Astronomy/Star Clusters & Gravity
QUESTION: Hi Philip,
Why doesn't the force of gravity cause open & globular star clusters to collapse to a central point? When I read about star clusters, it seems just the opposite happens, with the stars becoming "gravitationally detached" or the stars "dispersing."
The short answer to that (I will also give a longer, detailed answer) is that: a) gravity is too weak to do that for a system of objects in a system stretching over 4-5 parsecs or more, and b) distance considerations weaken gravity even more star-to-star - since F (force of gravitational attractions) diminishes as the inverse square of the distance.
We actually have two energies in competition in such bound stellar systems, the gravitational potential energy (call it φ)and the kinetic energy (call it T).
For any spherical stellar system, i.e. open cluster, where T is the kinetic energy
T = ½ S i mi[ui’ + vi’ + wi’]^2
Where S denotes 'sigma (summation) and mi denotes the masses belonging to the cluster, and u’, v’, w’ are the velocities of the member stars relative to the center of mass.
For equilibrium or stability of the cluster, we have from the virial theorem, see e.g.
2T + φ = 0
That is, twice the kinetic energy plus the gravitational potential energy = 0
The binding energy for holding the stars in a cluster is:
E(S) = T + φ
Combining the above with:
2T + φ = 0
2T = -φ or T = -φ / 2
E(S) = - φ/2 + φ = φ/2
I.e. the binding energy (holding the cluster together) is equal one half the gravitational potential energy
Given a cluster of N stars, each of mass m (say equal to 1 solar mass), moving at an average velocity V across a star cluster of radius R (say 5 parsecs) we get a total mass M = Nm, and total kinetic energy T = ½ nM V2 and a gravitational potential energy:
φ = G(Nm)^2 / R
Using the virial theorem form of T = -φ / 2 we can solve for V:
½ nM V^2 = G(Nm)^2 / 2 R
Or: nM V^2 = G(Nm)^2 / R
V = [G(Nm) / R]^½
From this it is possible to find what is called the “time of relaxation” which will mark that time by which the member stars will have effectively been dispersed.
The problem then for star clusters, is not that all the stars of a star cluster will collapse toward one central point or into one mass, but rather that over time – and on account of star to star ‘interactions’ within which momentum, energy is exchanged, that V will become such as to allow escape of enough stars to disperse the cluster.
For open clusters and globular clusters the time of relxation is about the same, roughly 100 million years – assuming 10,000 stars per cubic parsec for the globulars, and 10 per cubic parsec for the open clusters, and average velocities of 10 km/s (globular) and 1 km/s (open).
Hope this helps, and apologies if I got carried away!
---------- FOLLOW-UP ----------
QUESTION: Hi again,
I am sorry to report that your longer, detailed answer—for the most part—went over the head of this layman. Your short answer makes more sense to me. I think I underestimated the vast distances between the stars in a star cluster. Am I correct in thinking that the situation is somewhat analogous to the earth’s not falling into the sun because of the earth’s distance from the sun combined with the earth’s velocity and orbit around the sun?
P.S. Your John Archibald Wheeler quote at the end of your reply has me wondering where the observer was located when the phenomenon of earth’s origin occurred, the earth’s formation through the accretion of bits of matter orbiting the early sun.
Yes, sorry about the longer -detailed response. What I was trying to get at is that we are dealing with a complex, gravitationally -bound system of hundreds or thousands of stars. Thus, while in a first simplistic format one might use the Earth- Sun analogy (as you suggested) it is actually far too simple to apply to either open star clusters, or globular clusters. But I suppose as a first (primitive!) approximation it can do once one realizes that: a) Earth is not a star, and b) in actual star clusters hundreds of other gravitational interactions, 'encounters' -kinetic-dynamic interactions must be considered!
But the simpler version I provided is enough to grasp at a layman's level, specifically that gravity is not the fearsome force manifested, i.e. when a collapsar occurs from a supernova. The dispersal of gravitationally interacting objects over parsec-scale distances prevents that - given gravity is an inverse square force as well.
Re: The Wheeler quote, he intended it mainly for 'here and now' observations and phenomena, as they are linked to claims. In terms of Earth's formation, say via actual observation, clearly no one could have been around then, but this doesn't mean accretion was any less real. Or that accretion isn't a real phenomenon. In this case, we can use highly refined computer models to see how accretion operates with a mass such as the early solar system-primeval Sun would have, and subject to different rates of transfer for angular momentum (say for the rotating primitive Sun outward to the surrounding solar disk-nebula).
Since the computer models work, and show such accretion, we can accept them.....unless maybe we disagree and insist that computer models aren't 'real' or can't depict real scenarios.