TOOLBOX
Maybe have general stuff up above, then primary equations/relations for each topic.
(1) Units & Constants
| Primary psm Units and Conversions | |||
| Quantity | cgs | mks | psm |
| distance | cm | m = 102 cm | pc = 3.086 x 1018 cm |
| mass | gm | kg = 103 gm | M = 1.989 x 1033 gm |
| time | s | s | Myr = 3.156 x 1013 s |
| velocity | cm s-1 | m s-1 = 102 cm s-1 | pvu = pc Myr-1 = 9.778 x 104 cm s-1
 1 km/s |
| density | gm cm-3 | kg m-3 = 10-3 gm cm-3 | pdu = M pc-3 = 6.77 x 10-23 gm cm-3 = 40.5 mp cm-3 |
| G | 6.67 x 10-8 | 6.67 x 10-11 | G = 4.50 x 10-3 pc3 M-1 Myr-2 (km/s)2 pc M-1 |
| Less Commonly Used psm Units | |||
| Quantity | cgs | mks | psm |
| energy | erg = gm cm2 s-2 | J = kg m2 s-2 = 107 erg | peu = M pc2 Myr-2 = 1.90 x 1043 erg M (km/s)2 |
| luminosity | erg s-1 | W = J s-1 = 107 erg s-1 | plu = peu Myr-1 = 6.02 x 1029 erg s-1 = 1.56 x 10-4 L |
| acceleration | cm s-2 | m s-2 = 102 cm s-2 |
pau = pc Myr-2 = 3.10 x 10-9 cm s-2 km/s Myr-1 |
| force | dyne = gm cm s-2 | N = 105 dyne |
pfu = M pc Myr-2 = 6.16 x 1024 dyne M km/s Myr-1 |
| momentum | gm cm s-1 | kg m s-1 = 105 cgs | pmu = M pc Myr-1 = 1.94 x 1038 cgs M km/s |
| angular mom. | gm cm2 s-1 | kg m2 s-1 = 107 cgs | pamu = M pc2 Myr-1 = 6.00 x 1056 cgs M km/s pc |
| frequency | Hz | Hz | pfru = Myr-1 = 3.17 x 10-14 Hz km/s pc-1 |
| Astronomical Constants | |||
| Quantity | cgs | mks | psm |
| yr | 3.16 x 107 s | 3.16 x 107 s | 10-6 Myr |
| pc | 3.08 x 1018 cm | 3.08 x 1016 m | 1 pc |
| lyr | 9.46 x 1017 cm | 9.46 x 1015 m | 0.307 = 1 / 3.26 pc |
| AU | 1.49 x 1013 cm | 1.49 x 1011 m | 4.85 x 10-6 = 1 / 206265 pc |
| R | 6.96 x 1010 cm | 6.96 x 108 m | 2.26 x 10-8 pc [109 R ] |
| M | 1.99 x 1033 gm | 1.99 x 1030 kg | 1 M [3.33 x 105 M] |
| L | 3.85 x 1033 erg s-1 | 3.85 x 1026 W | 6400 plu |
| (M/L) | 0.516 gm / (erg s-1) | 5160 kg / W | 1.56 x 10-4 M / plu |
| H | 3.24 x 10-18 h Hz | 3.24 x 10-18 h Hz | 1.02 x 10-4 h Myr-1 ( 100h / 106) |
| tH = H-1 | 3.09 x 1017 h-1 s | 9.78 h-1 Gyr | 9778 h-1 Myr ( 106 / 100h) |
| rH = c H-1 | 9.25 x 1027 h-1 cm | 9.78 h-1 G lyr | 2.99 h-1 Gpc ( 3 x 105 x 106 / 100h) |
 crit = 3H2/8 G | 1.88 x 10-29 h2 gm cm-3 | 1.88 x 10-25 h2 kg m-3 | 2.77 x 10-7 h2 M pc-3 |
| Physical Constants | |||
| Quantity | cgs | mks | other |
| G | 6.67 x 10-8 cm3 gm-1 s-2 | 6.67 x 10-11 | 4.50 x 10-3 (psm) |
| c | 3.00 x 1010 cm s-1 | 3.00 x 108 m s-1 | 3.06 x 105 pc Myr-1 (psm) |
| mp | 1.67 x 10-24 gm | 1.67 x 10-27 kg | 938 MeV / c2 |
| me | 9.11 x 10-28 gm | 9.11 x 10-31 kg | 0.511 MeV / c2 |
| h | 6.63 x 10-27 erg s | 6.62 x 10-34 J s | 4.14 x 10-15 eV s |
| eV | 1.60 x 10-12 erg | 1.60 x 10-19 J | 1.78 x 10-33 gm c2 |
| kB | 1.38 x 10-16 erg K-1 | 1.38 x 10-23 J K-1 | 8.62 x 10-5 eV K-1 |
 | 5.67 x 10-5 erg cm-2 s-1 K-4 | 5.67 x 10-8 J m-2 s-1 K-4 | 3.54 x 107 eV cm-2 s-1 K-4 |
| a = 4/c | 7.56 x 10-15 erg cm-3 K-4 | 7.56 x 10-16 J m-3 K-4 | 4.72 x 10-3 eV cm-3 K-4 |
add Wein's constant (and/or have a separate table for the radiation relations, like in the cosmology section).
