Checklist Flashcards
What is a galaxy?
A large collection of stars, gas, and dust held together by mutual
gravitation.
Mass: 10⁶ ~10¹³ M☉︎
Size: ~1- 200+ kpc
What are a galaxy’s main constituents?
Gas, Dust, Stars, Dark matter halo
What are the principal types of galaxy?
Spirals, Ellipticals, Lenticulars, Irregulars, (“Peculiars”)
How are galaxies classified under the Hubble scheme?
- 3 classes: Ellipticals, Discs (Lenticular & Spiral), & Irregulars
- ‘Tuning fork’ diagram
- Ellipticals E0 - E7 (low to high ellipticity), LHS of diagram.
- S0 and SB0 are Lenticular, at ‘split’ of diagram
- Spirals split into barred and unbarred,
- Sa/SBa - Sc/SBc (larger bulge tighter arms less gas, to smaller bulge looser arms more gas)
- Irregulars also on diagram, but not in ‘fork’
- NOT an evolutionary sequence
- E0 ‘early’ type - Sc ‘Late’ type
https://courses.lumenlearning.com/suny-astronomy/chapter/types-of-galaxies/
How are galaxies classified under the de Vaucouleurs scheme?
Elliptical to irregular:
E, E⁺, S0⁻, S0⁰, S0⁺, Sa, Sb, Sc, Sd, Sm, Im
Unbarred : SA Barred : SB
Arms start on ring / has ring : (r) suffix
Arms start from centre / no ring : (s) suffix
External ring : (R) prefix
Can have mixture of ANY, eg (R)SABb(rs), sorta has bar, arms from central ring and from centre, external ring.
https://pages.astronomy.ua.edu/keel/galaxies/classify.html
What are the advantages and disadvantages of galaxy classification schemes?
- Adv:
1. Hubble retains main features, ignores superficial details, easier to see overall patterns.
2. De Vaucouleurs provides additional information and more accurately catagorises galaxies based on these characteristics. - Disadv:
1. Easy to disagree on classification
2. Orientation in sky can confuse
3. Appearence depends on how we look at galaxy (wavelength, sensitivity, image dynamic range, etc.)
4. Hubble assumes correlation between properties, eg spiral arms and central light concentration
5. Hubble ignoring certain features (rings, warped disks, stellar streams, etc.) may be detrimentak to our understanding of galaxy formation
What galaxy properties correlate with Hubble type?
Galaxy type (Elliptical, Disk, Irregular)
Ellipticity (0-7), Bars (B), increasing gas content, looser arms, smaller bulge (a-c)
Why does the Hubble classification scheme not correspond to an evolutionary sequence?
Hubble classification shows different types of galaxy as they are today, their evolutions into these forms are individual to the galaxy, influenced by things like dark matter and gravitational interactions.
What are the principal constituents of spiral galaxies?
Disk
* Contains metal-rich stars, gas (HI,H2 HII), dust
* Stellar motions dominated by rotation
Bulge
* Mixture of metal-poor and metal rich stars
* Slow (if any) rotation
* Strongly affected by bars
Stellar halo
* Metal-poor stars, globular clusters
* Hot x-ray emitting gas
* Little/no rotation
Dark matter halo
* Contains most of the galaxy mass
What are the principal constituents of elliptical galaxies?
Older, lower mass stars
Many globular clusters
Minimal star formation
What are the principal constituents of irregular galaxies?
Abundant gas and dust, with little structure
What are the main differences in characteristics of stellar populations?
Population I - Young stars, most still on main sequence, most luminous stars are main sequence blue giants
Population II - Older stars, many larger stars evolved off main sequence and now red giants, most luminous stars are red giants
What are the main differences in stellar populations within different types of galaxies?
Starburst - Pop I
Spiral - Pop I in arms, Pop II in bulge
Elliptical - Pop II
What is luminosity?
Luminosity, L, is total energy emitted by source
per second.
