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Unformatted text preview: Chapter 3 Stellar Interiors Our observations of stars are only skin-deep. The mass of the Sun’s pho- tosphere, chromosphere, and corona (the portions of the Sun we can see directly) only adds up to 10- 10 of the Sun’s total mass. We are not entirely ignorant of the 99.99999999% of the Sun that is opaque, however. Because the structure of the Sun, and other stars, is dictated by well-understood laws of physics, we can make models of stellar interiors, using the observed surface properties of stars as our boundary conditions. 3.1 Equations of Stellar Structure The internal structure of a spherical star in equilibrium is dictated by a few basic equations of stellar structure . The first equation of stellar structure is the familiar equation of hydrostatic equilibrium : dP dr =- GM ( r ) ρ ( r ) r 2 . (3.1) Make note of the assumptions that have gone into this equation: the star is spherical and non-rotating, the star is neither expanding nor contracting, and gravity and pressure gradients provide the only forces. Equation (3.1) is a single equation with three unknowns – P ( r ), M ( r ), and ρ ( r ) – so even with known boundary conditions, we can’t solve it to find a unique solution for the pressure and density inside the star. However, we can still extract interesting information from the equation of hydrostatic equilibrium. For instance, we can make a very crude estimate of the central pressure of the Sun. 51 52 CHAPTER 3. STELLAR INTERIORS A rough approximation to the equation of hydrostatic equilibrium is Δ P Δ r ∼ - G h M ih ρ i h r i 2 , (3.2) where Δ P is the difference in pressure between the Sun’s photosphere and its center, Δ r is the difference in radius between the Sun’s photosphere and its center, and h M i , h ρ i , and h r i are typical values of mass, density, and radius in the Sun’s interior. As a rough guess, we can set h ρ i ∼ ρ fl = 1400 kg m- 3 , the average density of the Sun. We can also guess that h M i ∼ M fl / 2 = 1 . × 10 30 kg and h r i ∼ r fl / 2 ∼ 3 . 5 × 10 8 m. The pressure at the photosphere will be much less than the central pressure, so we can rewrite equation (3.2) as- P c r fl- ∼ - G ( M fl / 2) ρ fl ( r fl / 2) 2 ∼ - 2 GM fl ρ fl r 2 fl . (3.3) With the numerical values of M fl , ρ fl , and r fl inserted, we find that P c ∼ 2 GM fl ρ fl r fl ∼ 5 × 10 14 N m- 2 ∼ 5 × 10 9 atm . (3.4) When we compare this to the pressure P phot ∼ 10- 3 atm in the Sun’s pho- tosphere (as computed in section 2.3), we see that the center of a star is a high-pressure place. The second equation of stellar structure is the equation of mass con- tinuity : dM dr = 4 πr 2 ρ ( r ) . (3.5) This simply tells us that the total mass of a spherical star is the sum of the masses of the infinitesimally thin spherical shells of which it is made. It tells us the relation between M ( r ), the mass enclosed within a radius r , and ρ ( r ), the local mass density at r . Combining equations (3.1) and (3.5) gives usCombining equations (3....
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This note was uploaded on 07/17/2008 for the course ASTRO 292 taught by Professor Ryden during the Winter '06 term at Ohio State.
- Winter '06