Star Formation & The Milky Way - pas.rochester.edukdouglass/Classes/Ast142/lectures/15_lect… ·...

Star Formation &The Milky Way

Free-Fall TimescalePre-MS stars and the Hayashi Track

The Shape of the GalaxyStellar Populations and Motions

Stars as a Gas

March 24, 2020

University of Rochester

Star formation & The Milky Way

Today’s lecture:I Collapse of a spherical cloud: free-fall timeI Synopsis of star formationI Pre-MS stars and the Hayashi trackI The shape of the GalaxyI Stellar populations and motionsI Stars as a gas: Scale height, velocities, and the

mass per unit area of the diskReading: Kutner Sec. 16.4, Ryden Sec. 19.1–19.3, ShuCh. 12

Wide-angle photo and overlay key of the Sagittarius region of the MilkyWay (from Bill Keel, U. Alabama).

March 24, 2020 (UR) Astronomy 142 | Spring 2020 2 / 30

http://pages.astronomy.ua.edu/keel/

Spherical free-fall time: Derivation

Follow a test particle that starts from rest at distance r0 from the center of the cloud attime t = 0. How long until it reaches the center, i.e., until the clump collapses to a point?

F = ma

−GMmr2 = m

d2rdt2

This is a 1D, nonlinear 2nd-order differential equation. M is the mass interior to r and isconstant. Multiplying both sides by dr

dt and integrate over t:

∫ t

0dt′

drdt′

d2rdt′2

= −GM∫ t

0

1r2

drdt′

dt′

Note that v = drdt and dv = d2r

dt2 dt and start solving.


Spherical free-fall time: Derivation∫ t

0

drdt′

d2rdt′2

dt′ = −GM∫ t

0

1r2

drdt′

dt′∫ v

0v′ dv′ = −GM

∫ r

r0

dr′

r′2

12

v2 =GM

r− GM

r0

Substitute back v = drdt :

12

(drdt

)2

=GM

r− GM

r0

drdt

= −√

2GMr− 2GM

r0

choosing the negative root so that the particle moves to smaller r.March 24, 2020 (UR) Astronomy 142 | Spring 2020 4 / 30

Spherical free-fall time: DerivationNow insert M = 4π

3 r30ρ0:

drdt

= −r0

√8πGρ0

3

( r0

r− 1)

Make a geometric u substitution r = r0 cos2 u:

drdt

= r0ddt

cos2 u = −2r0 cos u sin ududt

=⇒ dudt

=1

2 cos u sin u

√8πGρ0

3

(1

cos2 u− 1)

=1

2 cos2 u sin u

√8πGρ0

3(1− cos2 u)

=1

2 cos2 u

√8πGρ0

3


Spherical free-fall time: DerivationSeparate variables and integrate:∫ u

u0

cos2 u′du′ =12

√8πGρ0

3

∫ tff

0dt where

t = 0 : r = r0 r0 = r0 cos2 u0 =⇒ u0 = 0

t = tff : r = 0 0 = r0 cos2 u0 =⇒ u0 =π

2

=⇒∫ π/2

0cos2 u′du′ =

12

∫ π/2

0

[1 + cos 2u′

]du′ =

12

√8πGρ0

3

∫ tff

0dt[

u2+

sin 2u4

]π/2

0=

π

4=

tff

2

√8πGρ0

3

tff =

√3π

32Gρ0


Typical timescales

The number densities in the cores of molecular clouds can be quite large, of ordernmc = 106 cm−3.

If the gas is pure molecular hydrogen, then the typical density in the ISM is

ρ0 = µn = 2mpnmc ≈ 3× 10−18g/cm3

Plugging into the expression for the free-fall time gives a convenient expression in termsof these units:

tff =

√3π

32Gρ0= 40 kyr

(ρ0

3× 10−18 g/cm3

)−1/2

Question: If the Galaxy is ∼10 Gyr old and molecular clouds can collapse on such shorttimescales, why are any molecular clouds left?


Star formation synopsis

Molecular clump collapses gravitationally:I To a disk shape at first

I Collapse along the other two dimensions happens more slowly because of a rotation(angular momentum) and, possibly and to a lesser extent, magnetic forces

I Smaller scales — higher density regions — collapse faster (“inside-out collapse”)

I Central “core” gradually accretes some of the rest of the disk

I A bipolar outflow helps to get rid of the last of the accreted material’s angularmomentum.

The core becomes a star, and the remnants of the disk become a planetary system.


The initial mass function

The initial mass distribution is an empiricalfunction describing the distribution of stellarmasses that will form from a singlemolecular cloud.

Observations of numerous young stellarclusters reveal that their stellar massdistributions are nearly identical (theprimary difference due to the cluster’s age).