Measured in Watts/ Solar Luminosity units: L☉︎ = 3.8x10²⁶ Watts
Calculated by integrating over the most luminous frequency band.
L = 4πD²S where S=(0 - inf.) ∫ Sᶠ df [S is flux, D is distance]
Describe the concepts of apparent and absolute magnitude as measures of flux and luminosity.
Apparent magnitude is magnitude as observed from earth. Calculated using light recieved per unit area, so closer stars have higher flux, appear brighter, and have lower apparent magnitudes.
Absolute magnitude measures brightness regardless of distance/ at a fixed distance of 10 pc. Lower absolute magnitude corresponds to higher luminosity.
Differences in magnitudes can show differences in flux density, and therefore luminosity.
What are the relations between magnitude, flux, and distance?
m₁ - m₂ = -2.5log₁₀(S₁/S₂)
Magnitude difference N refers to flux ratio 10⁰ᶥ⁴N, eg difference of 5 is flux ratio of 100
S₁/S₂ = D₂²/D₁² (if both stars have same luminosity)
m₁ - m₂ = -5log₁₀(D₂/D₁) (fixed L)
M = m - 5log₁₀(D/10pc)
Visual luminosity of a star from absolute visual magnitude of sun, where M☉︎ = 4.83:
L/L☉︎ = 10^(0.4(4.83 - M))
What is surface brightness?
Galaxy brightness. Usually in magnitudes/arcsecond². Used because galaxies take up more area on the sky than stars do.
Given a luminosity and distance to an extended object like a galaxy, how would you calculate its surface brightness in magnitudes per square arcsecond?
- Calculate absolute magnitude using the sun as a guideline.
L/L☉︎ = 10^(0.4(4.83 - M)) - Use absolute magnitude to calculate apparent magnitude.
M = m - 5log₁₀(D/10pc) - Calculate fraction of galaxy area (solid angle) contained within 1as
1as² / area in degrees x 3600 x 3600 - Calculate surface brightness per arcsecond²
m - surface brightness = -2.5log₁₀(1/galaxy fraction)
What are the different distance determination techniques that comprise the “distance ladder”. What are the range and uncertainty of each these techniques?
- Parallax ~10kpc
- Spectroscopic parallax ~50kpc
- Period-Luminosity Relations: Cepheids ~20Mpc
- Tully-Fisher relation ~100Mpc
- Fundamental Plane relation ~100Mpc
- Type 1a Supernnovae ~Gpc
- Recession velocity (Hubble law) >1000Mpc
What is the Schechter luminosity function and its key parameters?
Formula that describes the number of galaxies as a function of their luminosities.
Total number density: (ᓫ is star, ∀ signifies over all range)
nₜₒₜ = ∀∫ ϕ(L) dL = ϕᓫ ∀∫ xᵅe⁻ˣ dx, where x = L/Lᓫ
Power law at low luminosity, L < Lᓫ, ϕ ∝ Lᵅ
Exponential cut-off at hight luminosity, L > Lᓫ, ϕ ∝ e⁻ᴸ
ϕᓫ is normalisation factor, scales to match number density of galaxies in universe.
Lᓫ is characteristic luminosity, signifies the turnover point in distribution of galaxy luminosities.
α is slope of function for low luminosities, hence descrbes distribution of faint luminosities.
Universal value for α estimated to be ≃ -1.3, implying many, many dwarf galaxies. Schecter diverges as L → 0 if α ≤ 0, so breaks down for low luminosities.
What are the complications with measuring luminosity functions?
- Velocities may be ambiguous
- Dwarfs very faint and hard to measure
- Peculiar velocities complicate simple Hubble relationship at large distances
- Galaxy distributions have structure within groups and clusters, not uniformly distributed.
- Incompleteness: surveys are surface-brightness limited (diffuse galaxies under-represented)
- Malmquist bias: at greater distances surveys detect only more luminous objects.
How does the luminosity function of galaxies vary with environment and with morphological type?
When normalised, S0 and spirals similar shape, less dim S0 than dim spirals. More bright ellipticals.
