Solar Physics - Atmosphere

1. Solar Physics - Atmosphere K.E.Rangarajan

2. Recommended Text : The Sun An introduction Second edition by Michael Stix Springer publication

3. Spectral Lines (e.g. 2D echelle image of optical Solar spectrum) NaI

4. Continuous Energy Distribution(e.g.UV optical spectrophotometry of Vega)

5. (Teff, log g) • Effective temperature (in K), is defined by L=4pR2Teff4related to ionization. • Surface gravity (cm/s2), g = GM/R2 , related to pressure. • The Sun has Teff=5777K, log g=4.44 – its atmosphere is only a few hundred km deep, <0.1% of the stellar radius. • The Solar atmosphere is most easily studied during total eclipse (lunar limb occults at rate of 0.5 arcsec/s or 300km/s at 1AU, so brightness variation during the final second before totality provides the physical size). • A red giant has log g~1 (extended atmosphere) whilst a white dwarf has log g~8 (effectively zero atmosphere).

6. Radiation Terms Black body radiation (Planck function) Effective Temperature (Stefan-Boltzmann) Specific and mean Intensity

7. The Black Body Imagine a box which is completely closed except for a small hole. Any light entering the box will have a very small likely hood of escaping & will eventually be absorbed by the gas or walls. For constant temperature walls, this is in thermodynamic equilibrium. If this box is heated the walls will emit photons, filling the inside with radiation. A small fraction of the radiation will leak out of the hole, but so little that the gas within it remains in equilibrium. The emitted radiation is that of a black-body. Stars share properties of the black-body emitter, in the sense that a negligibly small fraction of the radiation escapes from each.

8. Stefan – Boltzmann Law Blackbody radiation is continuous and isotropic whose intensity varies only with wavelength and temperature. Following empirical (Josef Stefan in 1879) and theoretical (Ludwig Boltzmann in 1884) studies of black bodies, there is a well known relation between Flux and Temperature known as Stefan-Boltzmann law: F=T4 with =5.6705x10-5 erg/cm2/s/K4 (Note that Bohm-Vitense refers to `astronomical flux’, F/, as `flux’).

9. Effective temperatures of stars Neglecting interstellar absorption, the total energy arriving above the Earth’s atmosphere is its observed flux, f, corrected for the distance to the star, i.e. L=4pd2f The same energy must be emitted by the star, i.e. L= 4pR2F where F is the surface flux, so F=f(d/R)2. For the Sun, the angular radius is 959.6 arcsec and f=1.367x106 erg/cm2/s so the radiant flux at Solar surface is F=6.317x1010 erg/cm2/s. The Stefan-Boltzmann law, F=T4eff, or alternatively L/(4pR2)=T4eff defines the `effective temperature’ of a star, i.e. the temperature which a black body would need to radiate the same amount of energy as the star. Teff is 5777K for the Sun.

10. The black body intensity is defined (following discovery by Max Planck in 1900) as either or where c=2.99x1010 cm, h=6.57x20-27 erg s, k=1.38x10-16 erg/s. Using cgs units ( in Angstroms) we have Planck formula

11. Wien’s displacement law For increasing temperatures, the black body intensity increases for all wavelengths. The maximum in the energy distribution shifts to shorter  (longer ) for higher temperatures. max T = 2.98978 x107 Ang K is Wien’s law for the maximum I providing an estimate of the peak emission (max=5175Ang for the Sun). (At long wavelengths, well beyond the peak in energy output, the Rayleigh-Jeans approximation is: I= 2kT2/c2 = 2kT/2 This is commonly referred to as the Rayleigh-Jeans `tail’) EXAMPLE

12. Specific Intensity Consider light passing through a surface area d in a narrow cone of opening solid angle d. Intensity relates to energy via Now consider the energy passing through a surface area d at an angle  with respect to the normal of this surface area, the effective beam width is reduced by cos The (specific) intensity is then a measure of brightness with units of erg/(s cm2 rad2 A). In model atmosphere calculations, I is obtained from the `transfer equation’ – which we shall introduce later on. I is proportional to energy/solid angle, and so is independent of distance.

