Structure of lectures I

„ Introduction and overview „ Core and interior: energy generation and standard solar model Introduction to „ Solar radiation and spectrum Solar Physics „ Solar spectrum „ Radiative transfer Sami K. Solanki „ Formation of absorption and emission lines „ Convection: The convection zone and IMPRS lectures granulation etc. January 2005 „ Solar oscillations and helioseismology „ Solar rotation

Structure of lectures II Structure of lectures III: next time

„ The solar atmosphere: structure „ Explosive and eruptive phenomena „ Photosphere „ Flares „ Chromosphere „ Transition Region „ CMEs „ Corona „ Explosive events „ Solar wind and heliosphere „ Sun-Earth connection „ „ The magnetic field „ CMEs and space weather „ Zeeman effect and Unno-Rachkovsky equations „ Longer term variability and climate „ Magnetic elements and sunspots „ Chromospheric and coronal magnetic field „ Solar-stellar connection „ The solar cycle „ Activity-rotation relationship „ Coronal heating „ Sunspots vs. starspots „ MHD equations & dynamo: see lectures by Ferriz Mas

The Sun: a brief overview The Sun, our star „ The Sun is a normal star: middle aged (4.5 Gyr) main sequence star of spectral type G2 „ The Sun is a special star: it is the only star on which we can resolve the spatial scales on which fundamental processes take place. „ The Sun is a special star: it provides almost all the energy to the Earth „ The Sun is a special star: it provides us with a unique laboratory in which to learn about various branches of physics.

1 The Sun: Overview The Sun: a few numbers 30 „ Mass = 1.99 10 kg ( = 1 M) „ Average density = 1.4 g/cm3 26 „ Luminosity = 3.84 10 W ( = 1 L) „ Effective temperature = 5777 K (G2 V) „ Core temperature = 15 106 K „ Surface gravitational acceleration g = 274 m/s2 „ Age = 4.55 109 years (from meteorite isotopes) „ Radius = 6.96 105 km „ Distance = 1 AU = 1.496 (+/-0.025) 108 km „ 1 arc sec = 722±12 km on solar surface (elliptical Earth orbit) „ Rotation period = 27 days at equator (sidereal, i.e. as seen from Earth; Carrington rotation)

The Sun’s Structure The solar surface Solar interior: „ Everything below „ Since solar material does not exhibit a phase transition (e.g. the Sun’s (optical) from solid or liquid to gaseous as for the Earth), a standard surface way to define the solar surface is through its radiation. „ Divided into „ The photons travelling from the core outwards make a hydrogen-burning random walk, since they are repeatedly absorbed and core, radiative and reemitted. The mean free path increases rapidly with radial convective zones distance from the solar core (as the density and opacity Solar atmosphere: decrease). „ Directly observable „ A point is reached where the average mean free path part of the Sun. becomes so large that the photons escape from the Sun. „ Divided into This point is defined as the solar surface. It corresponds to photosphere, optical depth τ = 1. Its height depends on λ. chromosphere, corona, „ Often τ = 1 at λ=5000 Å is used as a standard for the solar heliosphere surface.

Wide range of physical parameters

„ The Sun presents a wide variety of physical phenomena and processes, between solar core and corona. Solar physics in „ E.g. Gas density varies by ≈ 30 orders of magnitude, temperature by 4 orders, relevant time scales from 10-10 sec relation to other to 10 Gyr branches of physics

ÎDifferent observational and theoretical techniques needed to study different parts of Sun, e.g. helioseismology & nuclear physics for interior, polarimetry & MHD for magnetism, etc.

2 Solar Physics in Relation to Other Fields The Sun as a plasma physics lab. Sun-Earth relations: climate, space weather

Turbulence, Cosmic rays, local dynamos interstellar medium

Plasma physics Solar planets, extrasolar planets Sun

Atomic/molecular Cool stars: activity, physics structure & evolution Fundamental physics: Neutrinos, Gravitation

Solar Tests of Gravitational Physics The Sun and particle physics

„ Curved light path in solar gravitational field Î Test „ The fact that the rate of neutrinos measured by the of General Relativity Homestake 37Cl detector is only 1/3 of that predicted „ Red shift of solar spectral lines Î Test of EEP by standard solar models was for > 30 years one of the major unsolved problems of physics. „ Oblate shape of Sun Î Quadrupol moment of solar „ Possible resolutions: gravitational field: Test of Brans-Dicke theory (R. Mecheri) „ Standard solar model is wrong „ Neutrino physics is incomplete „ Comparison of solar evolution models with „ Recent findings from SNO and Superkamiokande: observations Î Limits on evolution of fundamental Problem lies with the neutrino physics constants ÎStandard model of particle physics needs to be „ Polarization of solar spectral lines: Tests revised gravitational birefringence Î Tests of equivalence „ Nobel prize 2002 for R. Davies for discovery of the principle & alternative theories of gravity (O. Preuss) principle & alternative theories of gravity (O. Preuss) solar neutrino problem.

Solar physics and astrophysics Which stars have magnetic fields or show magnetic activity?

