Lecture 26 - Protostars and Stellar Energy (3/22/96)

Seeds: Chapter 9

  1. 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.
  2. 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.
  3. 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.
  4. 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!
  5. Nuclear Energy in Stars
    • We showed in previous lectures that chemical energy was insufficient to power the Sun. Even gravitational energy, which could provide the Sun's luminosity by shrinking by 40 meters every year, could only do so for millions of years, not the billions that the Sun has lasted.
    • Since the gravitational and the electromagnetic forces are insufficient to provide the energy of the Sun, we turn to the remaining forces: the strong and weak nuclear forces. In particular, it is the strong nuclear force that can provide us this energy.
    • The strong nuclear force is the force that holds protons and neutrons together in the nucleus of the atom, overcoming the mutual electrical repulsion of the protons. The strong force is carried by particles called gluons, which hold together the quarks that make up protons and neutrons. (Electrons are not made up of quarks - they are a different sort of particle called a lepton.)
    • The weak nuclear force is a force that changes the kind of quark or lepton. The weak force is the kind of force that can change a proton into a neutron, and visa versa, for example.
    • Just like we can emit a photon (energy) by moving an electron into a more tightly bound orbit, we can liberate nuclear energy in the form of photons and other particles by making the nucleus more tightly bound.
    • It turns out that for elements ranging in mass from hydrogen to iron, the nuclei are more tightly bound as you increase the mass. For elements heavier than iron, the nuclei are less tightly bound the more massive they are.
    • This means the for elements lighter than iron, you can release energy by the fusion of more than one together to make a heavier nucleus.
    • Similarly, for elements heavier than iron, you can release energy by causing fission of a heavier nucleus into more than one lighter fragments.
    • The way to measure the energy released in the fusion or fission of nuclei is to measure the mass of the nuclei before and after the reaction.
    • Einstein's equation relates the mass and the equivalent energy:
      E = m c^2
    • If you could convert all of a proton's mass (1.673 x 10^-27 kg) into energy, you would get (1.67 x 10^-27 kg)(3 x 10^8 m/s)^2 = 1.506 x 10^-10 Joules. In more familiar units, this is equal to 938 x 10^6 eV or 938 MeV(millon-electron-volts). There is a tremedous amount of energy available in the mass of particles.
    • For comparison, the mass of the electron gives 511 x 10^3 eV of energy, or 511 keV (kilo-electron-volts).
    • Often the masses of particles are given in energy units of mc^2, so the proton mass is 938 MeV, and the electron mass is 511 keV. Note that you can also reverse the process, and given 938 MeV of energy you can create a proton! (It turns out you can't just create a proton, you have to create an anti-proton also. We will discuss this later on).
    • Thus, if you take four hydrogen nuclei (protons) and convert them into one helium nucleus (2 proton, 2 neutrons), you find that the sum of the 4 proton masses is 0.7% larger than the mass of 1 helium nucleus.
    • This difference in mass comes out in energy (photons and particles). The amount of energy is calculated by E = m c^2, where m is the mass difference, in this case 4.8 x 10^-29 kg which gives 27 MeV of energy. Compare this with the 50 eV or so of electromagnetic energy available from the ionization energy of helium.
    • It is hard to get the repulsive protons close enough together to let the strong force take over. You need very high temperatures (T > 10^7 K) in order for strong collisions to cause these reactions to happen. This is why this occurs in the very cores of stars that are massive enough to be that hot!
    • The reaction that supplies the energy for the Sun is:
      4 H -> He
    • The main way of doing this is through the proton-proton reaction. This occurs in 3 stages:
      H + H -> D + e+ + v (x2 , 1.4 x 10^10 yrs)
      H + D -> He3 + photon (x2 , 6 seconds)
      H + He3 -> He + H + H (x1 , 10^6 yrs)
      The funny things in the first reaction are a positron (e+), which is the anti-electron and is just like an electron but positively charged, and the neutrino (v), which is a very light particle. Note that the first reaction is very slow to happen, taking over 10 billion years on average. This is because it turns a proton into a neutron (D is deuterium a kind of hydrogen consisting of a proton and a neutron bound together) using the weak force (hence the positron and neutrino). The other reactions only use the strong force, and are easier.
    • This reaction provides 90% of the Sun's energy. It requires temperatures above 10 million K (10^7 K).

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Steven T. Myers - Last revised 27Mar96