Astr 5460 Wed., Sep. 29, 2004

This week: Finish Fundamental Plane

Spiral Structure

Follow-up WIRO Observing trip

Spiral Structure

(Ch. 5, Combes et al., parts)

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

Observing Project: Reductions+

Raw data to one accessible directory

Copies of logs to everyone

Everyone needs to do basic reductions:

Bias subtraction

Flat fielding

Combination of images and cosmic ray rejection

Flux calibration/image zero points

“Basic” Analyses (on several ellipticals and spirals)

Integrated magnitudes and colors, preferably with errors

Surface Brightness Profiles and fits (e.g., r^1/4 law)

“Pretty” three color pictures (presentation is important)

Additional Analyses (should try several of these)

Quantitatively measuring “diskiness/boxiness” of ellipticals (tricky)

Ellipse position angle, ellipticity as function of radius

Inclination angle of an intermediate spiral galaxy

Colors of bulge vs. dust lane across bulge

Colors of spiral arms/H II regions

Anything else clever you can think of – be creative!

Everyone should write up results as a paper using LaTeX with all the five standard parts of a scientific article

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?

Spiral Structure of Galaxies

Discuss basics only – too many topics to cover this semester.

Will be covered in more detail in “Stars and the Milky Way” next semester

Spiral Galaxies

Already discussed many observational properties (e.g., frequency, types, luminosity functions, rotation curves, light profiles, masses, etc.).

Now want to spend some time discussing their most notable feature – the spiral arms! It hasn’t been such an easy thing to understand. Why not?

Spiral Galaxies

Opening considerations:

Disk Stability Issues. Most of the mass of the galaxy is in a dark halo, and stars are treated as particles in the overall potential, so should disks even be stable?

Note that nearby interactions are unimportant: T_relax/t_c = (h/R)(N/(8logN)) – “collisionless” if large. For N = 10¹¹, the ratio of relaxation to crossing time is about 10⁸.

Spiral Galaxies -- Stability

Small Scale Stability

Must not satisfy the Jean’s criteria

Large Scale Stability

Rotational support

Spiral Galaxies -- Stability

Jean’s criteria governs “small scale” collapse due to self-gravitation

Treated as stability vs. perturbation in an infinite homogeneous medium in equilibrium

For fluid with a pressure P = ρ₀v_s², then perturbations with λ > v_s(G ρ₀)^-1/2 = λ_J are unstable.

Basically, does the disturbance have time to cross in a freefall time, t_ff = (Gρ)^-1/2?If so, then pressure forces are negligible.

The sound crossing time is r/v_s, which gives the λ_J criteria

For galaxies, we can write an effective pressure in terms of the velocity dispersion: λ > σ(G ρ₀)^-1/2 = λ_J

Spiral Galaxies -- Stability

Stability due to Rotation – large scales

See Toomre (1964) for details

Considered axisymmetric perterbations

Consider small region of size L on a disk with surface density μ (mass then is ~μL²), rotating at Ω, distance d from the center. Imagine a perturbation that locally increases the surface density.

Angular momentum conserved, so the angular velocity must also increase, creating a centrifugal force in the rotating frame. If this force can send the mass back to its original position, system is stable.

Stability: L > L_crit = 2Gμ/3Ω², so requires large scales. Order of magnitude, Ω²R = V²/R ~ GμR²/R² = Gμ, so L_crit ~ R

Spiral Galaxies -- Stability

Stability on all scales

Combine the criteria for large and small scales

Need a minimum velocity dispersion to make critical Jean’s length equal to L_crit.

Order of magnitude then, write Jean’s length in terms of surface density by equating free fall and crossing times: λ_J = σ²/Gμ = L_crit, or σ=Gμ/Ω

A more rigorous calculation gives σ_r=3.36Gμ/κ, where κ is the epicyclic frequency (see below)

Ratio of observed radial dispersion to critical dispersion is called Q, and stability requires Q > 1.

Near the Sun, Q is between 1 and 2.

Stellar Orbits

First-order Epicyclic theory (yes, epicycles!)

Just an overview here for context

Work with nearly circular orbits in a flattened axisymmetric gravitational potential. Write down the equations of motion in polar coordinates, using a Taylor series.

Basically just looking at the linear deviation from circular orbits. You wind up with equations that solve for the radial excursions and the precession of the orbit in terms of “epicycles” of frequency κ.

Stellar Orbits

κ=2Ω is an ellipse, for instance.

Stellar Orbits

Thinking of rotation curves, the core shows fixed rotation (Ω = constant, V=Ωr, and κ=2Ω)

Differential rotation where V = constant (Ω=V₀/r; then κ=√2Ω)

Ratio of κ/Ω stays between 1 and 2 for realistic potentials.

Linblad Resonances

There are resonances of interest between stellar orbits (Ω(r)) and spiral patterns (Ω_p).

Density Wave Theory

Grand Design Spirals

Winding Problem

What’s up with these spiral patterns anyway?

Spiral flat rotation curves

Differential rotation

Implication for the persistence of a pattern

Let’s work it out

Stochastic Spirals

AKA “Self-sustaining Star Formation” = SSF or “Contagious” star formation.

Precession–a clue to Grand Design Spirals?

Epicyclic orbits are basically precessing ellipses.

Precession–a clue to Grand Design Spirals?

Linblad: rate of precession of orbits Ω – κ/2 almost constant with r, so a 2-armed spiral wave would rotate without deformation.

Density Waves

Barred or spiral kinematic waves = density waves.

Trailing and Leading Spirals

Winding direction vs. sense of rotation.

