19 March 2013 Astronomy 142, Spring 2013 1

Today in Astronomy 142: the Milky Way

The shape of the Galaxy Stellar populations and

motions Stars as a gas:

• Scale height, velocities and the mass per area of the disk

• Missing mass in the Solar neighborhood

Wide-angle photo and overlay key of the Sagittarius region of the Milky Way. The very center of the Milky Way lies behind particularly heavy dust obscuration. (By Bill Keel, U. Alabama.)

The number of stars brighter than f0

Suppose stars are uniformly distributed in space, with number density n, and have typical luminosity L. How many are brighter (that is, have flux greater) than some value f0? Presuming that there is no extinction: to the flux f0

corresponds a distance r0 .

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( )

π ππ

ππ

−

= ⇒ =

> =

= ∝

20 0 0 0

30 0

32 3 2

00

4 443

43 4

f L r r L f

N f f r n

n L ff 0log f

3/20f−∝log N

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What is the shape of the Milky Way?

Herschel (1785), and later Kapteyn (1922), used this fact to characterize the shape of the Milky Way. Their idea was that if the Milky Way has edges, there would be points past which N would decrease faster than And indeed actual star

counts at low fluxes are less than predicted by this relationship, and the numbers at larger fluxes.

0log f

3/20f−∝

( )0

log Nf f>

Star counts in directions of the Milky Way disk (blue), and in the perpendicular directions (red).

−3/20 .f

Herschel’s Milky Way

“Section of our sidereal system” (Herschel 1785). The long axis of the figure runs roughly from 20h22m, +35° in Cygnus (left) to 8h20m, -35° in Puppis (right); the short axis points toward 12h24m,+58° (Ursa Major).

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The Great Rift in the summer Milky Way, interpreted as a dearth of stars – referred to as “an opening in the heavens,” since he was writing in English this time.

Sun

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Kapteyn’s “universe”

Kapteyn (1922) knew the distances to many nearby stars, and could therefore calibrate the star counts in terms of the stellar density n, and a shape of the stellar assemblage in terms of hydrostatic structure. This enabled him to hand-wave a position for the Sun

based upon the stellar density in the Solar neighborhood: r = 650 parsecs, z = 38 parsecs.

Sun

Shapley and Galactocentric distance

29 March 2011 Astronomy 142, Spring 2011 6

Dots: cluster positions projected onto the plane of the Milky Way. Circles: radii in integer multiples of 10 kpc. From Shapley 1919.

Harlow Shapley thought that globular clusters, massive as they are, would indicate Galactic structure better than stars. He was also armed with

Henrietta Leavitt’s discovery of the Cepheid period-luminosity relation (6 April 2013), which he applied to the RR Lyr stars.

This led to a Solar distance from the center of the cluster distribution of 100 parsecs (Shapley 1918).

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The Milky Way as an “island universe”

All these methods got the shape, and the Sun’s distance within from the center, quite wrong. As we have seen, interstellar extinction is substantial in

the plane of the Milky Way, and obscures most of the distant starlight.

Both Herschel and Kapteyn were therefore seeing only to the edge of the extinction, not to the edge of the stars.

By Kapteyn’s time many astronomers were on the right track by analogy: that the Milky Way is like the spiral nebulae, and we live well off the center, as Shapley found. • But not nearly as far: modern measurements of

parallax of water-vapor masers in molecular clouds near the Galactic center give 8.4 kpc for our Galactocentric radius. (Reid et al. 2009).

Halo, bulge and disk The globular clusters seem to trace a spherical halo around the galaxy, in which we are immersed ourselves. One can see non-cluster stars out there too, if one looks

hard enough. Once we know where to look for the Galactic center, we notice a couple of other features: the bulge, a thicker, brighter concentration of stars

surrounding the center…

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Bruno Gilli/ESO

Halo, bulge and disk

…and the disk, a belt of stars and extinction that passes through the center – this is the Milky Way proper.

Since the belt seems not to have ends, we are also immersed within it.

Distribution of stars thicker than that of dust everywhere along the belt: there is a thick disk and a thin disk.

Much like many other galaxies.

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Bruno Gilli/ESO

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The Milky Way’s central regions, in starlight (near-infrared), from the NASA COBE DIRBE experiment. NGC 891, also in starlight (near-infrared), from the 2MASS survey (U. Mass./NASA).

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Schematic structure of the Milky Way

Figu

re: C

hais

son

and

McM

illan

, Ast

rono

my

Toda

y

Halo, bulge and disk (continued)

As we will be seeing this week and next, the different components of the Galaxy owe their distributions to differences in motion. The disk is dominated by rotation. Objects which belong

to the disk have randomly-directed and rotational components of motion, but the rotational components are much larger.

The bulge and halo are dominated by random motion, with little to no trace of rotation.

Though our immersion within two of the components makes things complicated, we can use the motions to weigh the various components, and the Galaxy itself.

