Lecture 33 - Our Galaxy (4/8/96)

Seeds: Chapters 10, 12

1. Evolution of Binary Systems
   • Unless stars in binary system have the same mass, they will evolve off the main sequence at different rates.
   • The more massive star will exhaust its core hydrogen first, and expand into a red giant.
   • For close binary systems, this will overflow the Roche Lobe, which marks the region which is bound only to the massive star.
   • Mass will then be transferred to the less massive star (still on the main sequence) through the Lagrange point which marks the point between them where the gravitational forces from the two stars are equal.
   • While it expands up the giant branch, the massive primary will transfer a significant fraction of its mass onto the secondary.
   • Eventually the primary will eject what is left of its envelope and become a white dwarf orbiting a main sequence star.
   • The secondary will have gained mass from the primary, and will move "up" in spectral type, perhaps to an O or B star. This will present the strange sight of an "old" O or B star (since it spent a long time as a lower mass star).
   • Eventually the secondary will evolve off the main sequence, and transfer mass back onto the white dwarf after overflowing its Roche lobe.
   • This gas will form an accretion disk around the white dwarf, which will be heated to high temperatures and emit X-rays.
   • Some of this mass will make its way through the disk and be deposited on the surface of the white dwarf. This mass, which is mostly hydrogen, will build up until it is dense enough to undergo fusion, burning in a great flash called a nova.
   • Novae are less luminous than supernovae (hence the "super").
   • It is also possible that this transferred mass can push the white dwarf past the Chandrasekhar limit of 1.4 Msun. The star will collapse to a neutron star fusing its nuclei to iron, and causing a supernova. These sorts of supernova can be brighter than the usual sort of Type II Supernovae since they are without an envelope to absorb the energy, and are believed to explain Type I Supernovae.
   • Eventually the secondary will evolve to a white dwarf, leaving a white dwarf - white dwarf binary system.
   • Variations of this scenario can result in neutron-star / white dwarf, neutron-star / neutron-star binaries also. Black hole companions are possible also, though how such a system would survive the supernova of the primary is uncertain.
2. Survey of Our Galaxy
   • The milky way appears of a band of faint stars spanning the sky. It appears that we are in the middle of a disk or wheel of stars.
   • It was only early in this century that the true extent of our galaxy was realized - it is over 30,000 pc in diameter and contains about 100 billion stars!
   • Almost everything we can see with our unaided eyes in the sky is in our galaxy. The exceptions are the faint Andromeda galaxy (M31), and the Large and Small Magellanic Clouds near the south celestial pole.
   • There are a number of questions we would like answered about our galaxy:
     • What does our galaxy look like?
     • What are its physical properties?
     • What does it contain?
     • What is its history?
     • What is its neighborhood like?
3. The Appearance of the Milky Way
   • There are three main components to the Milky Way: disk, bulge, and halo.
   • We measure galactic distances in kiloparsecs, abbreviated kpc, which are thousands of parsecs: 1 kpc = 1000 pc.
   • The galactic disk, in which the Sun is embedded at a distance of about 8.5 kpc from the center, is over 15 kpc in radius.
   • The disk does not have a sharp edge at 15 kpc, but it fades away in luminosity fairly drastically at this radius.
   • The galactic bulge is a slightly flattened sphere of stars centered on the galactic center with a radius of about 2 kpc.
   • The bulge is bright, and has a high star density, but is difficult to see because of the large obscuration by dust in the inner disk of the galaxy.
   • The galactic halo is an extended tenuous assemblage of stars and globular star clusters, that extends out to more than 80 kpc from the center.
   • The disk is rotating, like a protostellar disk. The Sun is moving at a velocity of 220 km/s, which is the rotation velocity of the disk at the Sun's radius of 8.5 kpc.
   • The bulge may be slightly rotating, while the halo is not rotating at all, and the halo stars are moving in elliptical orbits with random orientations about the center.
   • The disk contains large amounts of gas and dust in addition to stars. All the gaseous nebulae are in the disk. The bulge and halo do not contain gas clouds.
   • The dust clouds in the disk are apparent as dark clouds, lanes, and nebulae against the brightness of the Milky Way. They obscure our view in optical wavelengths of most of the galaxy, especially toward the galactic center.
   • It is thus hard to measure the true luminosity of our galaxy, since we cannot just add up all the light. Comparison with other galaxies, and with galactic models lead us to estimate a total luminosity of about 10^11 Lsun for the Milky Way.
   • We can see only about 2 kpc from the Sun in the disk, except in some special "windows" where low dust content lets us see farther than this. Because of this, it was long thought that the galaxy was a wheel centered on the Sun (William Herschel 1785).
   • To find out what the galaxy really looked like, we needed to be able to measure distances.
4. Distance Indicators
   • Distances in the galaxy are too great to use parallax. The parallax of something 100 pc away is 0.01" and thus too small to measure from the ground.
   • We can use apparent and absolute magnitudes (luminosity) to find distances, using the formula for the distance modulus: m_v - M_v = 5 log d - 5