Comparing fields and clusters, clusters contain more luminous ellipticals.
Describe the Galactic coordinate system.
l = Galactic Longitude, 0° - 360°
Angular distance from Galactic Centre as viewed from Sun
b = Galactic Latitude, ±90°
Angular distance from mid-plane as viewed from the Sun
R₀ = distance of Sun from the Galactic Centre
https://commons.wikimedia.org/wiki/File:Galactic_coordinates.JPG
Describe the Galacto-centric coordinate system.
Cyclindical polar coordinates.
r is distance from galactic centre.
θ is angle from sun
z is dist. above plane of disk. (scale height)
R₀ = dist. to sun, ≃ 8kpc
V₀ = sun circular speed ≃ 220 km/s
What sort of galaxy are we in?
Disk galaxy (spiral)
Small-medium bulge
Bar
2 major spiral arm (+ minor arm fragments)
Likely Hubble type SBbc
What do we use to deterine the galactic plane?
Atomic Hydrogen (HI - radio wavelengths)
Molecules, eg carbon monoxide (at mm wavelengths)
Dust (Far infrared ~60 μm )
Old stars (Near infrared ~2 μm
Original observations were affected by dust and confusion with Gould’s belt (local feature, inclined ~20° to plane, likely part of spiral arm)
What are the principal components of the Milky Way Galaxy? What are approximate scale sizes and ages of each component?
Bulge:
R ~ 3kpc
~ 2kpc scale height
~10¹⁰ M☉︎
Vcirc ~ 100 km/s
Thin Disk:
Scale height ~100-300 pc
6 x10¹⁰ M☉︎
Young stars, 0-9 Gyrs
Thick disk:
Scale height ~1000 - 1300 pc
Older stars, 10-12 Gyrs
Stellar halo:
~ 15-20 kpc radius
~10kpc scale height
~2 x10⁹ M☉︎
13 Gyrs old
Pop II stars
~150 known globular clusters
What are the typical temperatures of different gas and dust components in our Galaxy?
no answer yet
How do the physical distributions of different gas and dust components differ in our galaxy?
no answer yet
How are we able to trace the distributions of atomic and molecular gas in the Galaxy?
no answer yet
What do we know about the nature of the black hole at the centre of our Galaxy and how do we obtain this information?
Sgr A * ~10⁸ km ~ 1AU
Sgr A * is stationary, occupies dynamical centre of Galaxy and lack of dynamics suggest it is massive.
Sgr A contains circum-nuclear rotating molecular disk, density 10⁵ cm⁻³, mass 3 x10⁴ M☉︎, Vrot ~100km/s, radius 2pc
Stars near central source Sgr A * observed over several years to determine orbits → Keplerian motion about Sgr A * . Velocity dispersion within 0.1pc is consistent with Keplerian V ∝ R^-0.5
Implies most mass of Sgr A is confined to centre (Sgr A * )
Orbits used to calculate mass of Sgr A * , current value M = 4.297 x10⁶ M☉︎
What methods do we use to measure our distance from the centre of the Galaxy?
no answer yet
How are stars distributed in our Galaxy, and how does their distributions depend on age and metallicity?
Extreme Pop I thin disk region:
* O and B stars
* Very young stars: ~ 10⁷ years
* metal rich: Z ~ 2-4% | [Fe/H] ~ 0 to 0.5
Thin disk:
* G, K and M type stars
* Young(ish) stars: 0-9 Gyrs
* metal rich: Z ~ 0.5-4% | [Fe/H] ~ -0.5 to 0.5
Thick disk:
* Older stars: 10-12 Gyrs
* less metal rich: [Fe/H] ≃ -0.6, tail down to ≃ -1.5
Bulge:
* similar ages as thick disk
* [Fe/H] ~ -1 to 0
Halo:
* age 12-14 Gyrs
* [Fe/H] ≃ -1 to -5
What are the two components of stellar motions?
