The Sun as a typical star: Central density, temperature, pressure The spectrum of the surface (atmosphere) of the Sun The structure of the sun s outer layers: convection, rotation, magnetism and sunspots Today in Astronomy 38 Figure: Solar eruptive prominence, seen in He II 30.4 nm from the EIT instrument on the NASA/ESA SOHO satellite (NASA/GSFC) 11 September 007 Astronomy 38, Fall 007 1
The Sun s interior, on the average Since we know its distance (from radar), mass (from Earth s orbital period) and radius (from angular size and known distance): 13 r = 1 AU = 1.4960 10 cm M R we know the average mass density (mass per unit volume): M 3M -3 ρ = = = 1.41 g cm V 3 4π R 10 33 = 1.98843 10 gm = 6.9599 10 cm = only 6% of Earth's. 11 September 007 Astronomy 38, Fall 007
The Sun s interior, on the average (continued) For the average pressure, we use the equation of hydrostatic equilibrium, and assume (very crudely) that the gas pressure varies linearly from some central value to zero at the surface: Then dp P Psurface Pcenter P = = dr r rsurface rcenter R GMρ GM ρ = = r P R Pcenter GMρ = = R ( 8)( 33 ) center 6.67 10 1.99 10 1.41 - = dyne cm 10 6.96 10 15-9 = 1.34 10 dyne cm = 1.3 10 atmospheres 11 September 007 Astronomy 38, Fall 007 3
The Sun s interior, on the average (continued) The Sun is an ideal gas: PV = NkT (N= number of gas particles, n= kt P = nkt = ρ number density: number of particles m per unit volume, m= avg. particle mass) Average particle mass is about the mass of the proton 4 ( 1.67 10 gm), so ( 4 )( 15 1.67 10 1.34 10 ) mp T = = K ρ k 16 1.41 1.38 10 ( )( ) Now for some less-crude estimates. 6 =11.5 10 K. 11 September 007 Astronomy 38, Fall 007 4
Central pressure in a star The star s center, being its densest and hottest spot, will turn out to be the site of virtually all of the star s energy generation, so we will make somewhat more careful estimates of the conditions there. From hydrostatic equilibrium equation again: R dp RGM ( r ) ρ ( r ) P( R) P( r) = dr = dr. r dr r r But the pressure at the surface, P(R), better be zero because the star s surface doesn t move and there s nothing outside the surface to push back, so R GM ( r ) ρ ( r ) P( r) = dr. r r 11 September 007 Astronomy 38, Fall 007 5
Central pressure in a star (continued) We can t do the integral unless we know the density as a function of position, so instead we make a crude approximation: ρ M V M R 3 6 r (ignoring dimensionless factors like 4π/3, because we re just trying to get the order of magnitude right), and the integral becomes 3 GM R M( r ) GM R 1 r P( r) dr M dr 3 r 3 r 3 R r R r R GM R R rdr 11 September 007 Astronomy 38, Fall 007 6
Central pressure in a star (continued) Central pressure (r = 0): P C GM R rdr = 6 R 0 GM R 4 z GM R Lo and behold, a complete calculation for stars of moderate to low mass (Astronomy 553 style) yields GM PC = 19 4 R so we have derived a pretty good scaling relation for P C. 6 R (still ignoring dimensionless factors) 11 September 007 Astronomy 38, Fall 007 7
For the Sun: Central pressure in a star (continued) M 33 10 R = 1.99 10 g = 6.96 10 cm P C 11 4 GM 19 =.1 10 dyne cm R > 10 atmospheres 17 - So, for other main sequence stars, to adequate approximation, P C 4 17 M R - 4 4 GM 19 =.1 10 dyne cm. R M R 11 September 007 Astronomy 38, Fall 007 8
Central density and temperature of the Sun Central pressure is a little more than a factor of 100 larger than average pressure. Guess: central density 100 times higher than average? (That s equivalent to guessing that the internal temperature does vary much with radius.) For the Sun, that s not bad; the central density turns out to -3 be 110 times the average density, ρ C = 150 gm cm. Thus, since ρc M R 3, 3 M M R -3 ρc = 150 gm cm 3 R M R As we will see in a couple of weeks, the average gasparticle mass in the center of the Sun, considering its composition and the fact that the center is completely 4 ionized, is = 1.5 10 gm. m C 11 September 007 Astronomy 38, Fall 007 9
Central density and temperature of the Sun (continued) But the material is still an ideal gas, so PV C PC Pm C C 6 TC = 15.7 10 K. N k = n k = ρ k = C C C Compare to : T doesn t vary very much with radius. T We can make a scaling relation out of this as well, to use in extrapolating to stars similar to the Sun but having different masses, sizes and composition: 3 Pm C C GM R 6 TC = m 15.7 10 K for the Sun; 4 C = ρc k R M 6 M R m TC = 15.7 10 K. M R m 11 September 007 Astronomy 38, Fall 007 10
Opacity and luminosity in stars At the high densities and temperatures found on average in stellar interiors, matter is opaque. The mean free path, or average distance a photon can travel before being absorbed, is about = 0.5 cm for the Sun s average density and temperature (given above). Photons produced in the center have to random-walk their way out. How many steps, or mean free paths, does it take for a photon to random-walk from center to surface? (PU problem 5.11.) Suppose photon starts off at the center of the star, and has an equal chance to go right or left after each absorption and re-emission. Average value of position after N steps is x = ( x + x + + x )/ N = N 1 N 0 11 September 007 Astronomy 38, Fall 007 11
