Ll Stellar energy sources

When material collapses to form a star, there is gravitational potential energy stored from the (negative) work done by gravity in bringing the material together. We might wonder how long this energy supply will last. We start by calculating the gravitational potential energy in a uniform sphere.

9.l.l Gravitational potential energy of a sphere

We are dealing with systems of large numbers of particles. Therefore, rather than thinking in terms of individual particles of mass m, we can think of a fluid of average density p. (The density is simply the mass per particle, multiplied by the number of particles per unit volume.) In this section, we will evaluate the potential energy for a uniform (constant density) sphere. Even though real objects might not be exactly uniform or spherical, the results will generally only change by numerical factors of order unity.

We begin by calculating the gravitational potential energy of a uniform sphere of mass M, radius R, and density p. These quantities are related by

The gravitational potential energy is the work required to bring all of the material from far away (infinity) to the final configuration. The final result does not depend on the order in which the various parts of the sphere are assembled, so we do the calculation in the easiest way that we can envision. We can think of the sphere as being made up of shells (Fig. 9.1). We can assemble the sphere one shell at a time, starting with the smallest.

Let's assume that we have already assembled shells through radius r. We now want to calculate the work done to bring in the next shell. The thickness of the shell is dr. The volume of the (thin) shell is its surface area multiplied by its thickness:

The mass contained in the shell is the volume multiplied by the density:

The total mass of material already assembled is

The quantity M(r) is important since the shell that ends up at radius r will only feel a net force from material inside it. Even after we bring in more material outside this shell, the net force exerted on any particle in the shell by any matter outside radius r is zero. Also, for mass in the shell, the mass M(r) exerts a force equal to that which would be exerted by the same mass all located at the center of the sphere.

We model stars by studying spherical shells.

For any two point masses, remember the gravitational potential energy (relative to infinity) is given by

9.1.2 Gravitational lifetime for a star

It is possible for stars to use their stored gravitational potential energy to power the star. In this section, we calculate how long this process can go on. For example, could the Sun be powered in this way even now? To see, we estimate the gravitational lifetime, tg. This lifetime is the stored energy divided by the rate at which the energy is being lost. The rate at which the energy is being lost is the luminosity. That is

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