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Galaxies > Semester 2 - Formulae > Flashcards

Semester 2 - Formulae Flashcards

1
Q

Tully fisher relation

A

LD ∝ Vmax^(Beta)

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2
Q

Mass

A

m(r) = ( r ∫ 0) dm

dm = 4πp(r)r^2dr

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3
Q

Time averaged kinetic energy

A

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

Nm = M
</K>

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4
Q

Velocity dispersion

A

3σ0^2 = <v^2>

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5
Q

Central radiation velocity dispersion

A

σ0^2 = GM/5R

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6
Q

Hubbles constant

A

d = v/H0

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7
Q

Eddington luminosity

A

L(edd) on formula sheet

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8
Q

Mass accretion rate

A

L = ηM(dot)c^2

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9
Q

Schwarzschild radius

A

RS = 2GM/c^2

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10
Q

Flux in terms of flux density

A

Flux = ∫ Sv dv

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11
Q

characteristic timescale

A

t = E/L

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12
Q

Strength of magnetic field

A

Umag = B^2/2μ0

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13
Q

Equipartition

A

Etotal = 2Emag

Emag = V Umag

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14
Q

Lorentz factor through synchrotron radio emission

A

vs = 3/2 γ^2 eB/2πme

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15
Q

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

A

t1,arr = t1 + d/c

t2,arr = t1 + Δt + d/c - ucosΦΔt/c

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16
Q

Binomial expansion

A

(1+x)^n = 1+nx

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17
Q

Drag force

A

Fdrag on formula sheet

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18
Q

derive the apparent transverse velocity and the lower limit.

A

v,app = vtesinΦ/Δt,arr

where β = v/c and βapp = vapp/c

lower limit found when dβ/dΦ = 0

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19
Q

Angular momentum

A

L = Mvr

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20
Q

Birth function

A

B(M,t) on formula sheet

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21
Q

Salpeter IMF

A

x = 1.35

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22
Q

Miller scale IMF

A

x = 0.8

23
Q

Number of stars formed

A

more generally N = ∫ ξ(M)dM

N = ∫ M^[-(1+x)] dM

where ξ can be various parameters

24
Q

When all the gas is gone

A

Ms = Mg(0)

Δf = ΔMs/Ms

25
Q

Mean stellar metallicity

A

<Z> = ( ∫ stars) Z df
</Z>

26
Q

Relation for Z when t = 0

A

Z = 0

27
Q

Line of sight velocity distribution LOSVD

A

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

28
Q

spectral velocity

A

u = c lnλ

29
Q

doppler shifted spectral velocity

A

Δu = cΔλ/λ = vlos

30
Q

composite spectrum

A

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

31
Q

Total spectrum

A

S(u) = Scont(u) + Sline(u)

Sline(u) > 0 emission

Sline(u) < 0 absorption

32
Q

To extract F(vlos)

A

F(vlos) = F^-1 [G(k)/S(k) ]

where they are all fourier transformed and F^-1 is the inverse fourier transform

33
Q

line of sight mean, vairance and velocity dispersion

A

vlos(bar) = (∞ ∫ -∞) vlos F(vlos) dvlos

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

σlos = (σ^2los)^1/2

34
Q

Cross correlation function

A

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

35
Q

dm/dr =

A

4πr^2p(r)

36
Q

disk mass to light ratio

A

η = MD/LD

37
Q

Faber Jackson relation

A

L ∝ σ0^4

38
Q

Radio luminosity

A

dL/dϵ ∝ ϵ^(-β)

39
Q

Quasars

A

dL/dϵ x ϵ ~ const

dL/dv x v ~ const

40
Q

derive the evidence for the blue bump in the quasar continuum

A

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

41
Q

The IMF function

A

ξ(M) ∝ M^[-(1+x)]

42
Q

derivative of sin and cos

A

sin => cos
cos => -sin

43
Q

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

A

δt = R/c

44
Q

Mr

A

Mr ∝ v^2r/G ∝ r^3

45
Q

Derive the Tully Fisher Relation

A

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

46
Q

Derive the Faber Jackson relation

A

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

47
Q

Derive a polar ring formation

A

2Kinit = -Uinit virial theorem

Kafter = Kinit + ΔK

E = -K

Eafter = Einit + ΔK

Kfinal = Kinit - ΔK

48
Q

Metallicity

A

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

49
Q

in outer halo

A

mv^2/r = GMrm/r^2

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

50
Q

torque

A

𝜏 = rFdrag = dL/dt

51
Q

Derive the drag force

F = K (GM)^2p/v^2

A

apply dimensional analysis

52
Q

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

A

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(<Z) = Mg(0) {1-exp[-Z-Z(0)/p]}

taken the derivative

dMs/dZ = Mg(0)/p exp{-Z-Z(0)/p}

53
Q

If ΔMh = pΔMs - ZΔMs

then derive ΔZ/ΔMg = -p/Mg

A

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

Galaxies (4 decks)

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