„ Phenomena happening all over the universe can be „ Best studied star: Sun best studied at close distances, where the relevant „ F, G, K & M stars (outer physical processes can be spatially resolved (same convection zones) show for solar planets). magnetic activity & have „ E.g. magnetic activity, is present on innumerable fields of G-kG. stars, in accretion disks, in jets, in the interstellar „ Early type stars: Ap, Bp, medium, etc., but the relevant spatial scales can (kG-100kG), Be (100G) generally not be resolved. The Sun provides a key. „ White dwarfs have B ≈ „ Radio astronomy, radiative transfer, kG-109 G, no activity spectropolarimetry, asteroseismolgy, etc. are „ Not on diagram: pulsars techniques first developed to study the Sun

3 The Sun compared with active stars Sun, Earth and planets

„ Solar output affects the magnetospheres and atmospheres of planets „ Solar energy is responsible for providing a habitable environment on Earth „ Solar evolution and liquid water on Mars...

The Sun’s core

„ In the Sun’s core mass is turned into energy. „ Nuclear reactions burn The solar interior 7x1011 kg/s of hydrogen into helium. „ Inside the core the particle density and temperature are so high, that individual protons ram into each other at sufficient speed to overcome the Coulomb barrier, forming heavier He atoms and releasing energy

Nuclear reactions in cores of stars Nuclear reactions of pp-chain

„ p=proton „ Sun gains practically all its energy from the reaction „ d=deuterium 4p → α + 2e+ + 2ν = 4He + 2e+ + 2ν „ α=Helium „ Two basic routes „ γ=radiation „ p-p chain: yields about 99% of energy in Sun „ ν=neutrino „ 2nd reaction „ CNO cycle : 1% of energy released in present day Sun 2 reaction (but dominant form of energy release in hotter stars) replaces step 3 of st „ Both chains yield a total energy Q of 26.7 MeV, 1st reaction rd nd mainly in the form of γ-radiation Qγ (which is „ 3 reaction replaces steps 2+3 of 2 reaction absorbed and heats the gas) and neutrinos Qν „ Branching ratios: (which escapes from the Sun). „ 1st vs. 2nd + 3rd 87 : 13 „ 2nd vs. 3rd → 13 : 0.015

4 Nuclear reactions of CNO-cycle Temperature dependence of pp- chain and CNO cycle „ C, N and O act only as catalysts: Basically the same things happens as with proton „ p-p chain in cool main- chain. sequence stars „ CNO cycle in hot main- sequence stars „ Triple alpha process in red giants: 3He → C

Solar neutrinos Solar neutrino spectrum

„ Neutrinos, ν, are produced at various stages of the pp-chain. „ Continua: „ Neutrinos are also produced by the reaction: p(p e- , ν)d , so- Continua: 2 called pep reaction. Being a 3-body reaction it is too rare to number/(cm2 s contribute to the energy, but does contribute to the number MeV) of ν. of ν. „ Lines: number/(cm2 s) „ Bars at top & shading: sensitivity of different materials to ν

Solar neutrinos II Results of various neutrino

„ Since 1968 the Homestake 37Cl experiment has experiments given a value of 2.1 ± 0.3 snu (1snu =1 ν / 1036 target atoms) „ Standard solar models predict: 7±2 snu ÎSolar Neutrino Problem! „ In 1980s & 90s water based Kamiokande and larger Superkamiokande detectors found that approximately half the rare, high energy 8B ν were missing. „ 71Ga experiments (GALLEX at Gran Sasso and SAGE in Russia) showed that the neutrino flux was too low, even including the p(p,e+ ν)d neutrinos.

5 Solar neutrinos III Solar Neutrinos IV

„ Sensitivity of H2O, „ Possible solutions to solar neutrino problem: 71Ga and 37Cl to ν increases ~ „ Standard solar model is incorrect (5-10% lower exponentially with temperature in core gives neutrino flux consistent increasing energy. with Homestake detector). ÎHomestake 37Cl detector and (Super-) Kamiokande „ Neutrino physics is incomplete (i.e. the standard see mainly high-energy ν from rare β+-decay of 8B. model of particle physics is wrong!) „ Branching ratios between the various chains: central „ Nuclear physics describing the pp-chain is 37 for predicting exact ν-flux detectable by Cl & H2O incorrect „ Branching ratios depend very sensitively on T(r=0), „ Nuclear physics describing interaction between while total ν-flux depends only linearly on luminosity. 37 71 neutrino and Cl is incorrect (Kamiokande & „ Even Ga experiments sensitve largely to high 71 energy ν. Ga showed that this wasn’t the problem)

Resolution of neutrino problem Resolution of neutrino problem II

„ SNO (Sudbury Neutrino Observatory) in Sudbury, Canada uses D2O and can detect not just the „ Lesson learnt: neutrinos have a multiple electron neutrino, but also µ and τ neutrinos personality problem (J. Bahcall) ÎThe neutrinos aren’t missing, e- neutrinos produced „ Other lesson learnt: the “dirty” and difficult in the Sun just convert into µ and τ neutrinos Other lesson learnt: the “dirty” and difficult ÎThe problem lies with the neutrino physics. solar model turned out to be correct, the -8 clean and beautiful standard theory of particle „ The neutrino has a small rest mass (10 me), which allows it to oscillate between the three flavours: e- physics turned out to be wrong, or at least neutrino, µ neutrino and τ neutrino (proposed 1969 incomplete (J. Bahcall) by russian theorists: Bruno Pontecorvo and Vladimir Gribov, … but nobody believed them) „ 2002: Raymond Davis got Nobel prize for „ Confirmation by measuring anti-neutrinos from uncovering the neutrino problem power plant (with Superkamiokande).

