Semester 1 - Formulae Flashcards
Number of stars per unit volume
N = ∫ ndV
Volume element of a sphere
dV = r^2 sinθdrdθdΦ
Galactic longitude
l = CGX
angle between the galactic centre the galactic North Pole and the star
Galactic latitude
b = 90° - GX
the stars velocity due solely to the rotation of the Milky Way
v = ω x R
the suns velocity due solely to the rotation of the Milky Way
v_sun = ω_sun x R_sun
the position of the star
R = R_sun + ds
Stars motion with respect to the sun
v = v - v_sun
Relation between Oorts constants A and B
A - B = Omega 0
A - B = Vc/R
Circular velocity
Vc = (GM/R)^(1/2)
The number of galaxies per Mpc^3
N = ( L ∫ 0) Φ(L)
The galactic luminosity function = Φ(L)
Relation between Φ(L) and Φ(M)
Φ(M) = - Φ(L) dL/dM
Surface brightness
I(R) = F/Ω
where Ω = S/D^2
and F and Ω are both proportional to 1/d^2
Flux in terms of density
F = fsun ∫ pdz . S
Luminosity
L =2π (∞ ∫ 0) I(R)RdR
3-dimensional luminosity density
I(R) = (∞ ∫ -∞) j(r) dz
z^2 = r^2 - R^2
to find dz
Schechter function
Φs(L)dL on formula sheet
Redshift
z = v/c = ∆λ/λ
Time for an encounter to occur
tencounter = 1/πr^2vn
Gauss’s law
∇ . F = - 4πGp
Virial theorem
2<T> + <u> = 0</u></T>
T = 1/2Nmv^2
U = GM^2/R
Crossing time
Tcross = 2R/V
Fraction of spirals
f = ns/ntot
where ntot = ne + ns
e for elliptical
A(m)
= dN/dm = dN/dM
= (∞ ∫ 0) Φ(M)r^2dr
Malmquist Bias
M0 - <M> = σ^2 dlnA(m)/dm</M>
Derive poisson’s equation
F = -∇Φ
∇.F = -4πGp
substitute F in giving
∇Φ^2 = 4πGp
Derive oorts constants
if vlos = ϴcosα - ϴ0sinl
and μ = ϴsinα - ϴ0cosl
Assuming circular and planar motion replace ϴ = Ω/R
Rcosα = R0sinl
Rsinα = R0cosl - d
Taylor expand to first order
Ω = Ω0(R0) + d Ω0/dR|R0 (R-R0)
Ω - Ω0(R0) = d Ω0/dR|R0 (R-R0)
Ω = ϴ/R so replace d Ω0/dR|R0
(R-R0) = -dcosθ
cos2θ= 2cos^2θ -1
Relaxation time
When Δv^2 = v^2
Strong encounters
<u> ≥ <T></T></u>
rs < 2Gm/v^2
derive a weak encounters
when r0 > rs > 2Gm/v^2
Force on formula sheet
Perpendicular component cos theta = b/r where r = b^2 + v^2t^2
F = M dv/dt
Integrate
1/(a^2+s^2)^(3/2) = 2/a^2
giving Δv(perp) = 2Gm/bv
minimum for a weak encounter
bmin = 2Gm/v^2
Schechter function - total number of galaxies
ntot = Φ* Γ(alpha + 1)
Schechter function - total luminosity
Ltot = Φ* L*Γ(alpha + 2)
Cylindrical coordinate system in terms of Π, Θ and Z
<Π> = dR/dt = 0
<Z> = R dθ/dt = 0
<Θ> = dz/dt ≠ 0
</Θ></Z></Π>
LSR coordinate system
ΠLSR = 0
ΘLSR = Θ0
ZLSR = 0
Axisymmetric drift
< v > = -C < u^2 >
calculating peculiar velocities Δu, Δv and Δw
Δu = u - usun
Δv = v - vsun
Δw = w - wsun
where usun = -<Δu>, vsun = -C<u^2> - <Δv> and wsun = - <Δw></Δw></Δv></Δu>
how do we find <v></v>
a least squares fit
similiarly vsun is the intersect of the graph Δv versus σu^2, where C is the slope
<Δv> = -C<u^2> - vsun
</Δv>
Velocity of the LSR
Θ0 = R0(A-B)
Mean rotation period
t = 2π/Ω0 where Ω0 = Θ/R0
de Vaucoulers law
I(R) = IE exp[-7.67(R/RE)^1/4-1]
where IE is the surface brightness at R = RE
and RE is the radius at which the integrated luminosity is half the total
Derive the potential of a uniform sphere
poisson’s equation and using spherical polar coordinates
r^2 dΦ/dr = b
Φ = -GM/r where b = GM
How do we calculate the potential of a flat disk?
