Astr 5460 Wed., Sep. 22, 2004

Today: Homeworks due – discussion?

WIRO Observing trip

More on Elliptical Galaxies

(Ch. 4, Combes et al. , Ch. 3, Longair)

Unless noted, all figs and eqs from Combes et al. or Longair.

Spectroscopy of Ellipticals

Top is K0 III star, M87 off center, M87 near center

Stellar Absorption Lines, note Ca II H&K

Properties of interest are dispersion and shift. Dispersion is especially complicated since it depends on an ensemble of stars with different absorption line widiths.

Spectroscopy of Ellipticals

Rotation (V) and velocity dispersion (σ) curves for some ellipticals (Davies et al. 1983)

Triaxial Elliptical Galaxies

So, are ellipticals simple to understand dynamically? Not so clear. We’re seeing a 2D picture of a 3D object.

In general, elliptical galaxies rotate too slowly for this to account for the flattening observed. In other words, their ratios of rotational to random kinetic energy is too low.

If ellipticity e were due to rotation (e.g., a pure oblate rotator with an isotropic velocity distribution) then (V_rot/σ)_iso ~ (e/(1-e))^1/2. Is it?

Triaxial Elliptical Galaxies

Ellipticity Profiles

Fit ellipses to surface brightness distribution.

Can define “boxy” and “disky” galaxies (see Kormendy & Djorgovski 1989, ARAA, 27, 235), which correlate with other properties.

Ellipticity Profiles

http://chandra.as.utexas.edu/~kormendy/tuningfork.html

Ellipticity Profiles

Top, e(r) plotted for some galaxies.

Bottom, e(r) and position angle as a function of radius. Proof of triaxalality? Or artifact of other?

Just FYI, e = 1-b/a and r = (ab)^1/2

Ellipticity Profiles

Deviations in r^1/4 light profile can be quantified into classes, and corresponds with the presence of close neighbor galaxies (T3).

Other Probes of Shape

Stellar “shells” from capture of smaller galaxies.

Dust lanes associated with captured material in equilibrium.

See text for more discussion.

Models of Elliptical Galaxies

Should follow equations of stellar dynamics.

Treat as “collisionless” ensemble

Relaxation time is important – this defines the time between “collisions”

Work out energetically to get

T_relax = V³/8πnG²m²log(R/b)

V is mean relative speed, n is stellar density, m is the mean stellar mass, R is radius of galaxy, and b is the minimum impact parameter.

Details will probably be a homework problem.

Models of Elliptical Galaxies

Crossing times: T_cross ~ R/V

Relaxation time then varies as

T_relax ~ 0.2(N/logN) T_crosswhere N is the number of stars in the system

For galaxies, crossing times ~1/100 the age of the universe (100 million years), and the relaxation times are on order of 10¹⁷ years – so close interactions between stars essentially never happen. “Collisionless” is a good approximation. This is because N is large, and the collision must be very close for another star’s potential to overwhelm the ensemble potential.

Note: this argument is NOT true for Globular Clusters! Also not true for galaxy clusters!

The upshot of this is that you can treat galaxies in terms of motions in a global potential.

On Model Approaches

Combes et al. approach the subject from the general case of distribution functions (f) – what are the positions, velocities, and time of the stars moving in a potential?

Can start with the continuity equation (mass) which is the collisionless Boltzmann equation, AKA the Vlasov equation:

-dU/dr is the gravitational force exerted. The potential can be obtained from the Poisson eq: ΔU(r) =4πGρ(r). Must find f(r,v) that solves these self-consistently. They discusses the isothermal sphere case:

f(E)=(2πσ²)^-3/2ρ₀e ^-E/^σ² (complications vanish in symmetry)

Should be familiar-looking function. Still, want density as a function of radius.

Isothermal Gas Spheres

Longair’s starting point is the Lane-Emden equation (4-7), which assumes spherical symmetry and hydrostatic equilibrium. Isothermal = same mean velocities everywhere. Can apply to stars, galaxies, clusters.

Isothermal Gas Spheres

Simple case, for which ideal gas law holds at all radii, p = ρkT/mass. In thermal equilibrium, 3/2 kT = ½ mass <v²>, therefore:

Non-linear, only has an analytical solution for large r using a power-series expansion and applying some reasonable boundary conditions.