Random: Gaussian distribution with width σ. This is the velocity dispersion. Probability of star having random velocity v is
P(v)dv = 1/σ√2π’ exp(-v²/2σ²) dv
Rotational component: orbital speed Vcirc of stars around a galaxy
How do rotational kinematics and velocity dispersion of stars differ between different components of the Galaxy
Thin disk:
σ ~ 20 km/s | Vcirc ~ 220 km/s
Thick disk:
σ ~ 40-60 km/s | Vcirc ~ 180 km/s
Bulge:
σ ~ 60-100 km/s | Vcirc ~ 100 km/s
Stellar halo:
σ ~ 100 km/s | Vcirc ~ 40 km/s
How do the dynamics of Population I and II stars differ?
Pop I - ordered motion, circular orbits in the disk plane.
Pop II - random motion, eccentric orbits passing through disk plane
What is the stellar initial mass function? How can it be used to make calculations of the number and average masses of stars?
Initial Mass Function Ψ(m) specifies the distribution in
mass of a newly formed stellar population and it is frequently
assumed to be a simple power law.
* Mass range from ~0.1 – 100 M⊙
* Number of stars of a certain mass range can be calculated by ∫Ψ(m)dm in desired mass range
* Total mass, ∫mΨ(m)dm
* Salpeter IMF, for example, takes form Ψ(m) = Cmˣ, x = -2.35
* Salpeter breaks down at low masses, and x = -1.3 for 0.08 M⊙ < m < 0.5 M⊙
Equation for stellar number density wrt scale height?
Stellar number density follows an exponential law
n(z) = n₀exp(-|z|/hᶻ)
hᶻ is disk scale height and n₀ is stellar number desity in disk plane.
How does the disk scale height, hᶻ of stars vary with stellar type?
O and B stars in extreme Pop I thin disk region
hᶻ ~ 60 pc
G, K, M stars in thin disk
hᶻ ~ 200-300 pc
What is asymmetric drift?
Difference between thick and thin disk rotation, ~ -40 km/s, thick slower than thin.
What is the Local Standard of Rest?
Average velocity of disk stars within ~50 pc
Sun’s velocity relative to LSR is ~17km/s towards l ~+56°, b ~+23°
What is the equation of motion of a star of mass m?
m d²r/dt² = -m∇ϕ, r in cylindrical polar.
ϕ is gravitational potential determined from Poisson’s equation: ∇²ϕ = -4πGρ
How can the vertical motions of stars in the disk plane be used to determine the local mass density of stars, gas and dark matter?
answer only really references dark matter
Studying motions of stars gives a value for density of matter in the solar vicinity. Comparing this to the mass of all observable matter shows discrepencies, which can be explained using dark matter.
The Oort limit, the dynamical mean density of matter in solar vicinity, has been determined to be
ρ₀ = 0.076±0.015 M☉︎ pc⁻³.
Observable matter estimated mass ρ ≃ 0.08 M☉︎ pc⁻³.
Observable matter accounts for all dynamical mass, so there is no evidence for dark matter in the disk.
How does the gas disk of spiral galaxies vary with Hubble type?
no answer yet
What differences are observed in our Galaxy between the spatial distribution of the gaseous disk and the stellar disk?
answer only references gaseous
Gaseous disk:
* Flares in outer regions of galaxy.
* HI: solar vicinity hᶻ ~ 150pc | outer galaxy hᶻ ~ 1kpc
* CO: solar vicinity hᶻ ~ 60pc | outer galaxy (16kpc) hᶻ ~ 200pc
* CO: Highly concentrated towards plane, mainly within inner 5kpc of galaxy. Used to trace H₂
* Dust: far-infrared emission, similar distribution to HI and CO
What is the equation for magnitude, factoring in optical extinction?
mᵥ - Mᵥ = 5log₁₀(d/10pc) + Aᵥ
Aᵥ is optical extinction.
Aᵥ ~ 1 mag/kpc (clumpy, not smooth)
Aᵥ ∝ N(HI) (hydrogen column density)
What are the scale heights of HI and molecular gas in the solar vicinity?