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Halo, bulge and disk (continued)

The visible mass of the galactic halo is small compared to that of the disk and bulge. As we will see, there are strong reasons to think that the true mass of the halo is similar to the others, leading to hypotheses of dark matter.

Dynamics and composition of stars are correlated: Population I: small dispersion of velocities (i.e. small

random velocities), absorption lines of heavy metals, confined to a very thin plane. Relatively young.

Population II: large dispersion of velocities, can lie further from the Galactic plane.

Population I lies predominantly in the disk, less in the bulge, not in the halo. Population II can be found in all three components.

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Motions of stars in the Galaxy, and their use in determining its mass distribution

How much does the Galaxy weigh? Stars move about in response to the gravitational potential of the rest of the stars in the galaxy, so… their systematic motions (rotation) can be used with

Newton’s Laws to measure masses within the Galaxy. • This works even better with interstellar gas than with

stars. Stars are too massive to be influenced by the pressure of

the interstellar medium, but they collide inelastically very rarely, so their random motions can be used with thermodynamics to measure masses within the Galaxy - the stars in this sense can be thought of as particles in a gas.

Mass per unit area of the Galaxy’s disk, in the Sun’s neighborhood

The radial structure of the disk is determined in the usual manner from centrifugal support: balancing the force on a test particle at radius r from the mass M(r) contained in interior orbits, with centrifugal force. Just like a protoplanetary disk, or any other astro-disk.

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( ) 2GM r m r

2mV r

V

Mass per unit area of the Galaxy’s disk, in the Sun’s neighborhood (continued)

The vertical structure is determined by hydrostatic equilibrium, as usual, but there are two main differences from protoplanetary disks: much of a galactic disk is self gravitating: the weight is

from the disk itself, not from a “star” in the middle. the pressure is that of the stars’ motions. Guess:

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V diskma

starsP A2

stars stars randomkinetic energy 1

volume 2P vρ≈ ≈

Mass per unit area of the Galaxy’s disk, in the Sun’s neighborhood (continued)

Weight. If the disk is thin and self gravitating, we can regard it locally as an infinite plane, and work out the weight of stars above and below the plane accordingly.

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r

µ h

dr

dF

m

( )2 2 2 2

2Gm rdr hdFh r h r

π µ=

+ +

Mass per unit disk area, not to be confused with mean particle mass in a gas.

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Mass per unit area of the Galaxy’s disk, in the Sun’s neighborhood (continued)

r

µ h

dr

dF

( )2

2

3/2 3/20 2 2

1/2

2

2

12 21/2

h

h

rdr duF G mh G mhuh r

uG mh G mh G mh

π µ π µ

π µ π µ π µ

∞ ∞

∞−

= =+

= = =−

∫ ∫

m

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Mass per unit area of the Galaxy’s disk, in the Sun’s neighborhood (continued)

Pressure. Recall formula for pressure in terms of particle number density, speed and momentum (12 February): Consider a certain class of stars to be gas particles, and consider the component of their motion perpendicular to the Galactic plane. Suppose the distribution of these stars extends above and below the plane by some scale height H/2. Consider stars lying on the ends of a cylinder of Galactic matter that extends one scale height above and below the plane.

2

1 1z

z z z

dpF nA zP pA A dt A tnv p v

δδ

ρ

≅ = ≅

= =

Number of particles that hit wall Typical momentum per particle Time interval in which they hit

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Mass per unit area of the Galaxy’s disk, in the Sun’s neighborhood (continued)

Weight of the cylinder, approximately: If stellar pressure balances gravity, then From above, for self gravitating disk: All terms on the right are observable!

H/2

dA

dW z

zg dmdWw g H

dA dAρ= = =

2 2, or .z z z zv g H g v Hρ ρ= =

2

2

2 2 .z

z z

F G m mg

g G v GH

π µ

µ π π

= =

⇒ = =

dW

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Mass per unit area of the Galaxy’s disk, in the Sun’s neighborhood (concluded)

Putting the numbers in, for the solar neighborhood: Star counts in the solar neighborhood enable us to estimate the luminosity per unit area L (also called the surface brightness) of the disk locally. This leads to the mass to light ratio: Thus the solar neighborhood on average emits light less efficiently than the Sun does – consistent with there being more lower-mass stars than higher-mass stars.

3 -21.5 10 gm cmµ −= ×

15 .M Lµ − =

L

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Dark matter I: the Galactic disk in the solar neighborhood

When this procedure was first applied to observations by Oort in the late 1940s, the resulting value of µ was greater than that of visible stars and interstellar gas in the Solar neighborhood by a factor of about 2. Missing mass, or dark matter: mass that emits no light

but can be detected by its gravity? Since then, Better (less biased) samples of stars have led to smaller

estimates of the total µ. The discovery of neutral atomic and molecular gas in the

ISM has increased the “luminous” mass a bit. Now the luminous and gravitating µ match precisely: this

form of dark matter has vanished (e.g. Kuijken and Gilmore 1991).