Standard solar model Equations describing solar interior Mass conservation : Equation of state : „ Ingredients: Conservation laws and material dependent ∂ r = 1 ρ = ρ(P,T ) equations 2 ∂m 4πρr or (for an ideal gas) : „ Mass conservation ρ = ρℜ gas density = T „ Hydrostatic equilibrium (= momentum conservation in a steady state) PG Hydrostatic equilibrium : µ „ Energy conservation = ∂ PG gas pressure „ Energy transport P = − Gm 2 ℜ = gas constant „ Equation of state ∂m 4πr µ = mean molecular weight „ Expression for entropy P = pressure Energy tra nsport : „ Nuclear reaction networks and reaction rates → energy production G = Gravitational constant L „ Opacity Energy balance: F = F + F + F = R C cond 4πr 2 „ Assumptions: standard abundances, no mixing in core or in ∂L ∂S F = total energy flux radiative zone, hydrostatic equilibrium, i.e. model passes = ε −T ∂m ∂T F = radiative flux through a stage of equilibria (the only time dependence is R ε = energy generation per unit mass F = convective flux introduced by the reduction of H and the build up of He in the C T = temperature F = conductive flux core). cond S = entropy L = luminosity

6 Internal structure of the Sun Solar evolution

„ Internal „ Path of the Sun in the models HR diagram, starting in PMS stage and shown for ending at ZAMS Sun „ Red giant stage (subscript „ White dwarf stage z) and for „ Note the complex present day patch during the red Sun (radius giant phase due to reaching various phases of out to 1.0, Helium burning and core contraction and subscript expansion, etc. ) „ η ~ mass loss rate

Evolution of Sun’s luminosity Faint young Sun paradox „ According to the standard solar model the Sun was approximately 30% less bright at birth than it is today

ZAMS Today „ Too faint to keep the Earth free of ice! „ Problem: Albedo of ice is so high that even with its current luminosity the Sun would not be able to melt all the ice away. „ Obviously the Earth is not covered with ice... „ So: Where is the mistake?

Possible resolution of the faint Evolution of solar luminosity young Sun paradox Runaway greenhouse effect „ The Earth’s atmosphere was different 4 Gyr ago. More methane and other greenhouse gases. Higher insulation through evaporation of oceans meant that even with lower solar input the Earth remained ice-free. „ As the Sun grew brighter life grew more abundant and Today changed the atmosphere of the Earth, reducing the greenhouse effect. „ Problem: what about Mars? Could it have had liquid water 4Gyr ago if Sun were so faint?

„ Alternative: Sun was slightly more massive (1.04-1.07M at birth and lost this mass (enhanced solar wind) in the course of time (Sackmann & Boothroyd 2003, ApJ). A more massive The future of the Earth? star on the ZAMS emits more light. Also agrees w. Mars data Sackmann et al. 1997

7 The Sun in white light: Limb darkening

„ In the visible, the Solar radiation and Sun’s limb is darker than the centre of the solar disk (Limb spectrum darkening) „ Since intensity ~ Planck function, Bν(T), T is lower near limb. „ Due to grazing incidence we see higher near limb: T decreases outward

Limb darkening vs. λ The Sun in the EUV: Limb brightening

„ In the EUV, the Sun’s „ Upper fig.: limb is brighter than C IV „ short λ: large the centre of the solar disk (Limb limb darkening; brightening) „ long λ: small „ Since the solar limb darkening atmosphere is optically thin at these „ departure from wavelengths, intensity straight line: ~ thickness of layer contributing to it. Due limb darkening to geometrical effects is more this layer appears thicker near limb complex than (radiation comes from I(θ) ~ cos(θ) roughly the same height everywhere).

The Sun in the EUV: Limb brightening Solar irradiance spectrum Irradiance = solar flux at 1AU „ Limb brightening in optically thin Spectrum is similar to, but lines does not not equal to imply that the Planck function Sun’s Î Radiation temperature comes from increases layers with diff. outwards temperatures. (although by Often used chance it does in temperature these layers....) 4 measure for stars: Effective temp: σT eff = Area under flux curve

8 Absorption in the Earth’s The solar spectrum: continua with atmosphere absorption and emission lines

„ The solar spectrum changes in character at different wavelengths. „ X-rays: Emission lines of highly ionized species „ EUV: Emission lines of neutral to multiply ionized species plus recombination continua „ UV: stronger recombination continua and absorption lines - „ Visible: H b-f continuum with absorption lines - „ FIR: H f-f continuum, increasingly cleaner (i.e. less lines, except molecular bands) „ Radio: thermal and, increasingly, non-thermal continua

Visible Solar UV spectrum

Note the transition from absorption lines (for λ>2000Å) to emission lines (for λ<2000Å)

EUV spectrum Detail of EUV spectrum by SUMER

„ The solar spectrum from 500 Å to 1600 Å measured by SUMER (logarithmic scale) Activity sensitive Lyman line: S VI continuum

Ly α Lyman continuum edge λ (Å) 800 1000 1200 1400 1600

9 Solar EUV irradiance Formation of the solar spectrum

„ Continuum „ Spectral energy distribution „ Centre to limb variation „ Spectral lines „ Absorption lines „ Emission lines „ Radiative transfer

Radiative transfer: optical depth Optical depth and solar surface

„ Axis z points in the direction of light „ Radiation escaping from the Sun is emitted

propagation mainly at values of τν ≈ 1.