HI ~ 150pc
H₂ ~ 60pc
What is the origin and distribution of ionised gas in our Galaxy?
Origin: Gas ionised from UV radiation emitted by massive O and B type stars (Photons E>13.6).
HII regions surround these stars, known as Stromgren spheres.
O & B stars typically located 0-60pc from central plane, so HII originates there. HII T~10^4 K, surrounding neutral gas T~100 K
Ionised pressure > neutral gas pressure, so HII region expands to form diffuse ionised medium along plane.
What are the Oort constants and what do their observed values tell us about the rotational dynamics of local stars?
Oort’s constant A expresses local shear, velocity relative to LSR.
A = ½(V₀/R₀ - (dV/dr)|R₀) = ½r(dω/dr)|R₀
Oort’s constant B expresses local vorticity, tangential velocities
B = -½(V₀/R₀ + (dV/dr)|R₀) = -½r(dω/dr + 2ω/r)|R₀
They can be determined by plotting Vᵣ/d and Vₜ/d against galactic longitude.
Vᵣ/d = Asin(2l) gives A = 14.8 ± 0.8 km s⁻¹ kpc⁻¹
Vₜ/d = Acos(2l) gives B = -12.4 ± 0.6 km s⁻¹ kpc⁻¹
They rule out solid body rotation, as A is not equal to 0.
Used to estimate local differntial rotation, A+B = -(dV/dr)|R₀
Flat rotation curve = V constant, so (dV/dr) = 0, aka A=-B
This is consisten with observed values, so rotation curve approximately flat in vicinity of Sun.
Also used to calculate local circular speed A-B = V₀/R₀
Vcirc ~ 210 km s⁻¹
|R₀ means evaluated at R₀
Distribution of atomic hydrogenin Milky Way?
Distribution of molecular, ionised gas, and supernova remnants in Milky Way?
Atomic hydrogen - broad distribution from 4 - 15 kpc
Molecular, ionised gas, and supernova remnants - peak between 4 - 7 kpc
Why can the radial velocity of HI clouds can be used estimate their distances?
We know how to model galactic rotation, so can calculate distance by comparing radial velocity to our model.
Limitations:
* assumes circular velocity, invalid for r < 3kpc
* requires small intrinsic (random) motion
* inaccurate near l = 0° and 180° as Vᵣ ~ 0 km/s
* potential distance ambiguity for r < R₀
* doesnt work for r ~ R₀ as Vᵣ ~ 0 km/s
Given a cloud radial velocity and direction, what calculations must be made to estimate the distance to the cloud?
if r > R₀:
Vᵣ = (ω(r) - ω₀)R₀ sin(l)
ω(r) = V(r)/r
combine to get
V(r)/r = Vᵣ/R₀sin(l)
sub in V₀ = 220 km/s | R₀ = 8 kpc | V(r) = V₀ (flat rotation curve)
if r < R₀:
ω(r) could have one of two values
Vᵣ = (ω(r) - ω₀)R₀ sin(l)
rearrange to get
V(r)/r = Vᵣ/R₀sin(l) + V₀/R₀
sub in V₀ = 220 km/s | R₀ = 8 kpc | V(r) = V₀ (flat rotation curve)
both:
galactocentric distance r = V(r)/ω(r)
can then use cosine rule to get two possibilities for d. can rule out a possibility if d is -ve
What are longitude-velocity plots and how do they provide evidence for a Galactic bar and for spiral arms?
Line spectra are shifted differntly depending on longitude.
for 0° < l < 90°, redshift corresponds to r < R₀, blueshift to r > R₀. Opposite way round for 90° < l < 180°.
Shift corresponds to radial velocities, and plotting these velocities against longitudes allows for a sort of “map” of gas.
Spiral arm and bar like structures can be seen .
Note that central 3kpc affected by non-circular motions.
