar system = 3.148 × 1050 gcm−2 s−1 Luminous intensity is defined as the luminous emission per sterad Candela (SI unit) cd = (1/60) luminous intensity of 1 projected cm2 black body at the temperature of melting platinum (2044 K) 29 Star, mv = 0, outside Earth atmosphere = 2.45 × 10 cd Luminous flux (both SI and CGS unit) = flux from 1 cd into 1 sr = flux from (1/60π) cm2 of black body at 2044 K Lumen of maximum visibility radiation = 1.470 × 10−3 W at 5550 Aû therefore1Wat5550 Aû = 680 lumens Jansky = 10−23 cm−2 s−1 Hz−1 = 10−26 Wm−2 Hz−1 Luminous energy Talbot (SI unit) = 1 lumerg (CGS unit) = 1 lumen second Surface brightness sb = 1cdcm−2 = π lambert = 1 lumen cm−2 sr−1 Lambert = (1/π) cd cm−2 = 1000 millilambert ≡ 1 lumen cm−2 for a perfectly diffusing surface Sp.-V/AQuan/1999/10/08:06:45 Page 21

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Apostilb = 1 lumen m−2 for a perfectly diffusing surface = 10−4 lambert Nit (SI unit) = 10−4 sb = cd m−2 Candle per square inch = 0.487 lambert = 0.155 stilb Foot-lambert = 1.076 × 10−3 lambert = 343 × 10−4 stilb −6 1mv = 0 star per square degree outside = 0.84 × 10 stilb atmosphere = 0.84 × 10−2 nit = 2.63 × 10−6 lambert −6 1mv = 0 star per square degree inside clear = 0.69 × 10 stilb unit airmass Luminous emittance (of a surface) Lumen per square meter (SI unit) = 10−4 lumen cm−2 (light received per unit surface) Phot (CGS unit) = 1 lumen cm−2 (SI unit) 1x = 1 lumen m−2 = 10−4 phot = 1 m-candle Foot-candle = 10.76 lux = 1.076 × 10−3 phot = 1 lumen ft−2 −10 Star, mv = 0, outside Earth atmosphere = 2.54 × 10 phot Electrical units The general inter-relations between electric and magnetic units are given in Sec. 2.6 Electrical charge Coulomb (SI unit) C = 2.997 925 × 109 esu = 0.10 emu =−6.241 51 × 1018 electrons Electron charge e =−4.803 25 × 10−10 esu e =−1.602 18 × 10−19 C Electrical potential Volt (SI unit) V = 3.335 64 × 10−3 esu = 108 emu Potential of electron at first Bohr orbit distance = 27.211 volt = 0.090 767 esu Ionization potential from first Bohr orbit = 13.606 volt = 0.045 384 esu Electric field Volt per meter (SI unit) = 3.335 64 × 10−5 esu = 106 emu Nuclear field at first Bohr orbit = 5.1402 × 1011 volt m−1 = 1.715 2 × 107 esu Resistance Ohm (SI unit)  = 1.112 65 × 10−12 esu = 109 emu Electric current Ampere (SI unit) A = 2.997 925 × 109 esu = 0.10 emu =−6.241 51 × 1018 electrons s−1 Current in first Bohr orbit = 1.054 × 10−3 A = 3.160 × 106 esu Sp.-V/AQuan/1999/10/08:06:45 Page 22

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Electric dipole moment Coulomb-meter (SI unit) C m = 2.9979 × 1011 esu = 10 emu Dipole moment of nucleus and electron in = 0.8478 × 10−29 Cm first Bohr orbit = 2.5417 × 10−18 esu Magnetic field Ampere-turn per meter (SI unit) = 4π × 10−3 [emu] = 3.767 × 108 esu Gauss (in free space) = 1 oersted = 79.58 amp-turn m−1 Gamma γ = 10−5 oersted 1/2 −1/2τ −1 = . × 7 Atomic unit (me a0 0 ) 1 715 10 gauss Field at nucleus due to electron in first Bohr orbit = 1.252 × 105 oersted α 1/2 −1/2τ −1 τ = ω−1 = . × −17 me a0 0 , 0 0 2 4189 10 s Magnetic flux density, Magnetic induction Tesla (SI unit) = 104 gauss = 1 weber m−2 Magnetic moment Weber-meter (SI unit) = (1/4π) 1010 emu = 0.02654 esu ( 1/2 5/2τ −1) = . × −18 −1 Atomic unit me a0 0 2 541 10 erg gauss −20 −1 Bohr magneton, magnetic moment of µB = 0.9274 × 10 erg gauss electron in first Bohr orbit 1 α 1/2 5/2τ −1 2 me a0 0 −24 −1 Nuclear magneton µK = 5.051 × 10 erg gauss µB(me/mp) Earth magnetic moment = 7.98 × 1025 emu Radioactivity Curie [4] = 3.700 × 1010 disintegrations s−1 Roentgen = exposure to radiation producing 2.082 × 109 ion pairs in 0.001293 g of air = 1 esu cm−3 = 2.58 ×10−4 Ckg−1 Rad = 10−2 Jkg−1