Opacity and luminosity in stars (continued) However, the average value of the square of the position is not zero. Consider step N+1, assuming the chances of going left or right are equal: 1 b g 1 b g xn+ 1 = xn + xn + 1 1 = xn xn+ + xn + xn+ N = x +. But if this is true for all N, then we can find by starting at zero and adding, N times (i.e. using induction): x = N. N x N 11 September 007 Astronomy 38, Fall 007 1
Opacity and luminosity in stars (continued) Thus to random-walk a distance xn = L, the photon needs to take on the average N = L steps. So far, we have discussed only one dimension of a threedimensional random walk. Three times as many steps need to be taken in this case, so to travel a distance R, the photon on the average needs to take N R = 3 steps. 11 September 007 Astronomy 38, Fall 007 13
Opacity and luminosity in stars (continued) For the Sun, and for a constant mean free path of 0.5 cm, N = e 3 6. 96 10 a f 05. j 10 = 581. 10 Each step takes a time t = c, so the average time it takes for a photon to diffuse from the center of the Sun to the surface is 3R 11 4 t = N t = = 9.7 10 s = 3.1 10 years. c Note that the same trip only takes R / c =.3 s for a photon travelling in a straight line. steps. 11 September 007 Astronomy 38, Fall 007 14
The solar spectrum By and large, the spectrum of the Sun resembles closely a blackbody. From the total energy flux at Earth (solar constant): f = 136. 10 6-1 - erg s cm we get the Sun s luminosity: 33-1 L = 3.86 10 erg s Setting solar luminosity equal to blackbody power gives Sun s effective temperature: 4 σ e L = 4πR T T = 5800 K e 11 September 007 Astronomy 38, Fall 007 15
The solar spectrum (continued) In detail: absorption lines are also seen in the solar spectrum; they match up with many known transitions of atoms, ions and molecules. (See Lab #.) Figure: the ultimate high-resolution spectrum of the Sun (Nigel Sharp, from data by Bob Kurucz et al. ( NOAO/NSO/Kitt Peak FTS/AURA/NSF) 11 September 007 Astronomy 38, Fall 007 16
Formation of absorption lines in the solar atmosphere Figure: Chaisson and McMillan, Astronomy Today 11 September 007 Astronomy 38, Fall 007 17
The outer layers of the sun Figure: Chaisson and McMillan, Astronomy Today Convection: Opacity from atoms rises as one moves from center to surface (more atoms there; all ionized in the center). Bubbles can neutralize, cool and sink. Hot material rises to take its place. 11 September 007 Astronomy 38, Fall 007 18
Solar granulation: the tops of convection cells Figure: Chaisson and McMillan, Astronomy Today 11 September 007 Astronomy 38, Fall 007 19
Further out: the corona and chromosphere Corona: observed to be well over 1000000 K. Theory of corona: heated by acoustic noise from boiling top of convection zone; diffuse enough that it can t cool very well, so it reaches very high temperatures. Figure: Chaisson and McMillan, Astronomy Today 11 September 007 Astronomy 38, Fall 007 0
Solar corona and atmosphere X-ray composite (NASA/SPARTAN) 11 September 007 Astronomy 38, Fall 007 1
Sunspots: solar magnetism Figure: Chaisson and McMillan, Astronomy Today Sunspots appear dark because they re slightly cooler than the rest of the solar surface. Zeeman effect measurements show that they are also maxima of magnetic field. Associated with other activity (e.g. prominences) 11 September 007 Astronomy 38, Fall 007
Sunspot progress, activity during solar cycle Figure: Chaisson and McMillan, Astronomy Today 11 September 007 Astronomy 38, Fall 007 3
11-year sunspot cycle, for the last 400 years Figure: Chaisson and McMillan, Astronomy Today 11 September 007 Astronomy 38, Fall 007 4
Sunspot formation and cycle: interaction of magnetism and differential rotation The Sun rotates, but not as a solid body; this differential rotation wraps and distorts an initially poloidal solar magnetic field. Occasionally, the field lines burst out of the surface and loop through the lower atmosphere, thereby creating a sunspot pair. The underlying pattern of the solar field lines explains the observed pattern of sunspot polarities. If the loop happens to occur on the limb of the Sun and is seen against dark space, a prominence is visible. The twisting and wrapping of the field lines eventually results in the production of a poloidal field again, but with north and south switched. Then the process repeats. years between identical field configurations, 11 years between sunspot-number maxima. 11 September 007 Astronomy 38, Fall 007 5
Sunspot formation and cycle: interaction of magnetism and differential rotation Figure: Chaisson and McMillan, Astronomy Today 11 September 007 Astronomy 38, Fall 007 6