What evidence suggests that the Galactic bulge is bar-like?
l - v plot: central ‘3 kpc arm’ structure implies bar motion
infrared: asymmetry in star counts about l = 0° indicates a ‘near’ and ‘far’ side, consistent with bar-like geometry
What is at the galactic centre?
Sgr A radio source - supernova remnant Sgr A East, spiral structure Sgr A West, central source Sgr A * , likely a massive black hole
How do galaxy rotation curves provide evidence for the existence of dark matter?
3 rotation candidates are solid body (Vcirc ∝ r), Keplerian
(Vcirc ∝ r^⁻½) and flat (Vcirc = constant)
Not Keplerian, not solid body, but flat-ish
Rotational velocity is much larger than that expected based on visible matter only, and this additional velocity is attributed to the dark matter halo.
What are the common functions used to model the surface profiles of the disk and bulge components of disk galaxies?
Disk: exponential law
I(r) = I₀ exp(-r/r₀)
where r₀ = characteristic length
and I₀ = central surface brightness
Bulge: follows r^1/4 law, ellipticals only have this part
I(r) = Iₑ exp{ -7.67( {r/Rₑ}^1/4 - 1)}
where Rₑ = radius containing half of total projected luminosity
and Iₑ = intensity at Rₑ
How is the total HI content ofa disk galaxy measured, and how does it vary with Hubble type?
HI mass can be measured if telescope beam > galaxy extent.
Mᴴᴵ/M☉︎ = 2.36 x 10¹⁹D² ±∞∫ Sᵥ dVᵣ
where Sᵥ is flux density in W/m^2/Hz
and Vᵣ is radial velocity in km/s
and D is distance in pc
Total HI mass % (wrt visible matter) increases from Sa to Sc
(~2-7%).
Irr galaxies have the most (10-15%)
Ellipticals have the least (≪ 1%)
How are HI emission and radial velocity measurements used to measure the mass of HI and the total mass of a disk galaxy?
More mass, faster rotation, broader HI line width (doppler shift).
Consider outermost HI gas cloud, assume all mass contained within its orbit, equate centripetal and gravitational forces.
mV²/R ~ GMm/R²
rearrange, M ~ RV²/G
using ΔV ~ 2V/sin(i) (i is inclination angle. i=0° face on, i=90° edge on, estimated assuming circular disk, i = cos⁻¹(b/a))
M ~ R(ΔVsin(i))²/4G
What are Tully Fisher, Luminosity-Dispersion (Faber-Jackson) and Fundamental Plane relations? What are their limitations?
All empirically derived, all useful for distance calculations.
Tully-Fisher:
* luminosity of a galaxy is directly proportional to the fourth power of its rotational velocity
* L ∝ Vcirc⁴ ∝ (ΔVsin(i))⁴
* Intrinsic luminosity gives absolute magnitude
* Comparison with apparent magnitude gives distance
* Scatter ±0.2 magnitudes
Luminosity-Dispersion:
* Relation between Luminosity L and the Stellar dispersion σ of elliptical galaxies.
* Scatter ±2 magnitudes
Fundamental Plane:
* Rₑ, Σₑ and σ₀ are related by Rₑ ∝ σ₀^1.4 Σₑ^-0.85
* surface in 3D ⇒ ‘The Fundamental Plane’
* Rₑ is the half light radius
* Σₑ is the mean surface brightness within Rₑ: Σₑ = 1/2 L/πRₑ²
* σ₀ is the central velocity dispersion
What is the “winding dilemma”?
Flat rotation curves mean that inner parts of galaxy have higher angular velocity than outer parts, hence spiral arms shoud ‘wind up’. Rotation period ~ 100-400 million years, spiral arms should wind up in ~10^8 years.
Implies Sa may be evolution of Sc, however not the case as timescales unrealistic. Sa much older than 10^8 years, Sa more massive than Sc, and Sa have significantly lower gas content (10^8 years too short for such a change)
What is the density wave theory for spiral structure, and how does it help to alleviate the “winding dilemma”?
