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PH507 The Hydrostatic Equilibrium Equations for Stars

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  1. PH507 The Hydrostatic Equilibrium Equations for Stars Dr J. Miao

  2. We will cover the following topics: • Equation of Mass distribution • Equation of hydrostatic support - Virial Theorem -Centre pressure and mean temperature of a star -Different time scales for stars -How do we know we are right? 3. Equation of Energy generation 4. Equation of Energy transportation

  3. The Equations for stars How does a star exist? Internal pressure gradient Force of gravitation • Stars are spherical and symmetric about their centers • Stars are in hydrostatic equilibrium Two fundamental assumptions: Four equations of stellar structure 1. Equation of mass distribution

  4. 2. Equation of hydrostatic support The gravitational mass m(r) situated at the centre gives rise to an inward gravitational acceleration equal to : Newton’s second law:

  5. r0 m0 r 2.1 What can we know from the equation of hydrostatic support i) What will happen if there is no pressure gradient to oppose the gravity? Each spherical shell of matter converges on the centre free fall of the star The mass of the thin shell is M with an initial radius of r0 The mass included in the sphere of radius r0 is m0, When the thin shell collapses to the distance r: KE = GE It follows that the free fall time to the centre of the sphere is given by (See Appendix) For the sun, tff ~ 2000s In fact, collapse under gravity is never completely unopposed. During the process, released gravitational energy is usually dissipated into random thermal motion of the constituents, thereby creating a pressure which opposes further collapseThe internal pressure will rise and slow down the rate of collapse. The cloud will then approach hydrostatic equilibrium

  6. ii) What will happen if a star is in hydrostatic equilibrium state? an element of matter at a distance r from the centre will be in hydrostatic equilibrium if the pressure gradient at r is (d2r/dt2 =0 in eq.1.3) The whole system is in equilibrium if this equation is valid at all radii. * Eq. (1.4) implies that the pressure gradient must be negative, or in other words, pressure decreases from the inner central region to the outer region * The three quantities m, r,  are not independent 2.2 From Hydrostatic equilibrium equation to Virial Theorem If m is chosen as the independent space variable rather than r, Divide [4r2] into two sides of (1.4)

  7. Integrating the left-hand side of above by parts, the equation can be written: Which is gravitational potential energy of the star Using the symbol: Note: Vc=0, and dm = dV If the star were surrounded by a vacuum, its surface pressure would be zero This is the general, global form of the Virial Theorem and will be used very often later on. It relates the gravitational energy of a star to its thermal energy.

  8. 2.3 What can Virial theorem tell us for classical idea gas system ? the equation of state of a classical gas is known as the internal energy per unit mass is the system is stable and bound at all points This is also the Virial theorem in another form Because the total energy (binding energy) E =  + U =  -  / 2 =  / 2 ,  and E are always negative, if a star is in stable. Virial theorem tells that in a contracting gas system (protostar ), the energy for radiation is provided by half of the decreased gravitational potential energy: E =  / 2. When there is no energy from contraction, the radiation of a star is provided by thermonuclear reactions.

  9. 2.4 Estimate the minimum pressure at the centre of a star: Integrating eq.(1.5) from the centre to the surface of the star On the right-hand side we may replace r by the stellar radius to obtain a lower limit for the central pressure: i.e, 1/r > 1/R The pressure at the centre of the Sun exceeds 450 million atmospheres

  10. 2.5 Estimate the minimum mean temperature of a star: We can use this result to estimate the average internal temperature of a star In the gravitational potential energy expression, r is less than R everywhere  between two stars of the same mass, the denser one is also hotter. For the Sun, Eq. (1.9) gives us T > 4 106 K if the gas is assumed to be atomic hydrogen

  11. 2.6 Estimate the importance of the radiation pressure: the corresponding expression for radiation pressure is with T =4 106 K and ( =1.4 103 kgm-3, a =7.5510-16 Jm-3K-4), We have: Therefore it certainly appears that radiation pressure is unimportant at an average point in the Sun! This is not true of all stars, however. We shall see later that radiation pressure is of importance in some stars, and some stars are much denser than the Sun and hence correction to the idea gas are very important.

