Formula Sheet for Midterm #2
ASTR 314
λobs − λrest
vr
=
λrest
c
hc
Kmax = hν − φ =
−φ
λ
h
∆λ = λf − λi =
(1 − cos θ)
me c
1
1
1
= RH
− 2 ,
λ
4 n
1
1
1
−
,
= RH
λ
m2
n2
µe4
1
1
En = −
= −13.6 eV 2
32π 2 20 ~2 n2
n
E = hν,
Kepler, Newton, Tidal Forces, and Ellipses
b2 = a2 (1 − e2 )
a(1 − e2 )
,
r=
1 + e cos θ
A = πab
Atoms and Light
0≤e<1
r + r0 = 2a
Mm
F~ = −G 2 r̂
r
Mm
U = −G
r
GM mR
∆F~ '
(2 cos θ î − sin θ ĵ)
r3
1/3
ρ¯P
r < fR
Rp
ρ¯m
GM µ
E = K + U = hU i/2, hU i = −
a
2
4π
P2 =
a3
G(m1 + m2 )
p
L = µ GM a(1 − e2 )
~ = ~r × p~ = ~r × (µ~v )
L
L2 /µ2
GM (1 + e cos θ)
dA
1 L
=
dt
2 µ
m1 m2
µ=
m1 + m2
m
r
~
+ m2 r~2
1
1
~ =
R
m1 + m2
r=
E = hc/λ,
n = 3, 4, 5, . . .
m < n(both integers)
Distances, Magnitudes, Spectrum of Light
1
pc
p00
L
F =
4πr2
d=
F2
= 100(m1 −m2 )/5 = 100.4(m1 −m2 )
F1
F1
m1 − m2 = −2.5 log10
F2
L
M = Msun − 2.5 log10
L
d
m − M = 5 log10 (d) − 5 = 5 log10
10 pc
L = E/t
c = λν
λpeak T = 0.002897755 m K
L = 4πR2 σT 4
Fsurf = σT 4
2hν 3 /c2
2hc /λ
Bλ (T ) = hc/λkT
, Bν (T ) = hν/kT
e
−1
e
−1
BC = mbol − V = Mbol − MV
2
5
Integrals
U − B = MU − M B , B − V = MB − MV
dx
Z ∞
= −a ln (a − x)
1 − x/a
U = −2.5 log10
Fλ SU dλ + CU
Z
0
dx
!
R∞
= a ln(x − a) + x
B
S
dλ
λ
U
1 − a/x
U − B = −2.5 log10 R0∞
+ CU −B
p
√
Bλ SB dλ
√
0
= x2 − ax + a ln x − a + x + a ln 2
Z ∞
Z
Z
dx
p
1 − a/x
Z
dx
p
1 − x/a
p
= −2 a2 − x
mbol = −2.5 log10
Fλ dλ + Cbol
0
–2–
Stars, Pressure
Relativity
γ=p
∆t = p
L0 = p
z=
mc2
1−
1 − u2 /c2
∆t0
1 − u2 /c2
L
1 − u2 /c2
λobs − λrest
= ∆λ/λrest
λrest
s
1 + vr /c
−1
z=
1 − vr /c
p~ = p
E=p
1
u2 /c2
,
m~v
1 − u2 /c2
E 2 = p2 c2 + m2 c4
p
νrest 1 − u2 /c2
νobs =
1 + (u/c) cos θ
p
νrest 1 − u2 /c2
=
1 + vr /c
p
∆tobs 1 − u2 /c2
∆trest =
1 + (u/c) cos θ
p
∆tobs 1 − u2 /c2
=
1 + vr /c
Nucleosynthesis and Radioactivity
Eb = ∆mc2 = [Zmp + (A − Z)mn − mnucleus ]c2
Rwd
Q = ∆mc2
5/3
(3π 2 )2/3 ~2
Z
ρ
P =
5
me
A mH
5/3
2
2/3
~
Z
1
(18π)
≈
1/3
10
A mH
Gme Mwd
dN
= −λN
dt
N (t) = N0 e−λt
ln(2)
λ=
τ1/2
−2/3 −33.80T61/3
W kg−1
−2/3 −152.28T61/3
W kg−1
pp = 0.241ρX 2 fpp ψpp Cpp T6
CN O = 8.67 × 1020 ρXXCNO CCNO T6
e
e
m1
a2
=
m2
a1
m1
v2r
=
m2
v1r
P (v1r + v2r )3
m1 + m2 =
2πG
sin3 i
3
P 3
m2
sin3 i =
v
2
(m1 + m2 )
2πG 1r
3/2 1/2
5kT
3
MJ '
GµmH
4πρ0
1/2
15kT
RJ '
4πGµmH ρ0
MV = −2.81 log10 Pd − 1.43
L
log10
= 1.15 log10 Pd + 2.47
L
Mr ρ
dP
= −G 2
dr
r
P
V
=
nkT
Z
1 ∞
P =
np pvdv
3
Z ∞0
1
P =
mnv v 2 dv
3 0
ρkT
Pg =
µmH
µ = m̄/mH
Z ∞
1
mnv v 2 dv
P =
3 0
ρkT
1
Pt =
+ aT 4
µmH
3
4πGc
LEd =
M
κ̄