Semester 2 - Formulae Flashcards
Tully fisher relation
LD ∝ Vmax^(Beta)
Mass
m(r) = ( r ∫ 0) dm
dm = 4πp(r)r^2dr
Time averaged kinetic energy
<K> = 1/2 Nm <v^2>
Nm = M
</K>
Velocity dispersion
3σ0^2 = <v^2>
Central radiation velocity dispersion
σ0^2 = GM/5R
Hubbles constant
d = v/H0
Eddington luminosity
L(edd) on formula sheet
Mass accretion rate
L = ηM(dot)c^2
Schwarzschild radius
RS = 2GM/c^2
Flux in terms of flux density
Flux = ∫ Sv dv
characteristic timescale
t = E/L
Strength of magnetic field
Umag = B^2/2μ0
Equipartition
Etotal = 2Emag
Emag = V Umag
Lorentz factor through synchrotron radio emission
vs = 3/2 γ^2 eB/2πme
Arrival times of two photons emitted by a highly relativistic blazer jet
t1,arr = t1 + d/c
t2,arr = t1 + Δt + d/c - ucosΦΔt/c
Binomial expansion
(1+x)^n = 1+nx
Drag force
Fdrag on formula sheet
derive the apparent transverse velocity and the lower limit.
v,app = vtesinΦ/Δt,arr
where β = v/c and βapp = vapp/c
lower limit found when dβ/dΦ = 0
Angular momentum
L = Mvr
Birth function
B(M,t) on formula sheet
Salpeter IMF
x = 1.35
Miller scale IMF
x = 0.8
Number of stars formed
more generally N = ∫ ξ(M)dM
N = ∫ M^[-(1+x)] dM
where ξ can be various parameters
When all the gas is gone
Ms = Mg(0)
Δf = ΔMs/Ms
Mean stellar metallicity
<Z> = ( ∫ stars) Z df
</Z>
Relation for Z when t = 0
Z = 0
Line of sight velocity distribution LOSVD
F(vlos)dvlos = fraction of stars contributing to spectrum with radial velocities between vlos and vlos + dvlos
spectral velocity
u = c lnλ
doppler shifted spectral velocity
Δu = cΔλ/λ = vlos
composite spectrum
G(u) ∝ = (∞ ∫ -∞) F(vlos) S(u-vlos) dvlos
Total spectrum
S(u) = Scont(u) + Sline(u)
Sline(u) > 0 emission
Sline(u) < 0 absorption
To extract F(vlos)
F(vlos) = F^-1 [G(k)/S(k) ]
where they are all fourier transformed and F^-1 is the inverse fourier transform
line of sight mean, vairance and velocity dispersion
vlos(bar) = (∞ ∫ -∞) vlos F(vlos) dvlos
σ^2los = (∞ ∫ -∞) (vlos-vlos(bar))^2 F(vlos) dvlos
σlos = (σ^2los)^1/2
Cross correlation function
CCF(vlos) = (∞ ∫ -∞) G(u) S(u-vlos) du
dm/dr =
4πr^2p(r)
disk mass to light ratio
η = MD/LD
Faber Jackson relation
L ∝ σ0^4
Radio luminosity
dL/dϵ ∝ ϵ^(-β)
Quasars
dL/dϵ x ϵ ~ const
dL/dv x v ~ const
derive the evidence for the blue bump in the quasar continuum
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
The IMF function
ξ(M) ∝ M^[-(1+x)]
derivative of sin and cos
sin => cos
cos => -sin
what is the upper limit on size of broad line regions.
δt = R/c
Mr
Mr ∝ v^2r/G ∝ r^3
Derive the Tully Fisher Relation
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
Derive the Faber Jackson relation
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
Derive a polar ring formation
2Kinit = -Uinit virial theorem
Kafter = Kinit + ΔK
E = -K
Eafter = Einit + ΔK
Kfinal = Kinit - ΔK
Metallicity
Z(t) = Mh(t)/Mg(t)
in outer halo
mv^2/r = GMrm/r^2
Mr = v^2r/G hence take derivative and equate to dm/dr
torque
𝜏 = rFdrag = dL/dt
Derive the drag force
F = K (GM)^2p/v^2
apply dimensional analysis
If Z(t) = Z0 + p ln[Mg(0)/Mg(t)] derive dMs/dZ
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}
If ΔMh = pΔMs - ZΔMs
then derive ΔZ/ΔMg = -p/Mg
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