Spiral arms are waves of stellar density.
Arm pattern moves around the galaxy at constant angular velocity (pattern speed) but different circular speed
Stars and gas move at approx. constant circular speed vcirc (km/s) and variable angular velocity w(r).
Perturbations cause radial oscillations in orbits, so they are slightly elliptical. Conservation of angular momentum means matter moves slightly faster and slower at differnt points, ‘slow’ regions have higher stellar density. A series of tilted ellipticals can create a spiral of density maxima.
What aspects of the physical properties of spiral galaxies does the density wave theory explain?
- Spiral arms are over-dense regions of the disk which move round at a different speed to the stars themselves
- Stars move in and out of the spiral arm, alleviating winding dilemma
- Existence of rings in centres of spirals (ILR)
- Prevalence of 2-armed spirals
How might the spiral structure of flocculent spirals form?
Stochastic self-propagative star formation.
Star formation triggers more star formation nearby, then galactic rotation creates appearence of spiral.
“Stochastic” because there is a small probability of random star formation in all areas in the disk
Why are stellar interaction important within globular clusters but not within galaxies?
Relaxation time for stellar interactions in galaxies is shorter than galactic lifetimes, but longer within globular clusters.
What is epicyclic frequency?
Stellar orbits are superposition of circular orbits (rad R, ang freq ω) with small retrograde epicycles (ang freq κ)
κ(r) is epicyclic frequency required for long-lived spiral structure
κ²(r) = (2ω)² (1+ r/2ω ∂ω/∂r)
For a circular orbit, R, θ = ωt, z = 0
For an epicyclic perturbation, x = small radial deviation, ε = small azimuth deviation, Φ(R,z) is gravitational potential
Epicyclic motion:
r = R + x
θ = ωt + ε
What is the connection between the epicyclic frequency in density wave theory and the Oort constants?
Oorts constants
A = ½(V₀/R₀ - (dV/dR)|R₀)
B = -½(V₀/R₀ + (dV/dR)|R₀)
A - B = V₀/R₀, A+B = -(dV/dR)|R₀
ω = v/r
dv/dr = d/dr(rω) = r dω/dr + ω
substituting gives
κ₀²(R₀) = -4B (A - B)
Typical spiral potential
Φ₁(r, θ, t) = Φ₁(r)exp( -i(mΩₚt - mθ + ψ(r))
where m = number of arms (usually m=2)
Ωₚ = pattern speed
ψ(r) = phase term determining pitch of arms
Total potential is symmetric potential + spiral potential (Φ₁)
Stars will experience gravitational potential varying with frequency m( ω(r) - Ωₚ )
What are the inner and outer Lindblad Resonances?
Stars and gas radially oscillate with epicyclic frequency κ.
Density wave produces periodic variation in global galactic potential Φₜₒₜ.
When frequencies are equal, κ = ±m(ω(r) - Ωₚ), (m normally 2)
Ωₚ = ω(r) ± κ/2
Inner Lindblad Resonance (ILR)
Ωₚ = ω(r) - κ/2,
Resonance is in inner part of galaxy
Stars and gas have higher velocity than the density wave and so ‘catch up’ with the density wave.
Often seen as a ring around the centre of a galaxy, spiral arms start from this ring
Outer Lindblad Resonance (OLR)
Ωₚ = ω(r) + κ/2,
Resonance is in outer part of galaxy
Stars and gas have a lower velocity than the density wave so the density wave ‘catches up’ the stars and gas.
Rarely observed, not as important as ILR.
Density waves can only propagate in the region between the ILR and OLR, and this region is largest for 2 armed spirals.
What is the corotation radius?
When Ωₚ = ω(r) stars and gas have the same angular velocity as the density wave. This occurs at the corotation radius
How does the star formation history of elliptical galaxies differ from that of disk galaxies?
Formed in early universe, mostly in first billion years.
Disk galaxies instead formed more uniformly over many billions (~10)
How are elliptical galaxies classified?
