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AST 353 Astrophysics

Lecture 2

Prof. Milos Milosavljevic, RLM 17.220,milos@astro.as.utexas.edu

The University of Texas at Austin

January 23, 2015

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Newtonian Gravitation in Astrophysics: Part II

Previous lecture (Part I):

Understand homogeneous isotropic expansion and collapse.Understand Hubble time, free fall time, dynamical time.

Goals for this lecture (Part II):

Understand gravitational potential energy.

Introduce virial equilibrium.

Goals for next lecture (Part III):

Derive the virial theorem.

Learn applications of the virial theorem in astronomy.

Understand the metaphor of negative specific heat.

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Gravitational Potential Energy of Point Masses

Given a pair of point masses m1 and m2 at locations r1 andr2, the

potential energy in their gravitational interaction is

E12= Gm1m2

r1 r2 (1)

You can obtain this by computing the (negative)1

work that isrequired to bring the second mass in from infinity to its finallocation atr2.

When we have more than two point masses, every mass interacts

gravitationally with every other mass. In the gravitational potentialenergy of the entire system, we must account for every pairofmasses.

1When work is negative, the agent performing work on the system derives

energy from the system.3 / 1 6

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The gravitational potential energy of a collection of point massesm1, ,mk at locationsr1, , rk, where k 2, is

E = k1i=1

kj=i+1

Gmimjri rj

(2)

Lets understand the two summation symbols. To account for everypair masses, we must sum over the first member of the pair, and

also over the second member that must be different from the firstmember. That would double countthe pairs, because it would, forexample, include the (i,j) = (3, 5) pair and also the (i,j) = (5, 3).The two sums in Equation (3) avoid double counting by ensuring

that only the pairs with i

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Gravitational Potential Energy of a Continuum

Continuum denotes the picture in which we do not observeindividual point masses (elementary particles). We keep track onlyof the composite properties of matter that vary smoothly frompoint to point in space.

In the continuum picture, a density function (r) describes thedistribution of mass in space. It is the limit in which the number ofpoint masses is large k , but each point mass is small mi 0,

while maintaining a given mass per unit volume at every location.

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Translating Equation (3) into the continuum picture gives

E = 1

2

G(r1)(r2)

r1 r2 d3r1d

3r2 (3)

Two integrals over all of space replace the two sums. In writingthe double integral, we have no easy way of preventing doublecounting. To compensate, the double integral is divided by 2.

This result is not very useful, but fortunately for us, stars, thesubject of this course, are approximately spherically symmetric (or

round). Why?We will explain the deep reasons for this, soplease be patient!

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Continuum with Spherical Symmetry

Assume that the spherical (or rotational) symmetry is around thecoordinate origin. Then the density depends only on the magnitudeof the radius vector, (r) =(r). We can also define the massenclosedwithin this radius

M(r) = r

0 (r

) 4r2

dr

(4)

Lets say we are assembling the mass gradually starting at theorigin. We have already brought in all the mass interior to radius r,and are just now bringing in a new, thin shell of matter with

thickness dr, volume dV = 4r2dr, and massdM=dV = 4r2(r)dr. By how much does the gravitationalpotential energy change as the new shell arrives?

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Since the shell is thin and the gravitational force it exerts on itselfis small, it is the gravitational force from the already assembledmass that is doing work. The gravitational acceleration at distancer is

g(r) = GM(r)

r2 r (5)

The work performed in bringing in the new shell is

dE=

r

g(r)dMdr = GM(r)dM

r

dr

r2 =

GM(r)dM

r(6)

where there were originally three different minus signs: one from

the path of the incoming shell being from larger to smaller radii,opposite of r, another one from flipping the limits of integration(, r) (r,), and a third minus sign from the gravitationalacceleration itself.

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The total gravitational potential energy is the combination of

those of all the thin shells that are being brought in, therefore

E =

dE

= GM(r)r dM

=

0

GM(r)

r 4r2(r)dr

= 4

0

GM(r)(r)rdr (7)

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E l 1

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Example 1

Take a uniform density sphere of mass Mand radius R. The

density equals

= M43 R

3 (8)

and the mass enclosed within radius R is

M(r) =r

R

3M (9)

Plugging these into Equation (7) and integrating we find that thegravitational potential energy equals

E = 3

5

GM2

R (10)

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Example 2

Now assume that the mass and radius are the same, but that thedensity inside is proportional the inverse radius squared

(r) 1r2

(11)

The density diverges as r , but the total mass is finite. Since

dM(r

)dr = 4r2(r) constant independent ofr (12)

the mass must rise linearly with radius. We can conclude that

M(r) = r

RM and (r) =

M

4r2R (13)

Substituting these in the expression for the gravitational potentialenergy we get

E= GM2

R

(14)

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A l i f th E l

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Analysis of the Examples

In both examples, the gravitational potential energy is

E GM2

R (15)

where the sign indicates that we are leaving out thedimensionless constant, whatever it is.

The dimensionless constant is 40% smaller in Example 1 withuniform density than in Example 2 with divergent density. This isbecause Example 2 is more concentrated, with larger mass residingat smaller radii, where its gravitational self-interaction is stronger.

If we observe a star with radius Rand measure its mass to be M,we can conclude that its gravitational potential energy is of theorder of the result in Equation (15), but to go further andconstruct a model if its internal structure that would tell us exactly

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I t f G it ti P t ti l E i A t h i

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Importance of Gravitation Potential Energy in Astrophysics

Since the work required to assemble a composite body isnegative, gravitational energy is converted into another,

positive form of energy, such as heat or radiation, in theassembly. The heat may enable nuclear fusion in the star. Theradiation may escape into space and heat interstellar clouds.The amount of positive energy produced can be estimated asminus the gravitational potential energy GM2/R.

Gravitational potential energy quantifies the degree to whichgravitational force binds mass together. Minus thegravitational potential energy is the amount of positive energy

that needs to be added to unbindthe mass. IfE GM2/Rinside a star, then a positive energy of GM2/Rneeds to beinjected into the star, e.g., by nuclear fusion, to unbind thestar (blow it up). The typical amount of energy required toblow up stellar size objects is 1051 g cm2 s2.

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E si C ll s d E ilib i

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Expansion, Collapse, and Equilibrium

We have learned about simple states that gravitating matter can

assume:Free (or Hubble) expansion.

Free fall.

In both of these, the density of matter evolves in time: it increases

in Hubble expansion and decreases in free fall.

Another simple state is possible

Virial equilibrium.

In virial equilibrium, the density remains constant in time. Clearly,something must counteract gravity to keep the system fromcollapsing. In virial equilibrium, it is the motion of the constituentbodies of the system that balances gravity.

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Star Cluster as an Example of Virial Equilibrium

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Star Cluster as an Example of Virial Equilibrium

Globular cluster M15, image by Haldan Cohn, WYIN, NOAO, NSF.

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What is Virial Equilibrium?

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What is Virial Equilibrium?

Equilibrium overall (or composite, macroscopic, coarsegrained, integrated, etc.) properties of the system remainconstant in time. The system always looks the same.Clearly, for equilibrium, a certain balance must be in place.

Virial the equilibrium reflects a balance of motion andgravity. Precisely, the total gravitational potential energy andtotal kinetic energy are related by the Virial Theorem.

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