  12. From Suppose: i.e: 2.7 How accurate is the Hydrostatic Assumption? If a mass element starts from rest with this acceleration, its inward displacement s after a time t : if we allow the element to fall all the way to the centre of the star, we can replace s in the above equation by r and then substitute The time t is that it would take a star to collapse if the forces are out of balance by a factor  But fossil and geological records indicate that the properties of the Sun have not changed significantly for at least 109 years(31016s) If t > 3*1016s   < (tff / t)2 < 10-27 most stars are like the sun and so we may conclude that: the equation of hydrostatic equilibrium must be true to a very high degree of accuracy !

  13. 2. 8 How valid is the spherical symmetry assumption? Departure from spherical symmetry may be caused by rotation of the star.  ~ 2  10-5  of the Sun is about 2.5 10 -6 Departures from spherical symmetry due to rotation can be neglected. This statement is true for the vast majority of stars. There are some stars which rotate much more rapidly than the Sun. For these stars, the rotation-distorted shape of the star must be accounted for in the equations of stellar structure. r (r, , )

  14. 3. Energy generation in stars 3.1 Gravitational potential energy It is a likely source of the stellar energy and has the form The total energy of the system : Assuming a constant density distribution the gravitational potential energy: the total mechanical energy of the star is: What can this tell us?

  15. Assuming that the Sun were originally much larger than it is today how much energy would have been liberated in its gravitational collapse? If its original radius was Ri, where Ri >> R, then the energy radiated away during collapse would be Further assuming Lsun is a constant throughout its lifetime, then it would emit energy at that rate for approximately Is the Kelvin-Helmholz time scale We have already noted that fossil and geological records indicate that the properties of the Sun have not changed significantly for at least 109 years (3 × 1016 s) But the Sun has actually lost energy: L * 3 * 1016 = 1.2 × 1043 J Gravitational potential energy alone cannot account for the Sun’s luminosity throughout its lifetime !

  16. 3.2 Nuclear reaction the total energy equivalent of the mass of the Sun, . If all this energy could be converted to radiation, the Sun could continue shining at its present rate for as long as is called nuclear timescale The Sun just have consumed its mass: Hence, for most stars at most stages in their evolution, the following inequalities are true td << tth << tn. (1.9)

  17. r r+dr 3.3 How do we include the energy source? Define luminosity L (r) as the energy flow across any sphere of radius r. The change in L across the shell dr is provided by the energy generated in the shell: where  (r) is the density;  (r) is the energy production rate per unit mass This is usually called the energy-generation equation The energy generation rate depends on the physical conditions of the material at the given radius.

  18. T+T P+P + r+r T+T, P+P, + T, P,  r T, P,  4. How is energy generated transported from center to outside? 4. 1 Convection. Energy transport by conduction (and radiation ) occurs whenever a temperature gradient is maintained in any body But convection is the mass motion of elements of gas, only occurs when. Consider a convective element of stellar material a distance r from the centre of the star define P, as the change in pressure and density of the element P, , as the change in pressure and density of the surroundings

  19. If the blob is less dense than its surroundings at r+r then it will keep on rising and the gas is said to be convectively unstable. The condition for this instability is therefore: Whether or not this condition is satisfied depends on two things: a) the rate at which the element expands (and hence decreases in density) due to the decreasing pressure exerted on it b) the rate at which the density of the surroundings decreases with height. We can make two assumptions about the motion of the element • The element rises adiabatically, i.e. it moves fast enough to ensure that there is no exchange of heat with its surroundings; PV=constant 2. The element rises with a speed < the speed of sound.. This means that, during the motion, sound waves have plenty of time to smooth out the pressure differences between the element and its surroundings and hence P = P at all times

  20. By using P/ =constant By the second assumptionP = P : For an ideal gas in which radiation pressure is negligible, we have: P = kT / m,  log P = log+ log T + constant. This can be differentiated to give relationship of the changes between P, T and  of the surrounding: Substitute (1.12) and (1.13) into (1.11) The critical temperature gradient for convection is given by