13. Mean Intensity and Flux One can alternatively define intensity in frequency units such that The two spectral distributions have different shapes for the same spectrum. The Solar Spectrum has a maximum in the green in I (5175A) but the maximum is in the far-red (8800A) for I. (Why? c=, d/d=-c/2 ,so equal intervals of  correspond to different intervals of  across the spectrum). The mean intensity J is the directional average of the specific intensity (over 4 steradians), whilst the flux, F, is the projection of the specific intensity in the radial direction (integrated over all solid angles). There is also a K-integral which we will use later on.

14. An example Let us consider a point on the boundary of a radiating sphere. From above, If no flux enters the surface & if there is no azimuthal dependence for F Finally, if Iis independent of direction over one hemisphere then F=I and J = ½ I. For the Sun, I=2 x 1010 ergs s-1 cm-2 ster-1 and J=1x1010 ergs s-1 cm-2 ster-1 (Note these are Bolometric quantities.) Important: I is independent of distance from the source, and can only be measured directly if we resolve the radiating surface. In contrast, F obeys the inverse square law and is all that may be measured for most stars.

15. Optical Depth • Consider radiation shining through a layer of material. The intensity of light is found experimentally to decrease by an amount dI where Here  is the density, and ds is a length and the so-called (mass) absorption coefficient (alias opacity) is . The photon mean free path is inversely proportional to  • Two physical processes contribute to the opacity ; (i) true absorption where the photon is destroyed and the energy thermalized; (ii) scattering where the photon is shifted in direction and removed from the solid angle under consideration. • The radiation sees a combination of  and ds over some path length, L given by a dimensionless quantity, the `optical depth’

16. Importance of optical depth We can write the change in specific intensity over a path length as using the definition of optical depth. This may be directly integrated to give the usual extinction law: An optical depth of =0 corresponds to no reduction in intensity (i.e. the top of photosphere for a star) An optical depth of =1 corresponds to a reduction in intensity by a factor of e=2.7. If the optical depth is large ( >>1) negligible intensity reaches the observer In stellar atmospheres, typical photons originate from =2/3 .

17. Source function We can also treat emission processes in the same way as absorption via an emission coefficient, j, with units of erg/s/rad2/Hz/g. Physical processes contributing to j, are (i) real emission – the creation of photons; (ii) scattering of photons into the direction being considered. The ratio of emission to absorption is called the Source function,

18. Local Thermodynamic Equil. In the study of stellar atmospheres, the assumption of Local Thermodynamic Equilibrium (LTE) is described by: • Electron and ion velocity distributions are Maxwellian f(v)=4v2(m/2kT)1.5e-mv^2/2kT • Excitation equilibrium given by Boltzmann equation • Ionization equilibrium given by Saha equation (introduced later) • The source function is given by the Planck function. i,e. Kirchoff’s law

19. Maxwellian velocity distribution

20. free states ionization limit bound states, „levels“ Eion 6 5 4 3 2 Energy 1 Boltzmann equation For excited levels u and l of e.g. atomic hydrogen, the Bolzmann equation relates their population (occupation) numbers as follows: where ul =Eu-El is the energy difference between the levels, gu & gl are their statistical weights (see next slide), k=8.6174x10-5 eV/K is the Boltzmann constant.

21. Hydrogen For H, orbital n has a statistical weight of gn=2n2 – the various permutations for n=1 and n=2 are listed here, with statistical weights g1=2 and g2=8, respectively. (l=electron orbit, ml=electron angular momentum with |ml|<l<ml, ms=electron `spin’ angular momentum ±1/2) Example I

22. Balmer lines An exceptionally high T is required for a significant number of H atoms to have electrons in their 1st excited states. Recall that the Balmer lines (involving an upward transition from n=2 orbital) reach a peak strength at subtype A (9000K) so why do the Balmer lines diminish in strength at higher temperatures? We need Saha equation to answer this question.

23. Derivation of Saha Equation We can extend the Boltzmann formula to states with +ve energies (the upper state is now an ion plus free electron, with energy ion+1/2mev2). Let us consider the simplest case of the lower and upper states being the ground states of the neutral atom (e.g. H0) and singly ionized ion (H+). Here me is the mass of the electron, ion the ionization energy, dN+1(v)is the number of ions in their ground state with the free electron having a velocity in the interval (v, v+dv), g1+ and ge are the statistical weight of the ground state of the ion & the electron.

