Physics 160 Fall 2013 Midterm Exam Equation Sheet Physical constants Quantity
Value
6.67x10-11
8.99x109
3.00x108
1.60x10-19
6.63x10-34
1.24
1.38x10-23
8.62x10-5
Stefan-Boltzmann constant (σ)
5.67x10-8
Radiation constant (a)
7.57x10-16
Electron mass (me)
9.11x10-31
0.511
Proton mass (mp)
1.6726x10-27
938.27
Neutron mass (mn)
1.6749x10-27
938.57
Atomic mass unit (u)
1.66x10-27
Avogadro's number (NA)
6.02x1023
Rydberg constant (RH)
1.10x107
Gas constant (R)
8.31
Bohr radius (a0)
5.29x10-11
Solar mass (M)
1.99x1030
Solar luminosity (L)
3.84x1026
Solar radius (R)
6.96x108
Solar effective temperature
5800
Solar abs. bolometric magnitude
4.74
Solar Spectral Type
G2 V
Earth mass (ME)
5.97x1024
Earth radius (RE)
6.38x106
λ0 for Lyα line (HI n=1 → n=2)
121.6
λ0 for Hα line (HI n=2 → n=3)
656.3
χH = hcRH (HI ionization potential) 13.6
Astronomical unit (AU)
1.50x1011
Light year (ly)
9.46x1015
Parsec (pc)
3.09x1016
Electron volt (eV)
1.60x10-19
Year (yr)
π x107
Atmospheric Pressure (bar)
105
π
3.14
e
2.72
log e
0.434
Gravitational constant (G)
Coloumb’s constant (ke)
Speed of light (c)
Electric charge (e)
Planck's constant (h)
hc
Boltzmann's constant (k)
Units
N m2 kg-2
N m2 C-2
m s-1
C
J s
eV µm
J K-1
ev K-1
W m-2 K-4
J m-3 K-4
kg
MeV
kg
MeV
kg
MeV
kg
mol-1
m-1
J mol-1 K-1
m
kg
W
m
K
kg
m
nm
nm
eV
m
m
m
J
s
Pa = N m-2
Physics 160 Fall 2013 Midterm Exam Equation Sheet Equations Definition of magnitude: m is the apparent magnitude, f is the flux Distance modulus: m1 – m2 = -­‐2.5 log10 (f1/f2) M is the absolute magnitude and d is distance; note that MVega = 0 in all bands M – m = -­‐5 log10 (d/10 pc) Bolometric magnitude: Mbol = 4.74 -­‐ 2.5 log10 (L/L) Bolometric correction: BCV = mbol – V = Mbol -­‐ MV Angular separation: θ (radians) = separation/distance θ (“) = separation (AU) / distance (pc) For radians, separation and distance must be in same units Parallax: π(“) = 1/d(pc) Tangential speed: Vtan (km/s) = 4.74 µ (“/yr) d(pc) µ is the magnitude of proper motion Doppler shift: λ0 is the rest frame wavelength, Vrad is the radial motion, positive for motion away from observer Stefan-­‐Boltzmann Law: F is the flux, T the temperature, L the luminosity and R the radius Blackbody distribution: B(T) is a spectral radiance: energy per unit time per unit area per unit wavelength per unit angular area Wein’s Displacement Law: λpeak is the peak wavelength of the blackbody distribution ∆λ = λ0
Vrad
c F = σT 4 L = 4πR2 σT 4 2hc2
1
Bλ (T ) = 5 hc/λkT
λ e
− 1 λpeakT = 2898 µm K Potential energy for e-­‐ orbital state n in a Hydrogen-­‐like atom: En = -­‐χH/n2 Degeneracy for orbital state n: gn = 2n2 Maxwell-­‐Boltzmann Distribution: n(v)dv = n
n(v) is the number density of particles with velocity v, n is the total number density, m is the particle mass ! m "3/2
2
e−mv /2kT 4πv 2 dv
2πkT
!
2
E −E/kT dE
n(E)dE = √ n
e
kT
kT π
Ideal gas pressure law: n and ρ are the number and mass densities, m = µmH is the individual mass M ≈ uNnucleon is the molar mass P = nkT =
ρRT
ρkT
=
µmH
M Physics 160 Fall 2013 Midterm Exam Equation Sheet 1 4
aT
3
!
j Nj Aj
µn = !
Mean molecular weight: j Nj for neutrals !
Aj = mj/mH, Nj is the number density and Nj Aj
j 1+zj
zj is the proton number of species j µi = !
j Nj for ionized gas Radiation Pressure Law P =
Boltzmann equation: Ni
gi
= e−(Ei −Ej )/kT
Nj
gj
Ni is the number of atoms in a given orbital state i Partition function: Z1 (T ) =
∞
!
gi e−(Ei −E1 )/kT
i=1
Saha equation: Nm is the number of atoms in ionization state m (m = 1 is a neutral atom), χm is the energy required to remove m electrons from a neutral atom 2 Zm+1
Nm+1
=
Nm
ne Zm
!
2πme kT
h2
" 32
e−
Mean free path: n and ρ are the number and mass densities of scatterers, σ is the cross section, κ is the absorption coefficient l = 1/nσ = 1/ρκ Definition of optical depth: dτλ = −nσλ ds = −ρκλ ds Optical depth is inward from surface Equation of Radiative Transfer: Sλ is the source function, jλ is the emission coefficient, κλ is the absorption coefficient, all wavelength-­‐dependent Eqn. of Hydrostatic Equilibrium: dIλ
= Iλ − Sλ
dτλ
Sλ ≡ jλ /κλ P is the local pressure, ρ the mass density, M(r) the mass within radius r, g(r) the surface gravity at radius r M (r)ρ(r)
dPr
= −ρ(r)g(r)
= −G
dr
r2
Equation of Mass Conservation: dMr
= 4πr2 ρ(r)
dr
Equation of Energy Transfer: L is the luminosity at radius r, ε is the energy generation rate per unit mass Equation of Radiative Transport: κ̄ is the mean opacity dLr
= 4πr2 ρ(r)#(r)
dr
3 κ̄(r)ρ(r) L(r)
dT
=−
dr
4ac T 3 (r) 4πr2 (χm+1 −χm )
kT