How about the Continuous ISM?

How about the clouds?

Flow of molecular clouds in a spiral potential, treated as a 1D mechanical oscillator. Clouds accumulate in the potential wells (spiral arms).

Excitation of Spiral Waves by Companion Galaxies

For Next Week

Chapter 7 in Combes et al. on interacting galaxies.

Start on Data Reduction and Analysis

New problem set next week likely.


	This week: Finish Fundamental Plane
	Spiral Structure
	Follow-up WIRO Observing trip

	Spiral Structure
	(Ch. 5, Combes et al., parts)



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


	Raw data to one accessible directory
	Copies of logs to everyone
	Everyone needs to do basic reductions:
		Bias subtraction
		Flat fielding
		Combination of images and cosmic ray rejection
		Flux calibration/image zero points
	“Basic” Analyses (on several ellipticals and spirals)
		Integrated magnitudes and colors, preferably with errors
		Surface Brightness Profiles and fits (e.g., r^1/4 law)
		“Pretty” three color pictures (presentation is important)
	Additional Analyses (should try several of these)
		Quantitatively measuring “diskiness/boxiness” of ellipticals (tricky)
		Ellipse position angle, ellipticity as function of radius
		Inclination angle of an intermediate spiral galaxy
		Colors of bulge vs. dust lane across bulge
		Colors of spiral arms/H II regions
		Anything else clever you can think of – be creative!
	Everyone should write up results as a paper using LaTeX with all the five standard parts of a scientific article


	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?


	Discuss basics only – too many topics to cover this semester.

	Will be covered in more detail in “Stars and the Milky Way” next semester


	Already discussed many observational properties (e.g., frequency, types, luminosity functions, rotation curves, light profiles, masses, etc.).

	Now want to spend some time discussing their most notable feature – the spiral arms! It hasn’t been such an easy thing to understand. Why not?


	Opening considerations:
		Disk Stability Issues. Most of the mass of the galaxy is in a dark halo, and stars are treated as particles in the overall potential, so should disks even be stable?
		Note that nearby interactions are unimportant: T_relax/t_c = (h/R)(N/(8logN)) – “collisionless” if large. For N = 10¹¹, the ratio of relaxation to crossing time is about 10⁸.


	Small Scale Stability
		Must not satisfy the Jean’s criteria

	Large Scale Stability
		Rotational support


	Jean’s criteria governs “small scale” collapse due to self-gravitation
		Treated as stability vs. perturbation in an infinite homogeneous medium in equilibrium
		For fluid with a pressure P = ρ₀v_s², then perturbations with λ > v_s(G ρ₀)^-1/2 = λ_J are unstable.
		Basically, does the disturbance have time to cross in a freefall time, t_ff = (Gρ)^-1/2?If so, then pressure forces are negligible.
		The sound crossing time is r/v_s, which gives the λ_J criteria
		For galaxies, we can write an effective pressure in terms of the velocity dispersion: λ > σ(G ρ₀)^-1/2 = λ_J


	Stability due to Rotation – large scales
		See Toomre (1964) for details
		Considered axisymmetric perterbations
		Consider small region of size L on a disk with surface density μ (mass then is ~μL²), rotating at Ω, distance d from the center. Imagine a perturbation that locally increases the surface density.
		Angular momentum conserved, so the angular velocity must also increase, creating a centrifugal force in the rotating frame. If this force can send the mass back to its original position, system is stable.
		Stability: L > L_crit = 2Gμ/3Ω², so requires large scales. Order of magnitude, Ω²R = V²/R ~ GμR²/R² = Gμ, so L_crit ~ R


	Stability on all scales
		Combine the criteria for large and small scales
		Need a minimum velocity dispersion to make critical Jean’s length equal to L_crit.
		Order of magnitude then, write Jean’s length in terms of surface density by equating free fall and crossing times: λ_J = σ²/Gμ = L_crit, or σ=Gμ/Ω
		A more rigorous calculation gives σ_r=3.36Gμ/κ, where κ is the epicyclic frequency (see below)
		Ratio of observed radial dispersion to critical dispersion is called Q, and stability requires Q > 1.
		Near the Sun, Q is between 1 and 2.


	First-order Epicyclic theory (yes, epicycles!)
	Just an overview here for context
	Work with nearly circular orbits in a flattened axisymmetric gravitational potential. Write down the equations of motion in polar coordinates, using a Taylor series.
	Basically just looking at the linear deviation from circular orbits. You wind up with equations that solve for the radial excursions and the precession of the orbit in terms of “epicycles” of frequency κ.


	Thinking of rotation curves, the core shows fixed rotation (Ω = constant, V=Ωr, and κ=2Ω)
	Differential rotation where V = constant (Ω=V₀/r; then κ=√2Ω)
	Ratio of κ/Ω stays between 1 and 2 for realistic potentials.


	There are resonances of interest between stellar orbits (Ω(r)) and spiral patterns (Ω_p).


	What’s up with these spiral patterns anyway?
		Spiral flat rotation curves
		Differential rotation
		Implication for the persistence of a pattern
		Let’s work it out


	AKA “Self-sustaining Star Formation” = SSF or “Contagious” star formation.


	Linblad: rate of precession of orbits Ω – κ/2 almost constant with r, so a 2-armed spiral wave would rotate without deformation.


	Flow of molecular clouds in a spiral potential, treated as a 1D mechanical oscillator. Clouds accumulate in the potential wells (spiral arms).


	Chapter 7 in Combes et al. on interacting galaxies.
	Start on Data Reduction and Analysis
	New problem set next week likely.