Eₙ: E0 - E7 where n = 10(a-b)/a
How is the 3D axis ratio of ellipticals used to determine if a galaxy is spherical, oblate, prolate or triaxial? How do these relate to possible 2D profiles as seen on the sky?
Ellipticals are isophotes, approx. elliptical in shape.
Spherical: a=b=c
Oblate: a=b > c, flat-ish
Prolate: a=b < c, pulled-ish in direction of axis
Triaxial: a ≠ b ≠ c
What are the key parameters in the r¼-law that determines the surface brightness profile of an elliptical galaxy?
log I(r) ∝ r¼
I(r) = Iₑ exp( -7.67( (r/Rₑ)^1/4 -1) )
Rₑ = radius within which half of the total light is contained
Iₑ = surface brightness at Rₑ
I is capital i
How do we measure the dynamics of stars in elliptical galaxies?
- Stellar absorption spectra
- Stars are old, with low mass, so have low luminosity.
- Strong absorbtion lines due to metals in stellar population.
- Average spectrum of many stars in line of sight
- Line width Δλ is measure of stellar velocity dispersion σ
- Variation of λ as a function of distance from the centre of the galaxy can tell us about net rotation
- Massive ellipticals show
– Low rotation velocities ~ 50-100 km/s
– High dispersion velocities ~200-300 km/s - Velocity/dispersion ratio v/σ typically ranges from 0 to 1
- To be supported by rotation, v/σ ~ (ε/1-ε) where ε = 1 - b/a
What is the virial theorem and its fundamental assumptions?
- Relates kinetic energy to its gravitational poteltial energy
- 2T + Ω = 0
- assumes system in a stable statistical equilibrium under its own gravitation
- assumes stellar interactions are negligible
How do we use the virial theorem to estimate the mass of elliptical galaxies and of galaxy clusters?
T = ½ Σmᵢvᵢ² = ½M⟨v²⟩
M is total mass of stellar system and ⟨v²⟩ is mean value of vᵢ²
Ω = -Σᵢ Σᵢ≠ⱼ Gmᵢmⱼ/rᵢⱼ = -αGM²/Rₑ
α is a factor of order unity
Combined,
M = Rₑ⟨v²⟩/αG ~ Rₑ⟨v²⟩/G (α ~ 1)
How do cD galaxies differ observationally from other elliptical galaxies?
- Large stellar halo
- Luminosities typically 10x Lstar
(recall Schechter luminosity function: Lstar = 2x1010 L⊙) - Halo extends 50-100kpc
- Origin – likely formed from mergers of galaxies
in the cluster core
How can one compute the tidal radius of a satellite galaxy?
r = (m/2M)^1/3 R
R is critical distance of approach, corresponds to the ‘Roche limit’ for M to disrupt m.
What are the principal differences between a group and a cluster of galaxies?
Groups:
* A few to 10 bright (Lstar = 2 x10¹⁰ L⊙) members
* ~100 dwarf galaxies
* Mass ~5 x 10¹² to 1 x 10¹⁴ M⊙
* Sizes ~ 1Mpc
Clusters:
* Around 100 bright (Lstar = 2x10¹⁰ L⊙) members
* Thousands of dwarfs
* Mass around 10¹⁴ to a few x 10¹⁵ M⊙
* Size typically around 3 Mpc
* Separation between clusters ~ 50 Mpc
What observations do we use to measure the star formation rate (SFR) within galaxies?
UV continuum - works with little dust, but high star formation often corresponds with high extinction
Far infrared - reemission of UV
Evidence for “Galaxy collapse” star formation
- Age correlates with vertical velocity
- Young stars formed close to the disk plane
- Old stars formed at range of heights above the plane
- must be fast collapse to produce eccentric orbits for old stars
Velocity and angular velocity
v = rω
Gravitational force
Fg = GMm/r^2
Centripetal force
Fc = mv^2/r
Gravitational potential
Ug = −GMm/r
kinetic energy
K = 1/2 mv^2
