**The Birth of Protostars**

Protostars are formed in star nurseries called nebulas, while nebulas are areas of higher dust and gas densities relative to the surrounding interstellar space. ^{[1]} Nebulas are made up of mostly hydrogen and a little helium, although more massive elements and even molecules are also ubiquitous. These more massive elements and molecules are from previous stars that have died and scattered some of their remnants into the nebula. These clouds of gas and dust can span hundreds of light years across and can be formed when enough gas and dust come together to become gravitationally bound or can be compressed by the explosion of stars known as supernova that also add their gases to the cosmos as planetary nebulas. ^{[2]} If the nebula has a high density, distubances of gas and dust may cause gravitational contraction to become significant. If gravity becomes significant enough to pull in the dust and gas from the cloud then the material trapped by gravity will collapse into itself, this is how a protostar is formed. ^{[3]} When gravity takes over and causes the material of the nebula to collapse in on itself this forms a sort of ball that will start to rotate. This rotation will cause gas and dust outside of the ball to start to rotate towards the ball, similar to when the drain of a tub is opened and all the rubber ducks at the other side of the tub start to move towards the drain getting caught in the whirlpool and finally sucked down the drain. This is how a protostar increases its mass; a protostar will start out as the small ball that begins to rotate (this can be thought of as the drain) as this ball rotates it will create an accretion disk (the whirlpool around the drain) this disk will suck dust and gas from the surrounding nebula and transfer it to the protostar. How fast this process happens can help determine the outcome of the new star. This process will stop when the protostar starts nuclear fusion of hydrogen. During the formation of the protostar and the accretion process the protostar becomes hotter and denser. The protostar becomes denser because the accretion disk is adding material to the star which is causing the gravity of the protostar to increase, thereby “pushing” the gas and dust from the accretion disk closer and closer towards the center of the protostar. This effect is correlated to the temperature of the protostar and as the density of the star increases so does the temperature. When the temperature at the center of the protostar reaches about 10^6 Celsius it will start to fuse hydrogen. ^{[1]} This is the start of the proton proton fusion chain that is the main fusion that supports the star and signifies the birth of a new star. The burning of hydrogen produces a solar wave that will blow the accretion disk away from the star allowing no new material to be added beyond that time.^{[3]}

**Simple Model of Protostar Formation**

The above is a description of the formation of new stars. We will now look at the early physics used to describe protostar creation. The first person to study protostars was Sir James Jeans; he studied globules and molecular clouds where protostar formation has been observed. Sir Jeans studied what conditions are needed in a molecular cloud or globule to induce collapse of material to form a protostar. During Jeans life time (1877-1946) the advanced computational powers of computers were not available so he had to simplify his calculations. The major simplifications Jeans made before he began his analysis was to assume that the effects of rotation, turbulence, and magnetic fields can be neglected. These assumptions are not true but Jeans calculations give a good starting point.

Jeans started with the Virial Theorem, 2K+U=0 equation 1

This theorem states that the total potential energy (U) of gravity is twice the absolute value of the total kinetic energy (K) of the system and when the two add to zero then the system is in equilibrium and the cloud will neither collapse nor dissipate. If the kinetic energy of the cloud is more than half of the potential energy then the cloud will expand and dissipate. If the kinetic energy is less than half its potential energy then the cloud will collapse into a protostar. We can write the potential energy as,

U= -(3/5)(GM2/R) equation 2

And kinetic energy can be written as,

K= (3MkT)/(μm_{H}) equation 3

where,

μ= mean molecular weight

T= temperature

M= mass of cloud

m_{H}= is mass of hydrogen

R= the radius of the cloud

G and k= gravitational constant and Boltzmann’s constant

rewriting R as,

R= [(3M)/(4πρ)]^{1/3} equation 4

ρ= the initial mass density of the cloud is assumed to be constant throughout the cloud

Sir Jeans then substituted the R equation into the Potential energy equation and then put both energy equations into the virial theorem and solving for the mass he found the minimum mass required for a cloud to collapse. This is called the Jeans mass and he found the minimum radius by placing the equation for R into the Jeans mass equation and solving this is called the Jeans length. If the cloud has a mass or radius larger than the values found by these equations then the cloud will collapse.

M_{J}= [(5kT)/(Gμm_{H})]3/2[3/(4πρ)]1/2 equation 5

R_{J}= [(15kT)/(4πGμm_{H}ρ)]1/2 equation 6

This was the first theoretical attempt to model the formation of a protostar. These equations give us good approximations of which molecular clouds will be able to form protostars. But observations of forming protostars and of molecular clouds show that the equations that Sir Jeans developed are not always accurate.^{[4]}>

**Constraining factors to Sir James Jeans Model**

The preceding section explained Sir Jeans’s model for the formation of protostars from the surrounding molecular clouds. Observations of molecular clouds and protostars have shown that this model is flawed. The model predicts that the entire cloud will collapse into the forming protostar; also the model predicts that if the mass or radius is higher than the Jeans mass or Jeans radius, than the cloud will collapse and form protostars. Astronomers have found molecular clouds and globules that do not follow this model very well. Observations have been made on globules and molecular clouds that have many stars forming in them and on others that are above the Jeans mass or Jeans radius and do not have a lot of protostar activity in them. Many astronomers have tried to determine what is wrong with the model and have found reasons to explain the observations made that contradict the model. One reason found was that the simplifications made by Sir Jeans could not be left out and that by including some of these previously excluded variables astronomers found that the model fit more closely to what they observed. Some of the variables that were excluded in Jeans model were cloud rotation, the presence of a magnetic field, temperature changes, mass density changes, external gas pressure, and fragmentation.^{[4]}

**Example**

Let us look at a diffused hydrogen cloud. Assume that the temperature is 50K and that the cloud is completely hydrogen with a density of 8.4x10^{-19} kg/m^{3}, and take µ to be 1. What then is the minimum mass necessary to cause the cloud to collapse? Using equation 5 from above with the given values we find that the mass necessary for collapse is roughly 1500 solar masses. The normal diffused hydrogen cloud ranges in mass from 1-100 solar masses therefore such clouds are stable since the Jeans Mass calculated above is greater than the mass of such clouds.^{[4]}

Now let’s look at what happens in the center of a dense giant molecular cloud (GMC). The typical temperatures for this cloud is 10K and we will take the density to be 3x10^-17kg/m^{3} and take µ to be 2. Again using equation 5 we find that the Jeans Mass is now only 8 solar masses. GMCs are approximately 10 solar masses. We can now reason that GMC cores are unstable. Consequently they will form stars because the Jeans Mass is lower than the mass of the cloud. This has been proven by astronomers through observations of GMCs in our night sky.^{[4]}

### ReferencesEdit

- ↑
^{a}^{b}Think Quest. Life cycle of stars., 2009, from http://library.thinkquest.org/17940/texts/star/star.html. - ↑ Bill Arnett. (1997). Types of nebulae., 2009, from http://astro.nineplanets.org/twn/types.html.
- ↑
^{a}^{b}Linda Hermans-Killam. The infrared universe., 2009, from http://coolcosmos.ipac.caltech.edu/cosmic_classroom/ir_tutorial/sform.html - ↑
^{a}^{b}Ostlie, D. A., & Carroll, B. W. (2007). In Black A. R. S. (Ed.), An introduction to modern stellar astrophysics (2nd ed.)San Francisco: Addison-Wesley.