24. ionization limit The (differential) statistical weight of the electron, ge, i.e. the number of available states in interval (v,v+dv) is Inserting this into Boltzmann’s equation reveals

25. Since we don’t care about v, we can integrate over all velocities, substituting x=v(2mekT) and using Finally, we arrive at: This relates the ground state populations of the atom and ion. To derive the ratio of the total number of ions (N+) to the total number of atoms (N0) we can use the conventional Boltzmann formula for each level n of the atom and ion, Nn/N1 and Nn+/N1+ i.e..

26. Partition function If N0 is the sum of all neutral particles in their different quantum states:. We find: Where we have introduced u0, the partition function of the atom. This is the weighted sum of the number of ways it can arrange its electrons with the same energy - e.g. all H is in the ground state for the Solar case, so u02 (the ground state statistical weight). Similarly for the ion, For H+, u+=1, since no electrons left. Tables listed overleaf

27. If we multiply N1+/N10 from earlier by N+/N1+ and N10/N0 we obtain the Saha equation (Meghnad Saha 1920): In logarithmic form Saha equation can be written as:. Where ion is measured in eV, =5040/T and Pe the electron pressure is related to the electron density via the ideal gas law (Pe=NekT). High temperature favours ionization, high pressure favours recombination. In stellar atmospheres, Pe lies in the range 1 dyn/cm2 (cool stars) to 1000 dyne/cm2 (hot stars). Note that 1dyn/cm2=0.1N/m2 (SI units), so for SI calculations the final constant is -1.48 instead of -0.48 Example II

28. Degree of ionization of H in stars We can use Saha to study the degree of ionization of H in general in stellar photospheres. The fraction of ionized hydrogen to the total is defined below. We find that H switches from mostly neutral below 7000K to mostly ionized above 11000K for typical Ne. This allows us to understand why hydrogen lines are strongest in A-type stars, with temperatures of 7500-10000K.

29. Strong lines in Solar photosphere

30. Ca II in the Sun The photosphere of the Sun has only two calcium atoms for every million H atoms, yet the Ca II H and K lines (produced by the ground state of singly ionized calcium, Ca+) are stronger than the Balmer lines of H (produced by the 1st excited state of neutral H). Why?

31. Saha-Boltzmann applied to Ca From the Saha equation we have found that H is essentially neutral in the Solar photosphere, yet from the Boltzmann formula H(n=2)/H(n=1)=5x10-9. i.e. very little H is available to produce Balmer absorption lines. For Calcium, ion=6.1eV, and Partition functions may be determined from tables via For =5040/T=0.872, the partition function of neutral Ca i.e. u0=1.3. Similarly, u+=2.3.

32. Essentially all Calcium is singly ionized. N(Ca+) in the first excited state relative to the ground state (g1=2, g2=4, =3.12eV) is 1/265 from Boltzmann eqn, so nearly all Calcium in the Sun’s photosphere is in the ground state of Ca+. Combining these results: N(Ca)/N(H) x N(Ca+ g.s.)/N(H n=2)=(1/500,000) x (0.99/5x10-9)=400 There are 400 times more Ca+ ions with electrons in the ground-state (which produce the Ca II H&K lines) than there are neutral H atoms in the first excited state (which produce Balmer lines). The Ca IIlines in the Sun are so strong due to T dependence of excitation and ionization (not high Ca/H abundance).

33. Ionization Potentials

34. Radiative transfer I Parallel-ray Transfer equation Plane-parallel Transfer equation Limb darkening

35. Radiative transfer equation The primary mode of energy transport through the surface layers of a star is by radiation. The radiative transfer equation describes how the physical properties of the material are coupled to the spectrum we ultimately measure. Recall, energy can be removed from (true absorption or scattered), or delivered to (true emission or scattered) a medium: From last lecture, collecting absorption () and emission (j) coefficients, the rate of change of (specific) intensity is: We can re-write this equation in terms of the optical depth and ratio of emission to absorption coefficients, the source function,

36. i.e. • This is the (parallel-ray) equation of radiative transfer. It will need a small modification before it is applicable to stars, but we can already gain some insight from its solution. • If S<I, the intensity will decrease, it will stay constant if S=I and increase if S>I. • One can readily solve this form of the transfer equation with an integrating factor exp(), i.e. so

37. i.e. With Inserting boundary conditions: Rearrange: The first term on the RHS describes the amount of radiation left over from the intensity entering the box, after it has passed through an optical depth , the second term gives the contribution of the intensity from the radiation emitted along the path.