„ Optical depth: ∆τν = -κν ∆z „ At wavelengths at which κν is larger, the radiation comes from higher layers in the where κν is the absorption coefficient and ν is the frequency of the radiation. Light only atmosphere.

knows about the τν scale and is unaware of z „ In solar atmosphere κν is small in visible and near IR, but large in UV and FIR Î We see ÎIntegration: τν = -∫κν(z)dz (note that the scales are floating, no constant deepest in visible and NIR, but sample higher of integration is fixed) layers at shorter and longer wavelengths.

Height of τ = 1, brightness Radiative Transfer Equation temperature and opacity vs. λ Equation of radiative transfer:

µ dIν/dτν = Iν – Sν

where Iν is the intensity (i.e. the measured quantity) and Sν is the source function. µ = cosθ is only 1.6µm important for non-vertical rays.

Sν = emissivity εν divided by absorption coefficient κν

The physics is hidden in εν and κν , i.e. in τν and Sν . These quantities depend on temperature, pressure, elemental abundances, and frequency or wavelength

10 Formal solution of RT equation When is an emission line formed, when an absorption line? For Sν = 0 the solution of the RTE in a slab with (optical) boundaries τν2 and τν1 is : „ Continua are formed deeper in a stellar

Iν(τν2) = Iν(τν1) exp(– τν2 + τν1 ) atmosphere than spectral lines at the same „ For general case formal solution reads (formal soln. wavelengths. assumes that we already know S ) ν „ A line is in absorption if Sν decreases with height, i.e. if the absorption at greater heights Iν(τν2) = Iν(τν1) exp(– τν2 + τν1 ) +∫Sν exp(– τν) dτν dominates over emission „ 1st term describes radiation that enters through lower boundary (only absorption, no emission in „ A line is in emission if Sν increases with slab), 2nd term describes radiation emitted in slab. height, i.e. if the emission at greater heights

„ In a stellar atmosphere τν1 = ∞, so that only the 2nd dominates over absorption term survives (lower boundary is unimportant).

Statistical equilibrium The assumption of LTE „ In the solar interior and photosphere (i.e. where density is large and collisions are common) we can „ In general both Sν and κν require a assume Local Thermodynamic Equilibrium (LTE) computation of the full statistical equilibrium „ Thermodynamic equilibrium (TE): a single for the species being considered. temperature everywhere (blackbody) Î Radiation emerging from object follows the Planck function Bν . „ This implies computing how much each „ LTE: Each layer of the solar atmosphere has its own atomic/ionic/molecular level is populated, i.e. temperature Î Replace Sν by Bν(T) in the RTE and solving rate equations describing transitions its solution. to and from each considered level. „ Problem of knowing Sν is reduced to knowing T(τ) in the atmosphere. „ Requires detailed knowledge of atomic, ionic, „ In addition, the statistical equilibrium can be solved molecular structure and transitions. simply by considering the Saha-Boltzmann equilibrium Î basically T and ne need to be known.

Elemental Elemental abundances: the FIP abundances effect

„ Photospheric values „ In TR and corona abundances differ from „ Logarithmic (to base 10) photospheric values abundances of the 32 lightest depending on the first elements on a scale on which ionization potential (FIP) H has an abundance of 12 of element. „ Elements with low FIP „ Heavier elements all have low have enhanced abundances abundance „ Note that in general the solar „ Apparently, acceleration through the chromosphere photospheric abundances are is more efficient for these very similar to those of elements (which are meteorites, with exception of ionized in chromosphere, Li, with is depleted by a factor while high FIP elements of 100. are not).

11 Spectrum synthesis by 1-D Diagnostic power of spectral lines radiative equilibrium model „ Doppler shift of line: (net) flows in the LOS direction. „ Line width: temperature and turbulent velocity „ Equivalent width: elemental abundance, temperature (via ionisation and excitation balance) „ Line depth: temperature and temperature gradient „ Line asymmetry: inhomogenieties in the solar atmosphere.

Heights of formation: on which layers do lines give information ?

“normal”lines Solar convection in visible UV, neutral EUV and strong visible lines

Multiply ionized EUV lines

The convection zone Scales of solar convection „ Through the outermost 30% of solar interior, energy is transported by convection instead of by radiation „ Observations: 4 main „ In this layer the gas is convectively unstable. The scales unstable region ends just below the solar surface. „ granulation I.e. the visible signs of convection are actually due „ mesogranulation to overshooting. „ supergranulation „ giant cells „ Due to this, the time scale changes from the time scale for a random walk of the photons through the „ Colour: radiative zone (due to high density, the mean free „ well observed path in the core is well below a millimeter) to the „ less strong evidence convective transport time: „ Theory: larger scales at greater depths. In „ t ~ 106 years >> t ~ months radiative convective detail a lot is unclear.

