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Newtonian Gravity

Gravitational Stability and Collapse

We see many gravitationally-stable systems in the sky, with the gravitational force within the system counterbalanced by the kinetic motion of the constituents: open star clusters, globular star cluster, galaxies, molecular clouds.  Within a star system, an equilibrium is established when the stars have an average kinetic energy equal to the gravitational potential energy.  Within a cloud of gas, equilibrium is established when the gravitational force is counterbalanced by gas pressure, which is effectively a balance between potential energy and thermal energy within the cloud.

This balance between gravitational potential energy and kinetic energy implies that only three parameters come into play in determining the stability of a group of stars or a cloud of gas: the mass density, the size, and the average velocity of the constituents.  The first two parameters determine the gravitational potential energy, and the third parameter determines the kinetic energy.  Because the mass of the constituent particles appears in both the gravitational potential energy term and the kinetic energy term, it drops out of the problem.  One finds that a stable gravitational system has a length scale that depends only on the  average mass density of the system and the average velocity of the constituents, whether they be stars or atoms.  This natural size is called the Jeans length.

One normally calculates the Jeans length for stars by assuming the stars are uniformly distributed throughout space with a particular average value for their random velocities.  This situation is not gravitationally stable.  Small fluctuations in density will cause the stars to cluster together.  One finds the Jeans length by imposing a sinusoidal density fluctuation on the stellar distribution and determining which wavelengths lead to the rapid growth of star clusters; the Jeans length defines the shortest wavelength that causes clustering.  The clusters that form at the Jeans length have a characteristic mass, called the Jeans mass, and they have a characteristic timescale for collapse, called the Jeans time.  These simple results for the evolution of a spatially-uniform distribution of stars provide an intuition of the scales for the clustering of stars that is accurate to within an order of magnitude.

The Jeans length for uniformly-distributed stars with mass density ρ and average random velocity V is

λJ = V(π/Gρ)1/2 = 27 pc (V/1 km s−1 )(ρ/1 solar mass cm−3)−1/2,

where G is the gravitational constant.  Taking as an example an average velocity of 0.5 km/s and a mass density of 10 solar masses per cubic parsec, which falls in line with the values associated with open clusters, one finds a Jeans length of about 4 parsecs, which is of order the size of an open cluster's core (The core of the Pleiades cluster has a 2 pc radius).  On the other hand, an average velocity of 15 km/s and a mass density of 0.08 solar masses per cubic parsec, which roughly describe the stars locally in the Galactic disk, gives a Jeans length of  1,400 parsecs, which is larger than the thickness of the Galactic disk.  This great length means that the stars in the Galactic disk will not form clusters, because the influence of the Galactic disk's differential rotation on the relative motions of the stars appears over a shorter length scale than the Jeans length.

The Jeans mass, which gives the characteristic mass of a cluster that forms from a uniform distribution of stars, is simply the mass density of stars times the volume enclosed by a sphere with a diameter equal to the Jeans length.  The Jeans mass is

MJ = 4πρ(λJ/2)3/3 = 1.0×104 solar masses (V/1 km s−1)3(ρ/1 solar mass pc−3)−1/2.

The Jeans mass for the density and velocity of an open cluster is of order 3,000 solar masses; this value is in line with more sophisticated observational estimates of open-cluster masses.

The timescale for the growth of a stellar cluster dependents only on the mass density of the stars in space; the average stellar velocity plays no role.  The Jeans time is

tJ = λJ/2πV = (4πGρ)−1/2 = 4.2×106 years (ρ/1 solar mass/pc3)−1/2.

Taking the mass density of 0.08 solar masses per cubic parsec of the local Galactic plane, one finds that the timescale for a cluster to grow is 15 million years.  Taking the density of 10 solar masses per cubic parsec as a typical value for an open cluster, one finds the cluster forms in 1 million years, which is very rapid compared to the 230 million years for the Sun to complete a single orbit of the Galactic center.

The stability of the interstellar medium against gravitational collapse is described by the same equations as the stability of a group of stars, but because we normally measure temperature rather than the average random velocity of gas particles, the velocity term in the Jeans length, which is the speed of sound in the gas, is expressed as a temperature relative to absolute zero divided by the mass of the gas particles, assuming that the gas is composed of only one chemical.  The Jeans length for a uniform cloud of gas is

RJ = (3πkT/Gρm)1/2 = 271 pc (T/100°K)1/2(m/1 H)−1/2(ρ/1 H cm−3)−1/2,

where k is the Boltzmann constant, T is the gas temperature, and m is the mass of the gas particles.  In the equation on the far right, the temperature is in degrees Kelvin, the mass is in units of the mass of hydrogen, and the density is in units of the mass of hydrogen per cubic centimeter.  Usually m is the mass of atomic hydrogen, but in a molecular cloud, it is the mass of molecular hydrogen (H2).

The warm regions of the galaxy, which have a characteristic temperature of several thousand degrees and a density of one atom per cubic centimeter, have a Jeans length of over 1 kpc; this value is about the scale height of the warm gas in the Galactic disk.  The Jeans length for cool regions, with their 100°K temperatures and 50 hydrogen atoms per cubic centimeter is around 40 parsecs.  This small size relative to the scale for the hot gas within the galactic plane permits the cooler regions to grow in size until they form gravitationally-bound clouds.  This small scale is also the source of the thickness of the Galactic disk, because the stars in the disk were born from the molecular clouds.  The cores of molecular clouds, which have temperatures of order 10°K and density greater than 105 hydrogen molecules per cubic centimeter, have a Jeans length of around 0.1 parsecs.

The Jeans mass for gas is given by

MJ = 2.5×105 solar masses (T/100°K)3/2(m/1 H)−3/2(ρ/1 H cm−3)−1/2.

The Jeans mass for  the cool interstellar medium is about 1 million solar masses.  This is the scale that should describe the molecular clouds.  The denser regions within molecular clouds given much smaller masses, with values of order 10 solar masses, which is of order the mass of the larger stars.

The Jeans time for the gravitational collapse of gas into a gravitationally-stable cloud is only dependent on the density of the material.  In terms of hydrogen atom masses per cubic parsec, the Jeans time is

tJ = 2.7×107 years (ρ/1 H cm−3)−1/2.

For the density typical of the warm regions of the Galaxy, the timescale is of order 27 million years.  The cool regions, with 50 hydrogen atoms per cubic centimeters, collapse over a timescale of 4 million years.  The cores of molecular clouds, with densities of order 105 molecules per cubic centimeter, have a Jeans time of order 85,000 years.

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