   • Thus, if we had a source of known luminosity, and thus known absolute magnitude, we could find its distance by comparing this to its apparent magnitude.
   • We can get a distance estimate for a main sequence star by using its temperature to find its luminosity on the H-R diagram. This method is called spectroscopic parallax. However, this is not very accurate, since it is hard to determine the effective temperature exactly, and the width of the main sequence in the H-R diagram can be as much as a magnitude.
   • It turns out there are a class of stars for which it is easy to find out their intrinsic luminosity, or absolute magnitude.
   • Certain stars are seen to vary in brightness with a regular pattern and period (not outbursts or novae). When plotted in the H-R diagram, these stars fall in a well-defined band above and to the right of the main sequence.
   • Stellar models predict that stars lying in this instability strip should indeed be variable because of an energy absorbing layer that forms in the envelope of the star. This layer can absorb and emit the energy like a resevoir, causing variations in the radius of the star, which pulsates alternately expanding and contracting.
   • Calculations and observations also show that the period of pulsation is dependent upon how far up the instability strip the star lies. Since the strip is rather narrow, and oriented at an angle to the luminosity-temperature axes, the luminosity of the star can be fixed by observations of the temperature and the period of the variations!
   • There are three varieties of these variable stars. Type I Cepheids are disk stars that lie in this strip, with periods from 1 day to over 100 days. These are the "classical" Cepheids first discovered by Harvard Astronomer Henrietta Leavitt in 1912.
   • The name Cepheid comes from the prototype star in this class: Delta Cepheii.
   • The Type II Cepheids are halo stars that are counterparts to the classical Type I Cepheids. They have similar ranges of periods, but have lower luminosities for a given period. It is easy to tell from the stellar spectra which sort of Cepheid a give star is (Type I have more elements heavier than helium).
   • There are also halo stars with shorter periods and lower luminosities than the Type II Cepheids. These are called RR Lyrae stars, after the first star in the class discovered.
   • If one of these types of variable stars is seen, then the period can be measured and the distance modulus, and thus the distance to the star, can be determined!
   • The astronomer Harlow Shapley used Leavitt's Cepheid relation, and measured the distances to a number of globular clusters in the halo.
   • In a series of papers published from 1915 to 1919, Shapley reported his findings that the system of globular clusters was not centered on the Sun, but on a point in the galactic plane in the direction of the constellation Sagittarius, and was about 15 kpc from the Sun!
   • This was identified as the location of the galactic center, and displaced us once again from the center of things in a post-Copernican revolution.
   • The best current estimates give the distance to the center of 8.5 kpc.
   • The scale of our galaxy given above was obtained with the use of Cepheids and RR Lyrae stars as distance indicators.
5. The Mass of the Galaxy
   • How do we find the mass of our galaxy? We cannot just add up the luminosity and use some approximate mass-luminosity relation (that would be incorrect as we see below).
   • As usual we use gravity - Kepler's 3rd Law! Note that since we can ignore the mass of any star in the rotating disk, Newton's version becomes: M(<R) = R³ / P² with R in AU and P in years giving M in Msun as usual. Note that the mass M corresponds to the mass enclosed by a sphere of radius R - the mass interior to R.
   • Using the Sun's velocity of 220 km/s at R=8.5 kpc, we get an mass of about 10^11 Msun inside the Sun's orbit in the disk.
   • It is easy to see the relation if we frame it in the form of Newton's law, with the centripetal velocity: v² = G M / R
   • Note that if all of the mass in the galaxy were at a point at the center, like the Sun in the solar system, the rotation velocities would fall off as the inverse square-root of the radius (since M is constant). This relation is called Keplerian rotation, since that is the relation Kepler found for the solar system.
   • However, measurements of the velocity as a function of radius from the center, v(R), show that after climbing up from zero in the center to about 200 km/s in the inner kpc, the rotation curve remains relatively constant out to the edge of the disk at 15 kpc!
   • Thus, it must be that the enclosed mass M(R) is growing as R, in order to keep v constant in our equation. Q: What must the mean density within radius R be doing as a function of R for a constant rotation curve?
   • Note that a solid body rotation curve, like a phonograph record, would have v proportional to R. The galaxy is somewhere between Keplerian and solid body.
   • Since the mass in the galaxy is growing with R in the outer parts of the galaxy, while the total enclosed luminosity is barely growing (the central parts are much brighter than the dim outer parts), most of the outer mass of the galaxy must be made of dark matter, or at least be made of stuff that has more mass per unit of luminosity than stuff in the inner galaxy.
   • Using the orbits of things in the outermost halo, we find that the total mass of the galaxy is about 10^12 Msun.
   • Thus, the average mass-to-light ratio of the entire galaxy is about 10^12 Msun / 10^11 Lsun or 10 Msun/Lsun. For comparison, the mass to light ratio in the solar neighborhood is about 1 Msun/Lsun.

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