Lecture 14 - Star Formation (3/2/99)

ISM --- | --- Stellar Energy

Reading:
Chapter 15-2, 15-3 (ZG4)

Notes:
pages 54 - 56

	Key Question:	Where do stars form?
	Key Principle:	Protostellar Collapse
	Key Problem:	What are the requirements for a cloud core to collape?

Investigations:

1. Dust Revisited
   • What is the physical cross section for a dust grain?
   • How is this related to the scattering cross section at a given wavelength?
   • What is Mie scattering theory?
   • What is the typical cross section at 5000 Angstroms?
   • What does 1 mag of extinction over 1 kpc path length imply for the dust density?
2. Pressure Equilibrium of Clouds in ISM
   • How can gas at different densities and temperatures co-exist in equilibrium in the ISM?
   • What is pressure equilibrium and how does it relate n and T between clouds?
   • Why are hot clouds more tenuous and cold clouds dense?
   • Does this have any relation to phase diagrams in thermodynamics of physical systems?
3. Protostellar Collapse
   • What will happen to a massive cold molecular cloud?
   • What happens when the gravitational energy overcomes the thermal energy of a cloud? What is the collapse condition?
   • For a 1 Msun cloud at 10K, what is the minimum size for collapse?
   • What is the free-fall time for this cloud, and why is it a minimum time for the collapse?
   • How many scale factors (powers of ten) does a 0.2 pc cloud collapse to form a star like the Sun?
   • How fast should the protostar spin conserving angular momentum?
   • What sort of magnetic field should it have conserving magnetic flux?
   • How does a protostar get rid of magnetic fields and angular momentum?
   • What are molecular bi-polar outflows?
   • Why do protostars form with large disks of gas around them? How do we see these disks? Are these related to protoplanetary disks?
   • What are T-Tauri stars and why do they have strong magnetic field activity?
   • How does a collapsing protostellar cloud cool? Why might the first stars to form be massive, while low-mass stars form predominantly now?
   • What tracks in the H-R diagram to protostars take?

Star Formation in Outline:

1. Under Pressure: The Ideal Gas Law
   • We have established that clouds will collapse under gravity until something stops the contraction. Some sort of pressure will do so - magnetic, radiation, or thermal gas pressure can do this.
   • To determine the pressure of a gas at a given temperature and density, we use what is called the ideal gas law. It is called an "ideal gas" law because it assumes the gas atoms or molecules are perfect elastic little spheres (not the "ideal" gas law because the law itself is perfect and just!).
   • By considering how atoms at velocity bounce off the walls of a container transmitting momentum, and remembering the definition of temperature as average gas molecule velocity, and the definition of pressure as force per unit area, we get the relation:
     p = n k T
     where n is the number density of atoms/molecules (per m^3), and k=1.38 x 10^-23 J/K is Boltzmann's constant.
   • What does this formula mean? First note that if you compress a gas holding the temperature T constant, you find that the pressure is inversely proportional to the volume - it takes force (pressure) to compress a gas.
   • Also note that if you take a gas in a container at constant pressure, say sitting at sea level at standard atmospheric pressure (10^5 N/m^2 = 14.7 pounds/square inch), then the density is inversely proportional to the temperature (volume proportional to the temperature.
   • What happens under gravity? The question of how the pressure, density and temperature of a gas behaves under gravitational contraction is at the core of the star formation problem. Because it is not under constant density, pressure, or temperature it is more complicated than we are prepared to derive quantitatively. Therefore, I will describe qualitatively the picture we have for star formation.
   • If we have two clouds in pressure equilibrium, that is co-existing at the same pressure p, then they can live in harmony as the pressures are balanced. If they had different pressures, then the lower pressure region would be compressed by the higher outside pressure until it reached equality with the other pressure.
   • The gas law tells us that for the regions 1 and 2:
     n_1 T_1 = n_2 T_2
     This is why cold clouds are denser than hot clouds at the same pressure in the ISM.
2. Gravitational Heating
   • As clouds contract under gravity, some of the gravitational potential energy gets turned into kinetic energy of the gas, which in turn gets turned into heat.
   • This is exactly analogous to the calculation we made for the kinetic and potential energies in the atom, we just use the masses instead of charges in the potential:
     E = mv^2/2 - GMm/r

   • Thus, as the cloud contracts r gets smaller and thus the potential energy more highly negative, to keep the total energy E the same, the kinetic energy gets more highly positive.
   • About 1/2 of the heat is radiated away from the cloud at its surface, the rest raises the temperature in the center of the cloud.
   • The more massive the cloud is, the more potential energy it has, and thus the hotter it gets in the center.
3. Gravitational Stability
   • As a protostellar cloud collapses, it heats up from the conversion of gravitational potential energy into heat. As the gas heats up, and at the same time becomes more dense, the pressure increases: p = n k T
   • If a cloud is not dense enough for its given temperature, or in other words is not cold enough for its given density, then the pressure wins, and the cloud will not collapse. Such a cloud is said to be gravitationally stable.
   • For such a cloud to collapse, it must get rid of the heat, that is, it must cool.
   • Cooling is a consequence of the laws of thermodynamics, heat flows from hot to cold. There are three main ways to transport heat in a cloud, star, or any other system:
   • radiation - photons carry away the energy, like in the Sun and stars. Atoms emit photons, thus losing energy of the electrons as they drop to lower orbits.
   • convection - movements of gas elements carries heat, this occurs in the photosphere of the Sun, as well as in a cup of coffee or pan of water being heated on the stove.
   • conduction - certain materials (solid metal and electrons for example) carry heat directly. Like a metal spoon in a cup of hot soup: you can feel the heat being conducted to the handle of the spoon.
   • Cooling a cloud usually takes a long time, since they are not very dense and so do not radiate efficiently (or convect at all).
   • Another way to cause a cloud to collapse is to "shock" it with a pressure wave, like from a nearby supernova. This compresses the gas suddenly, overcoming pressure, and can start the collapse process.
   • Very massive clouds that are not too hot collapse on their own easily - gravity is just too strong for pressure to oppose it.
4. Angular Momentum
   • Angular momentum is a property of motion having to do with rotation about some point in space. It also has to do with the concept of "spin"
   • The angular momentum L of a body of mass m moving at velocity v perpendicular to the line from the center to the body, at distance r is:
     L = m v r