12 Surface convection: Surface convection: granulation Supergranulation „ Typical size: 2 Mm „ 1 hour average of „ Lifetime: 6-8 min MDI Dopplergrams (averages out „ Velocities: 1 km/s (but (averages out peak velocities > 10 oscillations). km/s, i.e. supersonic) „ Dark-bright: flows towards/away from „ Brightness contrast: observer. 20% in visible observer. continuum (under „ No supergranules ideal conditions) visible at disk centre: velocity is mainly „ All quantities show a horizontal continuous distribution of values „ Size: 20-30 Mm, lifetime: days, 6 „ At any one time 10 horiz. speed: 400 m/s, granules on sun. no contrast in visible

Observing convection Line bisectors

„ Observing granulation „ Continuum images and movies at high spatial resolution: gives sizes, lifetimes and evolution (splitting and dissolving granules), contrasts „ Spectral lines (also at low spatial resolution). Line bisectors, line widths and convective blue shifts: contrasts, area factors, stratification

„ Observing supergranulation „ Images and movies in cores of chromospheric spectral lines, or magnetograms: observe the magnetic field at the edges of the supergranules instead of the supergranules directly „ Helioseismic techniques

Supergranules seen by SUMER Supergranules & magnetic field „ Si I 1256 Å full disk scan by „ Why are super- granules seen in SUMER in 1996 chromospheric and „ Bright network transition region lines? indicates location „ Supergranules are of magnetic related to the network magnetic network. „ Darker cells: „ Network magnetic supergranules fields are concentrated at edges of supergranules.

13 Illustration of convectively stable Onset of convection and unstable situations Schwarzschild’s instability criterion Convectively stable Convectively unstable Consider a rising bubble of gas: ρ* ρ z-∆z ↑ ρ ρ0 = ρ z bubble surroundings depth

Condition for convective instability: ρ* < ρ0 For small ∆z, bubble will not have time to exhange heat with surroundings: adiabatic behaviour. Convectively unstable if:

[dρ/dz – (dρ/dz)adiab] ∆z < 0 dρ/dz : true stellar density gradient,

(dρ/dz)adiab : adiabatic gradient

Onset of convection II Why an outer convection zone?

Rewriting in terms of temperature and pressure: ∆ „ Why does radiative grad exceed adiabatic gradient?

ad = (d log T/d log P) ad „ Mainly: radiative gradient becomes very large due ∆ ad ad to ionization of H and He below the solar surface. rad = (d log T/d log P) rad = gradient in an atmosphere with radiative energy transport „ Expression for radiative gradient (for Eddington

approximation): ∆

∆ Schwarzschild’s convective∆ instability criterion: 4 rad = (3Fr /16σg) (κgr Pg / T ) ad < rad „ Fr = radiative flux (≈ constant) „ σ = Stefan-Boltzmann constant „ g = gravitational acceleration (≈ constant)

„ κgr = absorption coefficient per gram. As H and He become ionized with depth, κgr increases rapidly, leading to large radiative gradient.

Ionisation of H and He Radiative, adiabatic & actual gradients „ Ionisation balance is described by Saha’s equation:

degree of ionisation depends on T and ne „ H ionisation happens just below solar surface

„ He → He+ + e- happens 7000 km below surface

„ He+ → He++ + e- happens 30’000 km below surface

„ Since H is most abundant, it provides most electrons (largest opacity) and drives convection most strongly

„ At still greater depth, other elements also provide a minor contribution.

14 The mixing length Why is mixing length interesting

„ As a gas packet rises, diffusion of particles and thermal exchange „ Gives an idea of the size scale of a with surroundings causes it to lose convection cell (height or depth of cell given its identity and to stop from moving by mixing length) on. „ horizontal extent of cell cannot be much „ Length travelled up to that point: bigger than mixing length due to mass mixing length l. conservation! „ Often used parameterization of l:

l = α Hp

„ Hp = pressure scale height „ α = mixing length parameter, typically 1-2 (determined empirically)

History of mixing length Convective overshoot „ Due to their inertia, the packets of gas reaching the „ Mixing length was introduced by L. Prandl, boundary of the CZ pass into the convectively stable who was director at Max Planck Institut für layers, where they are braked & finally stopped. Strömungsforschung in Göttingen. Îovershooting convection

„ It allowed him to describe convection in a „ Typical width of overshoot layer: order of Hp simple, but powerful way. „ This happens at both the bottom and top boundaries of the CZ and is important: „ Even today, most stellar interior models (and boundaries of the CZ and is important: many atmospheric models) use the mixing „ top boundary: Granulation is overshooting material. Hp ≈ 100 km in photosphere length to describe the effects of convection. „ bottom boundary: the overshoot layer allows B-field to be stored → seat of the dynamo?

