ASTR 498N                   Lecture 8                   The way forward

 

(online at www.astro.umd.edu/~drabin/)

 

Doroga k Zvjozdam Otkrita.

The way to the stars is open.

                Sergei Koroljov

 

 

The equations of stellar evolution

          (O&C §10.5)

 

We’ve already derived two of the four equations of static stellar structure, expressing mass conservation and hydrostatic equilibrium.  Although we still need to assemble more ingredientsconcerning the generation and transport of energylet’s write down a full set of evolution equations to get an overall sense of the enterprise at hand.

 

We’ll consider only spherically symmetric stars, in which all physical variables are constant on spherical shells of constant radius (or internal mass).  We also assume the star does not rotate.

 

What does it mean to specify stellar structure?

 

We will consider the instantaneous structure of a star to be specified if the following structural variables are known as functions of r or M :  

 

          Pressure                 P(r)    or      P(M)

          Density                  ρ(r)    or      ρ(M)

          Temperature          T(r)    or      T(M)

          Mass or radius       M(r)   or      r(M)

          Luminosity             L(r)    or      L(M)

          Composition          X(r)    or      X(M)

 

where X is an n-vector of element mass fractions with .  Since M and L are variables, we’ll need to use other symbols ( and ) for the total mass and luminosity of the star; for notational consistency, we’ll also use  for the total radius.  These choices are not universal.  For example, some authors use m or M_r in place of M, or L_r or  in place of L.

 

How many differential equations will be required?

 

There are n + 5 structural variables.   Of the n composition variables, only n1 are independent because they sum to unity.   We’ll show below that nuclear reaction rates determine n1 independent equations of the form , leaving five structural variables.  The pressure equation of state, P = P(ρ,T, X), provides one relationship these remaining variables.  We therefore expect to need four differential equations in addition to the composition equations.

 

 

The differential equations

 

Conservation of mass

 

                               

 

Conservation of momentum

 

                               

 

We earlier derived this in the form where the left hand side vanishes:  hydrostatic equilibrium.  If the pressure gradient does not exactly balance the gravitational force on the shell, the shell accelerates (Newton’s law).  Note that the time derivative of r = r(M) implies a Lagrangian rather than Eulerian description.

 

Conservation of energy

 

Write the first law of thermodynamics as , where δ is an infinitesimal Lagrangian change in a shell of mass dM, luminosity dL, and volume dV.  Substitute for the thermodynamic quantities to obtain

 

                               

 

Using  and , we can rewrite this as

 

                               

 

So, for an infinitesimal change over δt,

 

                               

 

For the static case (d/dt = 0), this just says that the luminosity dL added by shell dM is due to the energy generation rate per unit mass.  From the thermodynamic relation TdS = dE + pdV for quasistatic changes, the equation can also be expressed as

 

                               

 

where S is the entropy per unit mass.

 

Energy transport

 

In Lecture 5, we used dimensional arguments to show, using only the hydrostatic equilibrium equation and the perfect gas equation of state, that a star is much hotter at the center than it is near the surface.   From thermodynamics, we know that heat flows from higher to lower temperatures, the more so as the temperature gradient increases.  Thus, we expect that our fourth differential equation will relate the flow of energy to the temperature gradient:  dT/dr = f(L,…).  The form of this relationship will depend upon the dominant mode of energy transport.  We’ll find that when all the energy is transported by radiation,

 

                                   radiative transport

 

where  is a suitably frequency-averaged (Rosseland mean) opacity.  An equation of similar form can be used when conduction is important, as in white dwarfs.  This is not surprising, since optically thick radiative transport (photon “collisions”) and heat conduction (particle collisions) are both diffusive, random-walk processes.

 

However, we will also show that, when the temperature gradient exceeds a critical value known as the adiabatic gradient, the dominant mode of energy transport switches from radiation to convection (think of water coming to a boil when heated strongly from below).  Convection is so efficient that usually only a tiny “superadiabatic” temperature gradient is necessary to transport all the luminosity of a star, in which case, to an excellent approximation, the convective temperature gradient has the adiabatic value and

 

                                   convective transport

 

where Γ₂ (ρ,T, X) coincides, for an ideal gas, with γ ≡ C_p/C_v,  a ratio of specific heats.  For a monatomic ideal gas,  γ =5/3.  The adiabatic approximation to the convective temperature gradient is not good near the surface of a cool star, where a substantially superadiabatic gradient may be required to transport enough flux.

 

Only one of the two equations for dT/dr  is used at a given place and time.  If the gradient predicted by the radiative transport equation exceeds the adiabatic gradient, the convective equation applies.  The convective gradient is still related to luminosity in the sense that a larger gradient transports  more heat, but (dT/dr)_ad = f(L,…) is a very weak function of L (or, conversely, the heat flux is a very strong function of the superadiabatic gradient).

 

Composition changes

 

Recall that X_Z is the fraction by mass of element Z:  X_Z = m_Zn_Z/ρ.   It is most convenient to use the Lagrangian description, X_Z = X_Z (M,t), because then, if there is no particle diffusion between mass shells, only nuclear reactions can change X_Z.  Then

 

                               

 

Let r_jk give the rate, per unit volume and time, at which nuclear reactions transform nuclei of type j to type k.  The concentration of nucleus Z is increased by reactions r_jZ  that create Z and decreased by reactions r_Zk that destroy Z:

 

                               

 

where the sums are over all Z.  In practice, the number n of transmuting elements that play a significant role in influencing stellar structure at any one time is usually small.  However, if one is tracking elemental or isotopic abundances per se, n  may need to be large.

 

This expression of the composition equations is purely formal, in that we know nothing as yet about the reaction rates r_jk(ρ,T, X).  In fact, we won’t develop the composition equations furtherour interest in the reaction rates will center on their role in the energy equation.  However, it is important to understand that, although much can be learned from sequences of static stellar models (stellar structure), the composition equations are an essential ingredient of stellar evolution.

 

Timescales

 

We’ve previously introduced three important timescalesdynamical, thermal (Kelvin-Helmholtz), and nuclearthat are arranged in a hierarchy:

 

                               

 

In the Sun, for example, τ_dyn ~ 1 hr, τ_th ~ 10^6.5 y, τ_nuc ~ 10^10.5 y. 

 

If the star changes only on timescales much longer than τ_dyn, the momentum equation reduces to the hydrostatic (mechanical) equilibrium equation

 

                               

 

With the exception of pulsating stars, we will not consider situations in which the time-dependent momentum equation is required.

 

If the star changes only on timescales much longer than τ_th, the energy equation reduces to

 

                               

 

and the star is in thermal equilibrium.  The combination of mechanical and thermal equilibrium is sometimes known as complete equilibrium.  Unlike mechanical equilibrium, which is a good approximation for all but the most dynamic phases of stellar evolution, thermal equilibrium is not a good approximation during many evolutionary phasesfor example, during pre-main sequence evolution when the star derives energy from gravitational contraction rather than nuclear burning.

 

The nuclear timescale is quite longlonger than the estimated age of the solar system.  Therefore, we do not expect that the Sun is in nuclear equilibrium. 

 

 

The equations of static stellar structure

 

In the approximation of complete equilibrium, the evolution equations separate into two groups:  the structure equations, involving only spatial derivatives, and the composition equations, involving only time derivatives.  Thus, if X(M) is specified at some time t, the structure equations form a complete set of ordinary differential equations that determine the spatial structure of the star.  Here are the structure equations: