Semester 2 - Formulae

Question 1

Tully fisher relation

Answer 1

LD ∝ Vmax^(Beta)

Question 2

Mass

Answer 2

m(r) = ( r ∫ 0) dm

dm = 4πp(r)r^2dr

Question 3

Time averaged kinetic energy

Answer 3

<K> = 1/2 Nm <v^2>

Nm = M
</K>

Question 4

Velocity dispersion

Answer 4

3σ0^2 = <v^2>

Question 5

Central radiation velocity dispersion

Answer 5

σ0^2 = GM/5R

Question 6

Hubbles constant

Answer 6

d = v/H0

Question 7

Eddington luminosity

Answer 7

L(edd) on formula sheet

Question 8

Mass accretion rate

Answer 8

L = ηM(dot)c^2

Question 9

Schwarzschild radius

Answer 9

RS = 2GM/c^2

Question 10

Flux in terms of flux density

Answer 10

Flux = ∫ Sv dv

Question 11

characteristic timescale

Answer 11

t = E/L

Question 12

Strength of magnetic field

Answer 12

Umag = B^2/2μ0

Question 13

Equipartition

Answer 13

Etotal = 2Emag

Emag = V Umag

Question 14

Lorentz factor through synchrotron radio emission

Answer 14

vs = 3/2 γ^2 eB/2πme

Question 15

Arrival times of two photons emitted by a highly relativistic blazer jet

Answer 15

t1,arr = t1 + d/c

t2,arr = t1 + Δt + d/c - ucosΦΔt/c

Question 16

Binomial expansion

Answer 16

(1+x)^n = 1+nx

Question 17

Drag force

Answer 17

Fdrag on formula sheet

Question 18

derive the apparent transverse velocity and the lower limit.

Answer 18

v,app = vtesinΦ/Δt,arr

where β = v/c and βapp = vapp/c

lower limit found when dβ/dΦ = 0

Question 19

Angular momentum

Answer 19

L = Mvr

Question 20

Birth function

Answer 20

B(M,t) on formula sheet

Question 21

Salpeter IMF

Answer 21

x = 1.35

Question 22

Miller scale IMF

Answer 22

x = 0.8

Question 23

Number of stars formed

Answer 23

more generally N = ∫ ξ(M)dM

N = ∫ M^[-(1+x)] dM

where ξ can be various parameters

Question 24

When all the gas is gone

Answer 24

Ms = Mg(0)

Δf = ΔMs/Ms

Question 25

Mean stellar metallicity

Answer 25

= ( ∫ stars) Z df

Question 26

Relation for Z when t = 0

Answer 26

Z = 0

Question 27

Line of sight velocity distribution LOSVD

Answer 27

F(vlos)dvlos = fraction of stars contributing to spectrum with radial velocities between vlos and vlos + dvlos

Question 28

spectral velocity

Answer 28

u = c lnλ

Question 29

doppler shifted spectral velocity

Answer 29

Δu = cΔλ/λ = vlos

Question 30

composite spectrum

Answer 30

G(u) ∝ = (∞ ∫ -∞) F(vlos) S(u-vlos) dvlos

Question 31

Total spectrum

Answer 31

S(u) = Scont(u) + Sline(u) Sline(u) > 0 emission Sline(u) < 0 absorption

Question 32

To extract F(vlos)

Answer 32

F(vlos) = F^-1 [G(k)/S(k) ] where they are all fourier transformed and F^-1 is the inverse fourier transform

Question 33

line of sight mean, vairance and velocity dispersion

Answer 33

vlos(bar) = (∞ ∫ -∞) vlos F(vlos) dvlos σ^2los = (∞ ∫ -∞) (vlos-vlos(bar))^2 F(vlos) dvlos σlos = (σ^2los)^1/2

Question 34

Cross correlation function

Answer 34

CCF(vlos) = (∞ ∫ -∞) G(u) S(u-vlos) du

Question 35

dm/dr =

Answer 35

4πr^2p(r)

Question 36

disk mass to light ratio

Answer 36

η = MD/LD

Question 37

Faber Jackson relation

Answer 37

L ∝ σ0^4

Question 38

Radio luminosity

Answer 38

dL/dϵ ∝ ϵ^(-β)

Question 39

Quasars

Answer 39

dL/dϵ x ϵ ~ const dL/dv x v ~ const

Question 40

derive the evidence for the blue bump in the quasar continuum

Answer 40

v^2/r = GM/r^2 E = KE + PE = -GMm/2r dE = dE/dr dr dLring = dE/dt = GMM(dot)/2r^2 dr equating to = stefan's law x disk area

Question 41

The IMF function

Answer 41

ξ(M) ∝ M^[-(1+x)]

Question 42

derivative of sin and cos

Answer 42

sin => cos cos => -sin

Question 43

what is the upper limit on size of broad line regions.

Answer 43

δt = R/c

Question 44

Mr

Answer 44

Mr ∝ v^2r/G ∝ r^3

Question 45

Derive the Tully Fisher Relation

Answer 45

mv^2/r = GMrm/r^2 rearrange for v and R = αRD square both sides use the disk mass-to-light ratio and replace RD for disc luminosity assume I(0) and η are the same for all galaxies

Question 46

Derive the Faber Jackson relation

Answer 46

σ0^2 ∝ GM/5Re use the disk mass-to-light ratio η = M/L L ∝ IeRe^2 and assume I(e) and η the same for all ellipticals

Question 47

Derive a polar ring formation

Answer 47

2Kinit = -Uinit virial theorem Kafter = Kinit + ΔK E = -K Eafter = Einit + ΔK Kfinal = Kinit - ΔK

Question 48

Metallicity

Answer 48

Z(t) = Mh(t)/Mg(t)

Question 49

in outer halo

Answer 49

mv^2/r = GMrm/r^2 Mr = v^2r/G hence take derivative and equate to dm/dr

Question 50

torque

Answer 50

𝜏 = rFdrag = dL/dt

Question 51

Derive the drag force F = K (GM)^2p/v^2

Answer 51

apply dimensional analysis

Question 52

If Z(t) = Z0 + p ln[Mg(0)/Mg(t)] derive dMs/dZ

Answer 52

metallicity increases with time, as stars are formed and the gas in the ISM is steadily used up Z < Z(t) Mg(0) - Mg(t) rearranging Ms(

Question 53

If ΔMh = pΔMs - ZΔMs then derive ΔZ/ΔMg = -p/Mg

Answer 53

Z(t) = Mh(t)/Mg(t) ΔMh = Δ(MgZ) = ZΔMg + MgΔZ ΔZ = pΔMs - Z(ΔMs + ΔMg)/Mg ΔMs + ΔMg = 0 if no gas enters or leaves giving as required