Convection simulations Convection simulations II

„ 3-D hydrodynamic simulations reproduce a number of „ Simulations do not achieve the solar Reynolds number (Re = observations and provide new insights into solar convection. vl/ν) of 1010 , where v = typical velocity, l = typical length scale, ν = kinematic viscosity (R ~ ratio of viscous to „ These codes solve for mass conservation, momentum e advection time scales). conservation (force balance, Navier-Stokes equation), and energy conservation including as many terms as possible. „ For comparison with observations it is important that the simulations describe the surface layers well, i.e. code needs „ Problem: Simulations can only cover 2-3 orders of magnitude to consider: in length scale (due to limitations in computing power), while „ radiative transport of energy. Only few simulation codes do this the physical processes on the Sun act over at least 6 orders properly. of magnitude. „ partial ionization of many elements „ Also, simulations can only cover a part of the size scale of Also, simulations can only cover a part of the size scale of „ as low a viscosity as numerically possible solar convection, either granulation, supergranulation, or larger scales, but not all. „ The role of radiation is primarily to transport energy. At the solar surface the energy transported by radiation becomes comparable to that by convection.

15 Simulations of solar granulation Testing the simulations

Comparison between observed and computed 6 Mm bisectors for selected spectral lines (2 computed bisectors shown: one each with and Solution of Navier-Stokes equation etc. describing fluid without oscillations dynamics in a box (6000 km2 x 1400 km) containing the in the atmosphere) solar surface. Realistic looking granulation is formed.

Granule Granule evolution

structure „ Granules die in two ways: „ dissolve: grow fainter and smaller until they disappear Upflows are broad & (small granules) slow, downflows are „ split: break into two smaller granules (large granules) narrow and fast. „ Granules are born in two ways: Why? „ as fragments of a large splitting granule „ appearing as small structures and growing „ Initially most granules grow in size, some keep growing until they become unstable (see next slide) and split, others stop growing and start shrinking until they disappear (all within 5-10 min).

Granule evolution II Granule evolution III

„ Why do large granules split and small granules get „ Consider mass conservation: A larger granule squeezed out of sight? will build up a larger pressure above its centre „ Granules are overshooting convection structures, with an upflow in their centre, a radial horizontal flow because more mass needs to be accelerated over the whole granule and a downflow in the horizontally. surrounding lanes. „ The upflow builds up density and excess pressure „ At some point the pressure becomes so large above the granule Î pressure gradient relative to that the upflow is quenched. The centre of the flanks of granule. granule cools and a new downflow lane forms „ Pressure gradient = force. ÎGas is accelerated sideways. there. The granule splits „ Above intergranular lane: horizontally flowing gas meets gas from opposite granule Î build up pressure excess which decelerates flow.

16 Relation between granules and Increasing size of convective cells supergranules with depth

„ Downflows of granules keep going down to bottom of simulations. However, the intergranular lanes break up into individual narrow downflows. „ I.e. topology of flow reverses with depth: „ At surface: isolated upflows, connected downflows „ At depth: connected upflows, isolated downflows „ Idea put forward by Spruit et al. 1990: At increasingly greater depth the narrow downflows from different granules merge, forming a larger and less fine-meshed network that outlines the supergranules.

Convection on other stars „ F, G, K & M stars posses outer convection zones and show Oscillations and observable effects of convection (also WDs) helioseismology „ Observations are difficult since surfaces cannot be resolved. ÎUse line bisectors: independent of spatial resolution „ A,F stars show inverse bisectors: granulation has different geometry.

5-minute oscillations Solar Eigenmodes Hear the Sun sing! „ The entire Sun vibrates from „ The p-modes show a a complex pattern of acoustic Sound waves speeded up 42,000 times distinctive dispersion waves, with a period of Doppler shift relation (k-ω diagram: around 5 minutes k~ ω2) „ The oscillations are best seen „ Important: there is power as Doppler shifts of spectral only in certain ridges, i.e. lines, but also as intensity 2 2 2 for a given k (= kx + ky ), variations. only certain frequencies „ Identified as acoustic waves, contain power. ω called p-modes „ This discrete spectrum „ Spatio-temporal properties of oscillations best revealed by suggests the oscillations 3-D Fourier transforms. are trapped, i.e. eigenmodes of the Sun. eigenmodes of the Sun. k

17 Global oscillations Description of solar eigenmodes

„ The Sun's acoustic waves bounce from one side of the Sun to the other, causing the Sun's surface to oscillate up and „ Eigen-oscillations of a down. They are reflected at the solar surface. sphere are described by spherical harmonics „ Modes differ in the depth to which they penetrate: they turn 1/2 around because sound speed (CS ~ T ) increases with „ Each oscillation mode depth (refraction) is identified by a set of „ p-modes are influenced three parameters: by conditions inside the „ n = number or radial Sun. E.g. they carry info nodes on sound speed „ l = number of nodes on „ By observing these the solar surface oscillations on the surface we can learn „ m = number of nodes about the structure of passing through the the solar interior poles (next slide)

Illustration of spherical harmonics Spherical harmonics „ l = total number of nodes (in images: l = 6) = degree l = total number of nodes (in images: l = 6) = degree „ Let v(θ,φ,t) be the velocity, e.g. as measured at the solar „ m = number of nodes connecting the “poles” surface over time t. Then: ∞ l θ ϕ = m θ ϕ v( , ,t) ∑∑alm (t)Yl ( , ) l=−0 m= l

