Useful formulas for astronomy & astrophysics olympiad

1

Spherical astronomy

Law of sines and cosines (+ cyclic change) a sin α c2

b c = sin β sin γ = a2 + b2 − 2ab cos γ. =

Spherical law of sines and cosines (+ cyclic change) sin a = sin α cos c =

sin b sin c = sin β sin γ cos a cos b + sin a sin b cos γ.

Paralaksinis trikampis: spherical triangle, consists of zenith Z, celestial pole P + given star S. ZP = 90 − ϕ (ϕ - latitude), P S = 90−δ (δ - declination), ZS = z (z - zenith distance), angle ZP S is hour angle t (measured from south clockwise), angle SZP is 180 − A (A - azimuth, measured from south clockwise); it provides transformation between azimuthal (A, z) and equatorial coordinate system (t, δ) resp. (α, δ) where α = ϑ − t (ϑ - sidereal time, α measured from vernal equinox point counterclockwise) - just use sph. law of sines and cosines. The equation of time = tapp − tm (tapp - apparent solar time, tm - mean solar time). Precession: period approx. 26000 y, vernal equinox point moves in the opposite direction than right ascension increases. Aberration: tg α ≈ α = vc⊥ , v⊥ is the component of velocity perpendicular to comming rays. Refraction: near horizon approx 35.40 , Paralax: π(arcsec) = 1/r(pc) (Annual)

3

Resolution ability: α = 1.22λ/D. Magnification of the keplerian telescope: m = f1 /f2 (f1 - focal length of objective, f2 - focal length of eye-piece).

4

Celestial mechanics

Stellar astrophysics

4 Stefan-Boltzman law: L = 4πR2 σTef (you don’t have to know Planck’s formula I guess). Wien displacement law: λmax = b/Tef Pogson formula: m1 − m2 = −2.5 log(I1 /I2 ), m − M = = 5 log r − 5 (when extinction or bolometric correction given, use your head). Spectral classification: (W - Wolf-Rayet stars) Oh Be A Fine Girl and Kiss Me Like That! (Y - substellar objects) Luminosity classes: 0 - hypergiants, I - supergiants, II bright giants, III - normal giants, IV - subgiants, V - main sequence stars, VI - subdwarfs, VII - white dwarfs (Sun is G2V star). Chandrasekhar limit: (between white dwarf and neutron star) 1.4Ms , TOV limit: (between neutron star and black hole) 1.5 − 3Ms , Schwarzschild radius: R = 2GM/c2 Sun: ms = 1.99 · 1030 kg, Ls = 3.85 · 1026 W, Ms = = 4.83 mag, K = Ls /4πAU2 = 1368 W·m−2 , Ts = 5778 K, Rs = 6.96 · 108 m, Prot = 25 d. Hubble’s law: v = Hr, for small z: v = cz. H = = (71.0 ± 2.5) km·s−1 ·Mpc−1 (WMAP, 2010).

5 2

Telescopes

Math & Physics

Doppler effect: v/c = z = ∆λ/λ. Relativistic: z = p = (c + v)/(c − v) − 1. Gravitational: for weak fields (Sun) Conservation of: meachnical energy (kinetic + potential), can be derived from conservation of energy for photon (for pot. ~ = m~r × ~v = const., |L| ~ = mrv sin α (~r is energy use m = E/c2 = hf /c2 ) angular momentum L radius vector, ~v is velocity). v u Gravitation: u 1 − 2GM 1 − rGM 2 r1 c2 1c z=t − 1 ≈ − 1. 2GM GM m1 m2 m1 m2 m1 m2 1 − r2 c2 1 − r2 c2 , Etot = −G F = G 2 , Epot = −G r r 2a h Simple quantum formulas: E = hf = hc λ , λ = p (De Brog3rd and 2nd Kepler law: lie wavelength, p - momentum). 3 Spectroscopy: electron transitions in the hydrogen atom (a1 + a2 ) G (m1 + m2 ) 1 2 πab = , r ω = = const.   P2 4π 2 2 P 1 1 − 2 , ∆E = hf = R m2 n Elipse geometry: b2 + ε2 = a2 (ε is linear excentricity), e = = ε/a (e is numeric excentricity), rp = a(1−e) (rp - pericenter), R ≈ 13.6 eV, ∆E - energy absorbed by transition from m-th to ra = a(1 + e) (apocenter), S = πab p n-th level (thus f is the frequency of given spectral line, m = 1 Velocities: circular orbitp- vc = GM/r, parabolic orbit - Lyman, m = 2 - Balmer, m = 3 - Paschen). q (escape velocity) - vp = 2GM/r, eliptic orbit (don’t me2 1 − vc2 , t0 = γt, l0 = l/γ, m = Special relativity: γ = 1/ morize, derive it from conservation of mech. energy) v = el p = γm0 , p = mvγ. Energies: E = mc2 , E 2 − p2 c2 = m20 c4 , = GM (2/r − 1/a). 2 Sidereal period (orbital, prograde motion): (E - Earth’s Ekin = m0 c (γ − 1). nRT , U = f2 N kT (f - number of degrees of sid. period, S - planet’s synodical period) inferior (Mercury, Thermod.: pV = p p Venus) resp. exterior (Mars, Jup. etc.) planets 1/P = freedom), vRM S = 3kT /m, vprob = 2kT /m. = (1/E) + (1/S) resp. 1/P = (1/E) − (1/S). Optics: 1/f = (1/a) + (1/a0 ) (a - object distance, a0 - image Sidereal period (rotation, prograde motion): 1/T = distance), sin α/ sin β = n2 /n1 ,interf. max. cond.: ∆ = kλ (∆ = (1/S) + (1/P ) (P - sid. orbital period, S - syn. rot. period). - path difference) Retrograde motion would change signs, direction of Sun’s mo- Constants: c = 3 · 108 m, G = 6.673 · 10−11 N·m2 ·kg−2 , tion on the sky etc. - easy to derive (angular velocities). h = 6.626 · 10−34 J·s, k = 1.381 · 10−23 J·K−1 , σ = 5.67 · 11 6 Earth + Moon: ae = 1.496 · 10 m, Re = 6.371 · 10 m, 10−8 W·m−2 ·K−4 , b = 2.9 · 10−3 m·K (Wien) Me = 5.974 · 1024 kg, am = 3.844 · 108 m, Rm = 1.737 · 106 m, Approximations: (1 + x)n ≈ 1 + nx, sin x ≈ tg x ≈ x, Mm = 7.348 · 1022 kg. cos x ≈ 1 − x2 /2, ln(1 + x) ≈ x.