Bipolar jets and disks associated with young stars


Star formationThe outflows are supersonic (w.r.t. the molecular surroundings) and thus shocks aredriven into the ambient medium. The energy and momentum injected contributes to thesupport of the molecular cloud and can induce or hinder further star formation in theneighborhood.

L1641 VLA1/HH1-2 (Bally et al. 2002):


http://adsabs.harvard.edu/abs/2002AJ....123.2627B

Pre-Main Sequence stars

The young stars themselves are relativelyunextinguished at visible wavelengths unless thedisk is viewed edge-on; they have twodistinctive features:I The spectra show emission lines, notably

hydrogen recombination lines, and UVexcess emission, both produced in accretionshocks in material falling onto the star fromthe disk.

I Spectral type G/K/M: T Tauri stars. Earliertypes: Herbig Ae/Be stars.



I It takes lower-mass young stars millions of years to settle down to their final sizesand get fusion running.

I During this slow gravitational collapse, the conversion of gravitational potentialenergy dominates the luminosity of young stars. Recall that gravitationally-drivenluminosity is

L = −35

GM2

R2dRdt

I The gravitational collapse luminosity decreases with time and the star’s effectivetemperature changes little while this is going on. So the young star descends almostvertically at first through the H-R diagram. This path is called the Hayashi track.


https://en.wikipedia.org/wiki/Hayashi_track


I The Hayashi track happens to be the reverse ofthe path each star would take at the end of its lifeas it becomes a red giant.

I For a given age, the redder (lower mass, laterspectral type) stars lie further above the mainsequence than the bluer ones.

Right: T Tauri tracks (Stahler 1988). The Main Sequence is shown bythe diagonal line.


http://adsabs.harvard.edu/abs/1988PASP..100.1474S

The number of stars brighter than f0Suppose stars are uniformly distributed in space with number density n and typicalluminosity L. How many are brighter (i.e., have greater flux) than some value f0?

Presuming there is no extinction, there is adistance r0 corresponding to the flux f0:

f0 =L

4πr20→ r0 =

√L

4πf0

=⇒ N(f > f0) =4π

3r3

0n

=4πn

3

(L

4πf0

)3/2

∝ f−3/20

log f0

logN

∝ f3/20


Shape of the Milky Way

Herschel (1785) and Kapteyn (1922) usedthis idea to characterize the shape of theMilky Way.I If the MW has edges, N must decrease

faster than f−3/20 past the edges.

I Actual star counts at large fluxes areless than predicted by the N ∝ f−3/2

0relationship.

I Implication: the MW has a finite sizewith identifiable edges.

Red: star count in direction of Galactic diskBlue: star count ⊥ to Galactic disk

Observed counts:

log f0

logN

∝ f3/20


http://adsabs.harvard.edu/abs/1785RSPT...75..213H

http://adsabs.harvard.edu/abs/1922ApJ....55..302K

Milky Way structure: Halo, bulge, and diskI The globular clusters seem to trace out a spherical halo around the Galaxy, in which

we are immersed.I Non-cluster stars can be seen in the halo if you look carefully enough.I Once we know where to look for the Galactic center, we notice other features: a

bulge and a disk.

The Milky Way seen from Cerro Paranal (Bruno Gilli/ESO).March 24, 2020 (UR) Astronomy 142 | Spring 2020 17 / 30

http://www.eso.org/public/images/milkyway/

Halo, bulge, and disk

I Bulge: thick and bright concentration of stars surrounding the Galactic Center.I Disk: a belt of stars and extinction passing through the center (this is the “Milky

Way” proper).I Since the belt seems not to have ends, we are also immersed in it.I The distribution of stars is thicker than that of dust everywhere along the belt: there

is a thick disk and a thin disk (much like other galaxies).

IR image of the Milky Way from the 2MASS All-Sky Survey.


http://www.ipac.caltech.edu/2mass/gallery/showcase/

Schematic structure of the Milky Way

The different components of the Galaxy (Buser 2000).March 24, 2020 (UR) Astronomy 142 | Spring 2020 19 / 30

http://adsabs.harvard.edu/abs/2000Sci...287...69B

Mass of the halo, bulge, and diskThe different components of the Milky Way owe their distributions to differences inmotion.I The disk is dominated by rotation. Objects which belong to the disk have both

random and rotational components to their motion, but the rotational componentdominates.

I The bulge and halo are dominated by random motion with little or no evidence ofrotation.

What is the Galaxy’s mass? Stars move in response to the gravitational potential of therest of the galaxy, so we can use their motion to trace out the mass distribution.I The systematic motions (rotation) can be used with Newton’s Laws to measure

masses within the Galaxy.

I Note: This works even better with interstellar gas than with stars.


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