„ The temporal dependence lies in alm, the spatial dependence m in the spherical harmonic Yl . m θ ϕ = |m| θ ϕ Yl ( , ) Pl ( )exp(im ) |m| θ = Pl ( ) associated Legendre Polynomial

„ Due to the normalization of the spherical harmonic, the Fourier power is given by F(a)F(a)*

„ Here F(a) is the Fourier transform of the amplitude alm

More examples and a problem with Interpretation of k-ω or ν-l diagram

identifying spherical harmonics „ At a fixed l, different frequencies show „ General problem: Since we see only half of the Sun, significant power. Each the decomposition of the sum of all oscillations into of these power ridges spherical harmonics isn’t unique. belongs to a different order n (n = number of „ This results in an uncertainty in the deduced l and m radial nodes), with n n= 5 increasing from bottom 4 to top. 3 2 „ Typical are small values 1 of n, but intermediate to large degree l.

18 Accuracy of frequency A few observational remarks measurements „ 107 modes are present on the surface of the Sun at any given time (and interfering with each other). „ Plotted are identified frequencies and „ Typical amplitude of a single mode: < 20 cm/s frequencies and 7 error bars (yellow; „ Total velocity of all 10 modes: a few 100 m/s 1000σ for blue freq., „ Accuracy of current instruments: better than 1 cm/s 100σ for red freq. „ Frequency resolution ~ length of time series below 5 mHz and 1σ (Heisenberg’s uncertainty principle) ~ lowest for higher freq.) detectable frequency „ Best achievable freq. ÎLonger time series are better resolution: a few „ Gaps in time series produce side lobes (i.e. parts in 105; limit set spurious peaks in the power spectrum) by mode lifetime „ Highest detectable frequency ~ cadence of obs. ~100 d

Frequency vs. amplitude The measured low-l eigenmode signal „ Frequencies are the important parameter, more so „ Sun seen as a star: Due to cancellation effects, only than the amplitudes of the modes or of the power modes with l=0,1,2 are visible Î simpler power peaks. peaks. spectrum. „ The amplitudes depend on the excitation, while the „ Low l modes are important for 2 reasons: frequencies do not. They carry the main information „ They reach particularly deep into the Sun (see cartoon on on the structure of the solar interior. earlier slide). „ p-modes are excited by turbulence, which excites all „ These are the only modes measurable on other Sun-like frequencies. However, only at Eigenfrequencies of stars. the Sun can eigenmodes develop. „ These modes are sometimes called “global” modes.

„ Frequencies (being more constant) are also „ The different peaks of given l correspond to different measured with greater accuracy. n values (n=15...25 are typical).

Best current low-l power spectrum Mode structure of low l spectrum

„ GOLF/SOHO Note the regular spacing of the observations showing a modes: blowup of the l = 2 l = 0 small separation: νn,l -νn-1,l+2 power spectrum with large separation: νn,l –νn-1,l an l = 0 and an l = 2 mode. „ The noise is due to random re-excitation of the oscillation mode by turbulence

19 Types of oscillations p-modes vs. g-modes „ Solar eigenmodes can be of 2 types: „ p-modes propagate throughout the solar interior, but „ p-modes, where the restoring force is the pressure, i.e. normal sound waves are evanescent (later slide) in the solar atmosphere „ g-modes, where the restoring force is gravity (also called „ g-modes propagate in the radiative interior and in buoyancy modes) the atmosphere, but are evanescent in the „ So far only p-modes have been detected on the convection zone (their amplitude drops Sun with certainty. exponentially there, so that very small amplitudes are expected at the surface). Convection means „ They are excited by the turbulence associated with the convection, mainly the granulation near the buoyancy instability; oscillations require stability. solar surface (since there the convection is most „ g-modes are expected to be most sensitive to the vigorous). very core of the Sun, while p-modes are most „ Being p-modes, they travel with the sound speed sensitive to the surface CS. They dwell longest where CS. is lowest. Since „ Current upper limit on solar interior g-modes lies 1/2 CS ~ T , this is at the solar surface. below 1 cm/s.

Solar oscillations: simple treatment Solar oscillations: simplified treatment II „ Equations describing radial structure of adiabatic oscillations, neglecting any perturbations to the gravitational potential, „ Analytical solutions of these equations for an isothermal are: atmosphere are readily obtained: 1 d ξ 1  1 l(l +1)  −1/ 2 (r 2ξ ) − r +  − P = 0 ξ ρ 2 r 2 ρ 2 2ω 2 1 r (r) ~ 0 exp(ikr r) r dr c 0  c r  1/ 2 1 dP g P (r) ~ ρ exp(ik r) 1 + P − ()ω 2 − N 2 ξ = 0 1 0 r ρ ρ 2 1 r 0 dr 0c „ These oscillations are trapped in the body of the Sun. Since  1 dP 1 dρ  where N 2 = g − 0  = Brunt - Vaisala frequency the time scale on which the atmosphere reacts to  Γ ρ  disturbances is low, waves which are travelling in the solar  P0 dr 0 dr  interior are evanescent in the atmosphere → They are „ Here ξr is radial displacement and P1 is pressure perturbs. present only for discrete frequencies (similar to the bound Quantities with subscript 0 refer to the Γ =  d ln P  states in atomic physics). unperturbed Sun. Γ is the adiabatic exponent:    d ln ρ 