(2) Cosmology
| density match | condition | a @ equality | z @ equality | t @ equality |
| v = m |
0.73 = 0.27 a-3 | 0.72 | 0.39 | 9.43 Gyr |
| v = rel |
0.73 = 8.4 × 10-5 a-4 | 0.103 | 8.3 | 615 Myr |
b =  |
0.04 a-3 = 5.0 × 10-5 a-4 | 1.25 × 10-3 | 800 | 620 kyr |
| m = rel |
0.27 a-3 = 8.4 × 10-5 a-4 | 3.11 × 10-4 | 3200 | 57 kyr |
| m = |
0.27 a-3 = 5.0 × 10-5 a-4 | 1.85 × 10-4 | 5400 | 22 kyr |
Note that here rel refers to the sum of photons and neutrinos (relativistic matter)
Likewise m refers to the sum of baryons and CDM (non-relativistic matter)
| Useful Approximations [ see image ] |
Radiation era: log a < -3.8 t < 15 kyr E(a)
 r,o½ a-2 |
t = tH,o  da / a E(a) (0 to a) ½ tH,o
r,o-½ a2 741 a2 h72-1 Gyr = 2.34 x 1019 a2 h72-1 sec
|
| a 1.16 x 10-6 h72½ tyr½ 2.09 x 10-10 h72½ ts½
|
| T = 2.725 / a K = 2.39 x 10-4 / a eV 1.31 x 1010 h72-½ ts-½ K
1.14 h72-½ ts-½ MeV
|
|
rH = rH,o / E(a) c tH,o
r,o-½ a2 1480 a2 h72-1 Glyr = 2 c t
|
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rph = a rH,o da' / a'2 E(a') (0 to a) c tH,o
r,o-½ a2 1480 a2 h72-1 Glyr = 2 c t = rH
|
| Matter era: 0.002 < a < 0.8 1 Myr < t < 10 Gyr E(a)
m,o½ a-3/2 |
|
t = tH,o da / a E(a) (0 to a) 2/3 tH,o
m,o-½ a3/2 17.4 a3/2 h72-1 Gyr
|
| a 1.49 x 10-7 h722/3 tyr2/3
|
|
rH = rH,o / E(a) c tH,o
m,o-½ a3/2 26.1 a3/2 h72-1 Glyr = 3/2 c t
|
|
rph = a rH,o da' / a'2 E(a') (0 to a) 2 c tH,o
m,o-½ a3/2 52.3 a3/2 h72-1 Glyr = 3 c t = 2 rH
|
| Dark Energy Era: a > 0.8 t > 10 Gyr E(a)
v,o½ |
t - tnow =  t = tH,o da / a E(a) (1 to a) tH,o
v,o-½ ln(a) 15.9 ln(a) h72-1 Gyr
|
| a exp[
tGyr h72 / 15.9 ] (tnow = 0.988 tH,o)
|
|
rH = rH,o / E(a) c tH,o
v,o-½ 15.9 h72-1 Glyr
(a constant)
|
rph = a rH,o da' / a'2 E(a') (0 to a) =
|
|
reh = a rH,o da' / a'2 E(a') (a to ) c tH,o
v,o-½ 15.9 h72-1 Glyr = rH (a constant)
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| Epoch | Time | Scale Factor | Temperature | Energy |
| Planck | 10-42 s | 10-32 | 1032 K | 1019 GeV |
| GUT | 10-36 s | 10-28 | 1028 K | 1015 GeV |
| Electro-Weak | 10-10 s | 10-15 | 1015 K | 100 GeV |
| Quark-Hadron | 3 × 10-5 s | 10-12 | 2 × 1012 K | 200 MeV |
 -decouple |
1 s | 2 × 10-10 | 1.3 × 1010 K | 1 MeV |
| He4 synthesis | 200 s | 2.9 × 10-9 | 9.4 × 108 K | 80 keV |
| Equality | 57 kyr | 3.1 × 10-4 | 8700 K | 0.75 eV |
| Recombination | 380 kyr | 9.2 × 10-4 | 2900 K | 0.25 eV |
| Today | 14 Gyr | 1.0 | 2.72 K | 2.35 × 10-4 eV |
more robust, concordance cosmology relations (including subroutine for doing the main integral).
redshift conversions -- see peterson AGN p 156
(3) Radiation & Photometry
Blackbody curve/constants, Blam & Bnu
photometric conversions, including from UBV etc to Lsuns, zero points, wavelengths, Jansky etc
add Wein's constant (and/or have a separate table for the radiation relations, like in the cosmology section).
spectral regions defined: EUV (13.6 - 100eV = 91-12nm); soft X (0.1-1keV = 12 - 1.2nm); med (1-10keV = 1.2-0.12 nm); hard X (>10keV).
reddening curve
galactic reddening, link to NED calculator
(4) Kinetic Theory
Kinetic relations : pcle velocities vs T; sound speeds; shock velocity & post shock temps; Pressures nT; B2/8pi; radiation; Maxwell Bolzmann relation & expressions for mode(v) = sqrt(2kT/m); ave(v) = 2/sqrt(pi) * mode(v); and ave(v2) = 3kT/m and ave(E) = 1.5kT . (Flowers & Mendoza, p 109
column densities & emission measures; n = ntot = ne + np + nHI + nH2 = ??ne for ionized.
T(K) = 7740 E(eV)... (3/2 kT); lam(nm) = 1240/E(eV); nu(GHz) = 2.42 e5 E(eV)
(5) Coordinates/Sky Relations
All Sky = 4 Steradians = 41,253 deg2 = 5.34 x 1011 arcsec2
coordinate systems : RA/DEC, precession, l,b, and supergalactic L,B. link to NED calculator.
velocity corections: heliocentric; galactocentric; LSR; Local Group; Virgocentric; CMB
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