   • Angular momentum is conserved in systems where only gravity is operating (friction can destroy angular momentum).
   • It is conservation of angular momentum that causes a spinning skater to spin faster when they pull in their arms. As r gets smaller, then v must get larger to compensate. Similarly, planets move faster in their elliptical orbits when they are closer to the center of mass.
   • The rotation frequency f, in revolutions per second, is given by:
     f = v / 2Pi r

   • Thus:
     L = 2Pi m f r^2

   • Thus, as a cloud collapses, any rotation gets amplified by the factor: f1/f2 = (r2/r1)^2
   • A cloud will collapse from say 0.1 pc in diameter (3 x 10^12 km) to the size of the Sun (7 x 10^5 km), by a factor of 4 x 10^6. Thus, the cloud will spin up in rotation frequency by a factor of about 10^13! Even a slowly spinning cloud, as might be expected by random motions, will be amplified into a very fast rotation. This is not seen.
   • What in fact happens that as a cloud collapses, it rotates, and the gas friction causes it to form a spinning disk. These protostellar disks are seen around young stars, and may be precursors for the formation of planetary systems.
   • Binary stars can also form out of the spinning cloud, as this is a way of storing angular momentum.
5. Magnetic Fields
   • If a protostellar cloud has a magnetic field in it, as many are measured to have, then as the cloud collapses the magnetic field will be compressed and grow in strength.
   • The magnetic field, usually designated as B, will grow as the inverse of the volume of the cloud, that is, like the density of the gas: B2/B1 = (R1/R2)^3
   • The magnetic field exerts a pressure, and it turns out that the magnetic pressure is proportional to the square of the magnetic field, so: p2/p1 = (R1/R2)^6
   • Thus, as our cloud collapses by a factor of 4 x 10^6, the magnetic pressure increases by a factor of 4 x 10^39!!! Even tiny magnetic fields initially would grow tremendously and would overwhelm any other pressure.
   • Somehow, the magnetic field, along with excess angular momentum, must be gotten rid of during protostellar collapse.
   • Astrophysicists currently believe that this is done in a process called magnetic diffusion: the magnetic field, through the action of small amounts of ionized gas, will diffuse out of the spinning disk-like cloud, and will drag along gas removing angular momentum.
   • How ever it is accomplished, this is a crucial phase in star formation, and is an important area of study in astronomy.
6. Stages of Protostellar Collapse
   • Start with large cloud, say 0.1 pc in diameter, at around 100 K, containing 10 to 100 solar masses of gas.
   • At first, gravity is unimpeded and the collapse is free fall just like falling from the top of a building!
   • As the density grows and the cloud gets hotter, the pressure begins to slow the collapse. This is much slower than free-fall, and the outer gas is held up by the slowing inner gas core.
   • As collapse proceeds, the gravitational energy is turned into heat, about which 1/2 is radiated away from the cloud into space. The other 1/2 is able to heat up the core of the cloud.
   • The outer parts of the cloud spin-up to form a protostellar disk. This also slows the collapse since the protostar can only get new material through the inner parts of the spinning disk, which has lost angular momentum from gas friction and magnetic diffusion.
   • The protostar becomes hot enough to blow gas out from the "poles" perpendicular to the disk. The magnetic fields, which are twisted up by the spinning disk, also help spew gas out in bipolar jets, at speeds of 200 km/s or more!
   • Eventually the protostar gets hot enough to ignite nuclear fusion in the very center. It is now officially a star!
   • At some point the luminosity of the star is great enough to blow away most of the gas surrounding it, and the accretion stops. Very little of the original cloud mass makes it into the star itself. The star is now visible, since the gas and dust hiding it have been blown away.
   • Some of the protostellar disk remains. This might stay around long enough to form a planetary system!

Star Formation in our Galaxy

The Eagle Nebula M16, courtesy HST/STSCI

The Hubble Space Telescope has given us breathtaking views of the nebular sites of star formation on our galaxy. The gaseous pillars of the Eagle nebula shown above are spectacular. Similarly, the Great Nebula in Orion has a number of protostellar objects outlined againts the bright gas.

Also, see the spectacular Orion Mosaic MPEG Movie from HST!

Some examples:

• Young stellar objects showing accretion disks, outflows, winds and jets
• Gaseous pillars in the Eagle nebula
• The Orion Nebula Mosaic
• Proplyds in the Orion Nebula
• An explanation of the Orion region from Doug Johnstone (CITA)

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smyers@nrao.edu Steven T. Myers