through separation of variables J(R)Z(z).
Derive the star’s energy for orbiting stars
d/dt(mv) + m.∇Φ(x) = 0
multiply by v
where Φ(x) = -GM/x
and dΦ/dt = v.∇Φ(x)
d/dt[1/2mv^2] = mv(dot).v
we get d/dt [1/2mv^2 + mΦ(x)] = 0
i.e. KE + PE = 0
escape speed
ve^2 = -2Φ(x)
Φ(x) = -GM/x
speed of the Milky way at the distance of the Sun
v(R0) = Ω(R0)R0
rotational period of the Milky way at the distance of the sun
P(R0) = 2π/Ω(R0)
Derive the virial theorem
if I = (N Σ i=1) miri.ri
differentiate with respect to time
take the second derivative
at equilibrium the moment of inertia is constant = - ∑i∑j Gmimj/2|rij|
Derive the relaxation time for a weak interaction given Δv(perp)= 2Gm/bv.
V = 2πbvt db
N(collisions) = V x (N/Vsphere)
< Δv(perp) >^2 = (bmax ∫ bmin) Δv(perp)^2 . N(collisions)
Δv(perp)= 2Gm/bv
bmax = R
bmin = 1AU
Trelax = Δv^2 = v^2
Elliptical galaxies - ellipticity.
N = 10(1-b/a)
where the ellipticity ϵ = 1-b/a
Point-spread function P(d)
P(d) = 1/2πσ^2 exp[-d^2/2σ^2]
modified hubble law
Io = 2rojo
bulge fraction
B/T = LB/Ltot
where Ltot = LD + LB
disk-to-bulge ratio
D/B = (B/T)^-1
Where T is the total
Peculiar velocities
u = Π - ΠLSR = Π
v = Θ - ΘLSR = Θ-Θ0
w = Z - ZLSR = Z
Angular velocity curve
Ω(R) = Θ(R)/R
Derive the malmquist bias
if A(m) = (∞ ∫ 0) Φ(M)r^2 dr
Φ(M) = Φ0exp[-(M-M0)^2/2σ^2]
take the derivative with respect to M
and divide through by A(m)
rearranging gives as required
Derive the de Vaucouleurs surface brightness profile if
m = a + bR^1/4
me = a + bRe^1/4
bRe^1/4/2.5 = 3.33
subtract the two relations m - me.
use pogson’s equation
I(R) = Ie10^(-3.33[(R/R)^1/4 -1])
= Ie exp{-7.67[(R/Re)^1/4-1]}
Express the de Vaucoulers profile
I(R) = Ie exp{-7.67[(R/Re)^1/4 -1]}
in the form of the Sersic profile
I(R) = I0 exp{-(R/𝛼)^1/n}
Take the exp{-7.67} out
comparing the form we can see I0 = Ie exp{7.67} n = 4 and 𝛼 = Re/(7.67)^4
Intensity
I = F/θ^2
ϴ0
ϴ0 = ϴ(R0) where R0 is the galactocentric distance
Volume swept out by a star
V = πr^2vt