  21. Note that the temperature and the pressure gradients are both negative in this equation, we can use modulus sigh to express their magnitudes: Eq.(1.15) can also be written as: Convection will occur if temperature gradient exceeds a certain multiple of the pressure gradient. The criterion for convection derived above can be satisfied in two ways : a) The ratio of specific heats, , is close to unity In the cool outer layers of a star, the gas is only partially ionized, much of the heat used to raise the temperature of the gas goes into ionization and hence cv and cp are nearly same ~1. A star can have an outer convective layer b) the temperature gradient is very steep a large amount of energy is released in a small volume at the centre of a star, it may require a large temperature gradient to carry the energy away  A star can have convective core.

  22. 4.2 Conduction and radiation Conduction and radiation are similar processes because they both involve the transfer of energy by direct interaction, the flux of energy flow Which of the two - conduction and radiation - is the more dominant in stars to transport energy? particles photons Energy: ~ Number density: nparti nphoton > mean free path: parti~ 10 -10 m photon~ 10-2m << Photons can walk more easily from a point where the temperature is high to one where it is significantly lower before colliding and transferring energy, resulting in a higher transport of energy.

  23. Conduction is therefore negligible in nearly all main sequence stars and radiation is the dominant energy transport mechanism over conduction in most stars. 4.3 Equation of Radiative transport If we assume for the moment that the conditions for the occurrence of convection is not satisfied we can write down the fourth equation of stellar structure, The energy carried by radiation in the flux Frad, can be expressed in terms of the dT/dr and a coefficient of radiative conductivity, rad, where the minus sign indicates that heat flows down the temperature gradient. The radiative conductivity measures the readiness of heat to flow

  24. Astronomers generally prefer to work with an inverse of the conductivity, known as the opacity, which measures the resistance of the material to the flow of heat. Detailed arguments (see Appendix 2 of Taylor’s book ‘stellar evolutions’) show that the opacity where a is the radiation density constant and c is the speed of light Combining the above equations we obtain: Recalling that flux and luminosity are related by the equation of radiative transport

  25. It is the temperature gradient that would arise in a star if all the energy were transported by radiation It should be noted that the above equation also holds if a significant fraction of energy transport is due to conduction, but in this case, Lr Lr + Lcond. Then (1.22) can be written as Clearly, the flow of energy by radiation/conduction can only be determined if an expression for is available.

  26. 4. 4 Radiation of Neutrinos In massive stars late in their lives, the amount of energy that must be transported is sometimes larger than either radiation of photons or convection can account for . In these cases, significant amounts of energy may be transported from the center to space by the radiation of neutrinos. This is the dominant method of cooling of stars in advanced burning stages, which also plays a central role in events like supernovae associated with the death of massive stars.

  27. Summary: 1. Based on two fundamental assumptions: we derived the four equations of stellar structure There are four primary variables M(r ), P(r), L(r ), T(r ) in these equations, all as a function of radius We also have three auxiliary equationsP: equation of state, P=P(,T, Xi): opacity (,T,Xi): nuclear fusion rate, (,T,Xi). These are three key pieces of physics and we will discuss them in detail

  28. 2. From the most important hydrostatic equilibrium equation: -- Drive the global form of Viral theorem. With the gravitational potential energy of a stat If the density of the system is a constant, Drive another form of Viral theorem: Which tells us: a star in hydrostatic equilibrium is stable and bound at all points - E = U = -  / 2. .–only half of the released potential energy can be used as radiation during the collapse process inside a star! -- estimate the minimum center pressure in a star : -- estimate the minimum mean temperature of a star: 3. Criteria for convection: 4. Three important time scale: td << tth << tn. (1.9)

  29. 5. Show that the radiation pressure is not important in Sun-like stars 6. Radiation is more efficient way to transport energy from place to place than conduction

  30. Appendix: This may be simplified by introducing the parameter to give