Isothermal Gas Spheres

Boundary conditions:

Smoluchowski’s Envelope at large radii where density, timescales go to extremes.

In astrophysical systems, outmost radii are subject to interactions with other clusters, hence a tidal radius.

Isothermal Gas Spheres

Then recast our equation as:

Where we have the dimensionless quantities x=r/α and ρ = ρ₀y, & A, structural length α:

Isothermal Gas Spheres

As usual in real astronomy things are complicated because we only see a projection rather than a 3D distribution. The projection surface density distribution onto a plane is:

Where now q is the projected distance from the center.

Isothermal Gas Spheres

Tabular solution:

Isothermal Gas Spheres

Graphical solution:

Isothermal Gas Spheres

Fit the function to an observed distribution.

N(q) = ½ at q=3, or core radius R_1/2 =3 α.

To get a central mass density then also need to measure the velocity distribution. From equipartition ½ m<v²>=3/2 kT. So:

King Models

Isothermal spheres have infinite extent. More complex and realistic version based on Fokker-Planck equation by King (1966). Assumes no particles present with escape velocity+. Sharp E cutoff. See texts for more.

Other Models

Real galaxies are not necessarily spherical and isotropic. Eddington and Michie models:

Fundamental Plane

Luminosities, surface brightness, central velocity distribution, (and others), are correlated, hence the term “fundamental plane.” Ellipticals populate a plane in parameter space. BIG area of research – very useful tool and helps us understand galaxies.

Faber & Jackson (1976) is a classic in this area (you might want to look up and read this one):

L ~ σ^x where x ≈4

So, get dispersion from spectrum, get luminosity, and with magnitude get distance!

Fundamental Plane

Dressler et al. (1987) include all three of the plane parameters and find a tight relationship:

Can also substitute in a new variable Dn (a diameter chosen to match a surface brightness) which incorporates L and Σ.

Can get distances then to various accuracies.

Fundamental Plane

Three views of the relationship from Inger Jorgensen et al. (1997). The top is “face-on” and the other two views are projections.

As the relationship involves Luminosity it is a distance indicator.

Physical origin of interest for understanding early type galaxies.

Fundamental Plane

Physical origin:

If Virial Theorem applies, then the FP means that M/L ratio depends on the three variables.

The orientation of the plane in parameter space implies that M/L depends on the mass (M/L ~ L^0.2)

Metallicity also?


	Today: Homeworks due – discussion?
	WIRO Observing trip

	More on Elliptical Galaxies
	(Ch. 4, Combes et al. , Ch. 3, Longair)



	Unless noted, all figs and eqs from Combes et al. or Longair.


	Top is K0 III star, M87 off center, M87 near center
	Stellar Absorption Lines, note Ca II H&K
	Properties of interest are dispersion and shift. Dispersion is especially complicated since it depends on an ensemble of stars with different absorption line widiths.


	Rotation (V) and velocity dispersion (σ) curves for some ellipticals (Davies et al. 1983)


	So, are ellipticals simple to understand dynamically? Not so clear. We’re seeing a 2D picture of a 3D object.

	In general, elliptical galaxies rotate too slowly for this to account for the flattening observed. In other words, their ratios of rotational to random kinetic energy is too low.

	If ellipticity e were due to rotation (e.g., a pure oblate rotator with an isotropic velocity distribution) then (V_rot/σ)_iso ~ (e/(1-e))^1/2. Is it?


	Fit ellipses to surface brightness distribution.
	Can define “boxy” and “disky” galaxies (see Kormendy & Djorgovski 1989, ARAA, 27, 235), which correlate with other properties.


	Top, e(r) plotted for some galaxies.
	Bottom, e(r) and position angle as a function of radius. Proof of triaxalality? Or artifact of other?
	Just FYI, e = 1-b/a and r = (ab)^1/2


	Deviations in r^1/4 light profile can be quantified into classes, and corresponds with the presence of close neighbor galaxies (T3).


	Stellar “shells” from capture of smaller galaxies.

	Dust lanes associated with captured material in equilibrium.

	See text for more discussion.


	Should follow equations of stellar dynamics.
	Treat as “collisionless” ensemble
	Relaxation time is important – this defines the time between “collisions”
	Work out energetically to get
	T_relax = V³/8πnG²m²log(R/b)
	V is mean relative speed, n is stellar density, m is the mean stellar mass, R is radius of galaxy, and b is the minimum impact parameter.
	Details will probably be a homework problem.


	Crossing times: T_cross ~ R/V
	Relaxation time then varies as
	T_relax ~ 0.2(N/logN) T_crosswhere N is the number of stars in the system
	For galaxies, crossing times ~1/100 the age of the universe (100 million years), and the relaxation times are on order of 10¹⁷ years – so close interactions between stars essentially never happen. “Collisionless” is a good approximation. This is because N is large, and the collision must be very close for another star’s potential to overwhelm the ensemble potential.
	Note: this argument is NOT true for Globular Clusters! Also not true for galaxy clusters!
	The upshot of this is that you can treat galaxies in terms of motions in a global potential.


	Combes et al. approach the subject from the general case of distribution functions (f) – what are the positions, velocities, and time of the stars moving in a potential?
	Can start with the continuity equation (mass) which is the collisionless Boltzmann equation, AKA the Vlasov equation:


	-dU/dr is the gravitational force exerted. The potential can be obtained from the Poisson eq: ΔU(r) =4πGρ(r). Must find f(r,v) that solves these self-consistently. They discusses the isothermal sphere case:
		f(E)=(2πσ²)^-3/2ρ₀e ^-E/^σ² (complications vanish in symmetry)
	Should be familiar-looking function. Still, want density as a function of radius.


	Longair’s starting point is the Lane-Emden equation (4-7), which assumes spherical symmetry and hydrostatic equilibrium. Isothermal = same mean velocities everywhere. Can apply to stars, galaxies, clusters.


	Simple case, for which ideal gas law holds at all radii, p = ρkT/mass. In thermal equilibrium, 3/2 kT = ½ mass <v²>, therefore:



	Non-linear, only has an analytical solution for large r using a power-series expansion and applying some reasonable boundary conditions.


	Boundary conditions:
		Smoluchowski’s Envelope at large radii where density, timescales go to extremes.
		In astrophysical systems, outmost radii are subject to interactions with other clusters, hence a tidal radius.


	Then recast our equation as:


	Where we have the dimensionless quantities x=r/α and ρ = ρ₀y, & A, structural length α:


	As usual in real astronomy things are complicated because we only see a projection rather than a 3D distribution. The projection surface density distribution onto a plane is:



	Where now q is the projected distance from the center.


	Fit the function to an observed distribution.
		N(q) = ½ at q=3, or core radius R_1/2 =3 α.
	To get a central mass density then also need to measure the velocity distribution. From equipartition ½ m<v²>=3/2 kT. So:


	Isothermal spheres have infinite extent. More complex and realistic version based on Fokker-Planck equation by King (1966). Assumes no particles present with escape velocity+. Sharp E cutoff. See texts for more.


	Real galaxies are not necessarily spherical and isotropic. Eddington and Michie models:


	Luminosities, surface brightness, central velocity distribution, (and others), are correlated, hence the term “fundamental plane.” Ellipticals populate a plane in parameter space. BIG area of research – very useful tool and helps us understand galaxies.
	Faber & Jackson (1976) is a classic in this area (you might want to look up and read this one):
	L ~ σ^x where x ≈4
	So, get dispersion from spectrum, get luminosity, and with magnitude get distance!


	Dressler et al. (1987) include all three of the plane parameters and find a tight relationship:



	Can also substitute in a new variable Dn (a diameter chosen to match a surface brightness) which incorporates L and Σ.
	Can get distances then to various accuracies.


	Three views of the relationship from Inger Jorgensen et al. (1997). The top is “face-on” and the other two views are projections.
	As the relationship involves Luminosity it is a distance indicator.
	Physical origin of interest for understanding early type galaxies.


	Physical origin:
	If Virial Theorem applies, then the FP means that M/L ratio depends on the three variables.
	The orientation of the plane in parameter space implies that M/L depends on the mass (M/L ~ L^0.2)
	Metallicity also?