38. Solution to transfer equation • Imagine first the case in which I0=0, i.e. solely emission from the volume of gas. We have two limiting cases: • Optically thin case (<<1) • EXAMPLE: • Hot, low density nebula • Optically thick case (>>1) • EXAMPLE: • Black body, S=B(T) Opacity  versus  Intensity versus 

39. Hot nebular gas: emission lines –optically thin

40. Absorption versus emission • Imagine now I00, again with two extreme cases: • Optically thin case (<<1) • If I0>S, so there is something subtracted from the original intensity which is proportional to the optical depth – we see absorption lines on the continuum intensity I. EXAMPLE: stellar photospheres • If I0<S, we will see emission lines on top of the background intensity. • Example: Solar UV spectrum • Optically thick case (>>1): • Planck function as before. Opacity  versus  Intensity versus 

41. Outward decreasing temperature In a star absorption lines are produced if I0 > S i.e. the intensity from deep layers is larger than the source function from top layers. In LTE, the source function is B(T), so the Planck function for the deeper layers is larger than the shallower layers. Consequently the deeper layers have a higher temperature than the top layers (since the Planck function increases at all wavelengths with T). (Instances occur where LTE is not valid, and the source function declines outward in parallel with an increasing temperature). Solar Spectrum (4300-4320Ang)

42. Transfer Equation for Stars The plane-parallel transfer equation (for stars with thin photospheres) is identical to the parallel-ray transfer equation (for ISM studies), except for (a) the cos term and (b) sign change, since we are now looking from the outside in, along an arbritary direction x, i.e. d=-dx. The full spherical geometry transfer equation is necessary for supergiants.

43. Surface Intensity • To derive the intensity at the surface, we multiply the plane-parallel transfer equation by an integrating factor exp(- sec)=exp(-u), • This can be written as • Integrating du from 0 to infinity

44. Limb darkening Let us assume a linear source function: We derive, Recall u=sec, so =u cos and we can use the standard integral We obtain In the linear approximation for the Source function, the surface optical depth lies between 0 and 1. From the centre of the star we see radiation leaving the star perpendicular to the surface: I(0,0)=a+b, whilst at the limb the starlight leaves the surface at an angle I(0,90)=a. Limb darkening (less light from the limb versus the centre).

45. Solar limb darkening This optical image of the Sun clearly shows limb darkening. We see into the atmosphere down to a depth of =1. Limb darkening exists because the continuum source function decreases outward. As we look towards the limb, we see higher photospheric layers, which are less bright.

46. Schematic of limb darkening Schematic illustration of limb darkening – penetration of different lines of sight (heavy lines) to `unit optical depth’ (dashed lines) corresponds to different depths in the photosphere, depending on . Radiation seen at 2 is characteristic of higher (cooler) layers than the radiation seen at position 1

47. Linear vs Quadratic source function Up to now we assumed a linear source function. More generally, if: Then We still get S(0) at the limb, but a more complicated result at the centre. For example, a quadratic term requires the solution of At =90 degrees, =0, whilst at =0, 1+2a1/a2 providing a2<<a1. The ratio of the limb-to-centre intensity is

48. Example for Solar Case: The measured centre to limb variation of the solar intensity (Table 4.17, AQ 4th edition) is

49. Wavelength dependence Limb darkening is observed to be greatest at shorter wavelengths in the Sun. The temperature distribution of the upper atmosphere of the Sun can be obtained fromlimb darkening measurements, carried out via e.g. multi-filter images of the Solar continuum (between the lines). Until recently, the Sun was the only star for which limb darkening was observed, since one needs to spatially resolve the disk (most other stars appear as point sources!) to measure limb darkening. Other methods are now possible: (Pierce & Waddell 1961). Centre Limb

50. Limb darkening for other stars • Direct interferometry, via high spatial resolution `imaging’ – eg. COAST array, providing star is very large and nearby (a cool supergiant), • The light curve from an eclipsing binary system during secondary eclipse allows us to study limb darkening of the primary, although non-trivial! Similar approach followed by extra solar planets occulting parent star (e.g. HD209458) • The light curve due to the gravitational micro-lensing of a background (generally Galactic bulge or Magellanic Cloud) star by a foreground source (e.g.PLANET team)