2.6 ELECTRIC AND MAGNETIC UNIT RELATIONS

Table 2.4 on pages 24 and 25 is adapted from the previous Astrophysical Quantities edition. Many of these quantities are now superseded by the SI units, but the older esu and emu units are still frequently used for special cases in astrophysics. For the SI units, the permittivity (0) and permeability (µ0) of free space are defined to be exact as (1/4πc2) × 1011 Fm−1 = 8.854 187 817 ...Fm−1 and 4π × 10−7 NA−2, respectively. Here F is the farad capitance unit, N is the newton force unit, and A is the ampere current unit. Sp.-V/AQuan/1999/10/08:06:45 Page 23

2.6 ELECTRIC AND MAGNETIC UNIT RELATIONS /23

REFERENCES

1. Astrophysical Quantities. 1, ¤7 AJ, 115, 635 2. Astrophysical Quantities, 3, ¤10 9. Standish M. 1995, Report of the IAU WGAS Sub- 3. Abramowitz, M. & Stegun, I.A. 1965, Handbook of Group on Numerical Standards, in Highlights of Astron- Mathematical Functions, (Dover, New York), p. 2 omy edited by Appenzeller (Kluwer Academic, Dor- 4. Anderson, H.L. 1989, A Physicist’s Desk Reference, the drecht) second edition of Physics Vade Mecum (American In- 10. Fairhead L., Bretagnon, P. & Lestrade, J.F. 1998, IAU stitute of Physics, New York) Symposium 128 (Kluwer, Dordrecht) p. 419 5. Cohen, E.R., & Taylor, B.N. 1998, 11. Hirayama, Th. et al. 1987, Proc. IAG Symposia I., http://physics.nist.gov/cuu/Reference/versioncon.html IUGG XIX General Assembly, Vancouver 6. Seidelmann, P.K. 1992, Explanatory Supplement to the 12. Astronomical Almanacs (USNO, GPO) Astronomical Almanac (University Science Books, Mill 13. Liu, Bao-Lin, & Fiala, A.D. 1992, Canon of Lunar Valley, CA) Eclipses 1500 B.C.ÐA.D. 3000 (Willmann-Bell, Rich- 7. Dehnen, W., & Binney, J.J. 1998, MNRAS, 298, 387 mond) 8. Metzger, M.R., Caldwell, J.A., & Schechter, P.L. 1998, 14. Nelson, R.A. 1998, Physics today, BG11 Sp.-V/AQuan/1999/10/08:06:45 Page 24

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Table 2.4. Electric and

Quantity SI symbol and unit in esu

Charge Q coulomb C = c × 10−1 esu Current I ampere A = c × 10−1 esu Potential, EMF V volt V = (1/c) × 10−12 esu Electric field E volt/m = (1/c) × 10−14esu Resistance R ohm  = (1/c2) × 109 esu

Resistivity ρ ohm m = (1/c2) × 10−11 esu Conductance G siemens, mho = c2 × 10−9 esu Conductivity σ mho/m = c2 × 10−11 esu Capitance C farad F = c2 × 10−9 cm Electric flux  coulomb C = 4πc × 10−1 esu

Electric flux density, displacement D coulomb/m2 = 4πc × 10−5 esu Polarization P coulomb/m2 = c × 10−5 esu Electric dipole moment coulomb/m = c × 101 esu Permittivity, dielectric constant  farad/m = 4πc2 × 10−11 esu 2 11 Permittivity of free space 0 (1/4πc ) × 10 F/m = 1 esu

