Semester 1 - Formulae

Question 1

Number of stars per unit volume

Answer 1

N = ∫ ndV

Question 2

Volume element of a sphere

Answer 2

dV = r^2 sinθdrdθdΦ

Question 3

Galactic longitude

Answer 3

l = CGX

angle between the galactic centre the galactic North Pole and the star

Question 4

Galactic latitude

Answer 4

b = 90° - GX

Question 5

the stars velocity due solely to the rotation of the Milky Way

Answer 5

v = ω x R

Question 6

the suns velocity due solely to the rotation of the Milky Way

Answer 6

v_sun = ω_sun x R_sun

Question 7

the position of the star

Answer 7

R = R_sun + ds

Question 8

Stars motion with respect to the sun

Answer 8

v = v - v_sun

Question 9

Relation between Oorts constants A and B

Answer 9

A - B = Omega 0

A - B = Vc/R

Question 10

Circular velocity

Answer 10

Vc = (GM/R)^(1/2)

Question 11

The number of galaxies per Mpc^3

Answer 11

N = ( L ∫ 0) Φ(L)

The galactic luminosity function = Φ(L)

Question 12

Relation between Φ(L) and Φ(M)

Answer 12

Φ(M) = - Φ(L) dL/dM

Question 13

Surface brightness

Answer 13

I(R) = F/Ω

where Ω = S/D^2

and F and Ω are both proportional to 1/d^2

Question 14

Flux in terms of density

Answer 14

F = fsun ∫ pdz . S

Question 15

Luminosity

Answer 15

L =2π (∞ ∫ 0) I(R)RdR

Question 16

3-dimensional luminosity density

Answer 16

I(R) = (∞ ∫ -∞) j(r) dz

z^2 = r^2 - R^2

to find dz

Question 17

Schechter function

Answer 17

Φs(L)dL on formula sheet

Question 18

Redshift

Answer 18

z = v/c = ∆λ/λ

Question 19

Time for an encounter to occur

Answer 19

tencounter = 1/πr^2vn

Question 20

Gauss’s law

Answer 20

∇ . F = - 4πGp

Question 21

Virial theorem

Answer 21

2<T> + <u> = 0</u></T>

T = 1/2Nmv^2

U = GM^2/R

Question 22

Crossing time

Answer 22

Tcross = 2R/V

Question 23

Fraction of spirals

Answer 23

f = ns/ntot

where ntot = ne + ns

e for elliptical

Question 24

A(m)

Answer 24

= dN/dm = dN/dM

= (∞ ∫ 0) Φ(M)r^2dr

Question 25

Malmquist Bias

Answer 25

M0 - <M> = σ^2 dlnA(m)/dm</M>

Question 26

Derive poisson’s equation

Answer 26

F = -∇Φ

∇.F = -4πGp

substitute F in giving

∇Φ^2 = 4πGp

Question 27

Derive oorts constants

if vlos = ϴcosα - ϴ0sinl
and μ = ϴsinα - ϴ0cosl

Answer 27

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

Question 28

Relaxation time

Answer 28

When Δv^2 = v^2

Question 29

Strong encounters

Answer 29

<u> ≥ <T></T></u>

rs < 2Gm/v^2

Question 30

derive a weak encounters

Answer 30

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

Question 31

minimum for a weak encounter

Answer 31

bmin = 2Gm/v^2

Question 32

Schechter function - total number of galaxies

Answer 32

ntot = Φ* Γ(alpha + 1)

Question 33

Schechter function - total luminosity

Answer 33

Ltot = Φ* L*Γ(alpha + 2)

Question 34

Cylindrical coordinate system in terms of Π, Θ and Z

Answer 34

<Π> = dR/dt = 0
<Z> = R dθ/dt = 0
<Θ> = dz/dt ≠ 0
</Θ></Z></Π>

Question 35

LSR coordinate system

Answer 35

ΠLSR = 0

ΘLSR = Θ0

ZLSR = 0

Question 36

Axisymmetric drift

Answer 36

< v > = -C < u^2 >

Question 37

calculating peculiar velocities Δu, Δv and Δw

Answer 37

Δu = u - usun
Δv = v - vsun
Δw = w - wsun

where usun = -<Δu>, vsun = -C<u^2> - <Δv> and wsun = - <Δw></Δw></Δv></Δu>

Question 38

how do we find <v></v>

Answer 38

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>

Question 39

Velocity of the LSR

Answer 39

Θ0 = R0(A-B)

Question 40

Mean rotation period

Answer 40

t = 2π/Ω0 where Ω0 = Θ/R0

Question 41

de Vaucoulers law

Answer 41

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

Question 42

Derive the potential of a uniform sphere

Answer 42

poisson’s equation and using spherical polar coordinates
r^2 dΦ/dr = b

Φ = -GM/r where b = GM

Question 43

How do we calculate the potential of a flat disk?

Answer 43

through separation of variables J(R)Z(z).

Question 44

Derive the star’s energy for orbiting stars

Answer 44

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

Question 45

escape speed

Answer 45

ve^2 = -2Φ(x)

Φ(x) = -GM/x

Question 46

speed of the Milky way at the distance of the Sun

Answer 46

v(R0) = Ω(R0)R0

Question 47

rotational period of the Milky way at the distance of the sun

Answer 47

P(R0) = 2π/Ω(R0)

Question 48

Derive the virial theorem

if I = (N Σ i=1) miri.ri

Answer 48

differentiate with respect to time

take the second derivative

at equilibrium the moment of inertia is constant = - ∑i∑j Gmimj/2|rij|

Question 49

Derive the relaxation time for a weak interaction given Δv(perp)= 2Gm/bv.

Answer 49

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

Question 50

Elliptical galaxies - ellipticity.

Answer 50

N = 10(1-b/a)

where the ellipticity ϵ = 1-b/a

Question 51

Point-spread function P(d)

Answer 51

P(d) = 1/2πσ^2 exp[-d^2/2σ^2]

Question 52

modified hubble law

Answer 52

Io = 2rojo

Question 53

bulge fraction

Answer 53

B/T = LB/Ltot

where Ltot = LD + LB

Question 54

disk-to-bulge ratio

Answer 54

D/B = (B/T)^-1

Where T is the total

Question 55

Peculiar velocities

Answer 55

u = Π - ΠLSR = Π
v = Θ - ΘLSR = Θ-Θ0
w = Z - ZLSR = Z

Question 56

Angular velocity curve

Answer 56

Ω(R) = Θ(R)/R

Question 57

Derive the malmquist bias

if A(m) = (∞ ∫ 0) Φ(M)r^2 dr

Φ(M) = Φ0exp[-(M-M0)^2/2σ^2]

Answer 57

take the derivative with respect to M

and divide through by A(m)

rearranging gives as required

Question 58

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

Answer 58

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]}

Question 59

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}

Answer 59

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

Question 60

Intensity

Answer 60

I = F/θ^2

Question 61

ϴ0

Answer 61

ϴ0 = ϴ(R0) where R0 is the galactocentric distance

Question 62

Volume swept out by a star

Answer 62

V = πr^2vt