Inductance L henry H = (1/c2) × 109 esu Magnetic pole strength m weber Wb = (1/4πc) × 108 esu Magnetic flux  weber Wb = (1/c) × 108 esu Magnetic field H ampere turn/m = 4πc × 10−3 esu Magnetomotive force, magnetic potential F ampere turn AT = 4πc × 10−1 esu

Magnetic dipole moment M weber m = (1/4πc) × 1010 esu Electromagnetic moment m ampere m2 Mag. flux density, induction B tesla T = (1/c) × 104 esu Intensity of magnetization J weber/m2 T = (1/4πc) × 1016 esu Magnetic energy density B × H joule/m3

Permeance  henry H = (1/4πc2) × 109 esu Reluctance 1/henry = 4πc2 × 10−9 esu Permeability µ henry/m = (1/4πc2) × 107 esu −7 2 Permeability of free space µ0 4π × 10 H/m = (1/c ) esu Sp.-V/AQuan/1999/10/08:06:45 Page 25

2.6 ELECTRIC AND MAGNETIC UNIT RELATIONS /25

magnetic units. Dimensions in emu, etc. ESU EMU SI esu LMT κ LMT µ emu L M T I = −1 / / − / / / − / / 10 emu 3 2 1 2 1 1 2 1 2 1 2 0 1 2 1 c 0011 = −1 / / − / / / − − / / 10 emu 3 2 1 2 2 1 2 1 2 1 2 1 1 2 1 c 0001 = 8 / / − − / / / − / − − 10 emu 1 2 1 2 1 1 2 3 2 1 2 2 1 2 c 213 1 = 6 − / / − − / / / − / − − 10 emu 1 2 1 2 1 1 2 1 2 1 2 2 1 2 c 113 1 = 109 emu −101−110−11c2 21−3 −2

= 1011 001−120−11c2 31−3 −2 = 10−9 emu 1 0 −11−101−11/c2 −2 −132 = 10−11 emu 0 0 −11−201−11/c2 −3 −132 = 10−9 emu 1 0 0 1 −102−11/c2 −2 −142 = π × −1 / / − / / / − / 4 10 emu 3 2 1 2 1 1 2 1 2 1 2 0 1 2 1/c 0 0 1 1 = π × −5 − / / − / − / / − / − 4 10 emu 1 2 1 2 1 1 2 3 2 1 2 0 1 2 1/c 2011 = −5 − / / − / − / / − / / − 10 emu 1 2 1 2 1 1 2 3 2 1 2 0 1 2 1 c 2011 = / / − / / / − / / 10 emu 5 2 1 2 1 1 2 3 2 1 2 0 1 2 1 c 1011 = 4π × 10−11 emu 0 0 0 1 −202−11/c2 −3 −142 = (1/c2) emu 1/c2

= 109 cm −102−11001c2 21−2 −2 = ( / π)× 8 / / − / / / − / − − 1 4 10 emu 1 2 1 2 0 1 2 3 2 1 2 1 1 2 c 212 1 = 8 / / − / / / − / − − 10 (Mx) 1 2 1 2 0 1 2 3 2 1 2 1 1 2 c 212 1 = π × −3 / / − / − / / − − / / − 4 10 oersted (Oe) 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 c 1001 = π × −1 / / − / / / − − / / 4 10 (Gb) 3 2 1 2 2 1 2 1 2 1 2 1 1 2 1 c 0001 = ( / π)× 10 / / − / / / − / − − 1 4 10 emu 3 2 1 2 0 1 2 5 2 1 2 1 1 2 c 312 1 = 3 / / − / / / − − / / 10 emu 7 2 1 2 2 1 2 5 2 1 2 1 1 2 1 c 2001 = 4 − / / − / − / / − / − − 10 gauss (Gs) 3 2 1 2 0 1 2 1 2 1 2 1 1 2 c 012 1 = ( / π)× 4 − / / − / − / / − / − − 1 4 10 emu 3 2 1 2 0 1 2 1 2 1 2 1 1 2 c 012 1 = 40π Gs Oe −11−20−11−201−11−20

= (1/4π)× 109 Mx/Gb −102−11001c2 21−2 −2 = 4π × 10−9 Gb/Mx 1 0 −21−100−11/c2 −2 −122 = (1/4π)× 107 emu −202−10001c2 11−2 −2 = 1 emu c2 Sp.-V/AQuan/1999/10/08:06:45 Page 26