Regimes of oscillation Deducing internal structure from solar oscillations „ In regimes of acoustic and 2 gravity waves kr > 0, while in „ Global helioseismology: Gives mainly the radial regime of evanescent waves 2 dependence of solar properties, although latitudinal kr < 0 (exponential damping). 2 dependence can also be deduced (ask R. Mecheri). The solid lines show kr = 0. „ Evanescent waves occur „ Radial structure of sound speed when the period is so long „ Structure of differential rotation that the whole (exponentially stratified) medium has time to „ Local helioseismology: Allows in principle 3-D adapt to the perturbation, Cutoff frequency for imaging of solar interior. E.g. time-distance achieving a new equilibrium. acoustic waves in a helioseismology does not measure frequencies, but Therefore the wave does not rather the time that a wave requires to travel a propagate, but rather the stratified medium: rather the time that a wave requires to travel a certain distance (relatively new) medium as a whole oscillates. ωC = CS/2H

20 Global helioseismology Testing the standard solar model: results of forward modelling „ Use frequencies of many modes. „ Relative difference between C 2 obtained from „ Basically two techniques for deducing information S inversions and from standard solar model plotted vs. on the Sun’s internal structure radial distance from Sun centre. „ Forward modelling: make a model of the Sun’s internal structure (e.g. standard model discussed earlier), „ Typical difference: compute the frequencies of the eigenoscillations of the 0.002 Î good! model and compare with observations model and compare with observations „ Typical error bars „ Inverse technique: Deduce the sound speed and rotation from inversion: by inverting the oscillations (i.e. without any comparison 0.0002 Î poor! with models) 0.0002 Î poor! „ Problem areas: „ Note that forward modelling is required in order to first identify the modes. Only after that can „ solar core inversions be carried out. „ bottom of CZ „ solar surface

Variation of the solar rotation Local excitation of wave by a flare velocity with period of 1.3 years „ Clear example of wave being triggered. „ The wave is not travelling at the surface, but rather reaching the c increases s surface further out at later times. Note how it travels ever faster. Why?

Local helioseismology Local helioseismology II

„ Temperature and velocity „ Does not build upon measuring frequencies of eigenmodes, but rather measures travel times of waves through the solar structures can be interior, between two “bounces” at the solar surface (for distinguished, since a flow particular technique of time-distance helioseismology). directed with the wave will affect it differently than a „ The travel time flow directed the other way between source and first (increase/decrease the bounce depends on the sound speed). structure of CS below the „ By considering waves surface. By considering passing in both directions it waves following different is possible to distinguish paths inhomogeneous between T and velocity. distributions of C can be S „ At right: 1st images of determined. convection zone of a star!

21 Time-Distance Helioseismology of Time-Distance Helioseismology of a sunspot a sunspot II

„ Subsurface structure of sunspots „ Sunspots are good targets, due to the large temperature contrast. „ Major problem: unknown influence of the magnetic field on the waves. Kosovichev et al. 2000 Zhao et al. 2004

Subsurface Structure of Sunspots Seeing right through the Sun

„ How can sunspots last „ A technique, called two- for several weeks skip far-side seismic holography, allows without flying apart? images of the far side of „ Models: inflows the Sun to be made. „ Waves from front go to „ Observations at back and then return. surface: outflows (moat flow) „ Acoustic waves speed Zhao et al.: 2001, ApJ 557, 384 up in active regions ¾ Strong inward flows right underneath the surface (hotter in subsurface layers) ¾ Sunspots surprisingly shallow: become warmer than „ The delay of the sound surroundings already some 4000 km below surface waves is about 12 sec in „ Results to be confirmed (unkown influence of B) a total travel time of 6 hours

Seeing right through the Sun II Helioseismology instruments „ Needed: „ uninterrupted, long time series of observations „ Either high velocity sensitivity, or high intensity sensitivity (and extremely good stability) „ Low noise „ Spatial resolution better than 1” (for local helioseismology) „ Instruments are either: „ Ground based global networks (GONG+, BiSON) „ Space based instruments in special full-Sun orbits (advantage of lower noise relative to ground-based networks; MDI, GOLF, VIRGO on SOHO, HMI on SDO) „ Usually filter instruments with high spectral or intensity Real-time far side images: fidelity http://soi.stanford.edu/data/farside/index.html

22 Instruments and projects Asteroseismology First reliable detection of oscillations on the near solar analogue, „ Ground (networks of 3 or more telescopes aimed at α Centauri, and other Sun-like stars. Note the shift in the p- reducing the length and number of data gaps) mode frequency range to lower values for α Centauri, which is older than the Sun (note also factor 103 difference in ν scale) „ GONG+

„ BISON „ TON Sun α Centauri

„ Space (uninterrupted viewing, coupled with lack of noise introduced by the atmosphere) „ SOHO MDI, GOLF and VIRGO (running) „ SDO HMI (being built) „ Solar Orbiter VIM (planned)

Projects

„ Major asteroseismic Space missions: Solar rotation „ COROT „ Kepler „ Ground based: „ ESO 3.6m (HARPS) „ ESO VLT (UVES) „ Networks of smaller Telescopes

Solar rotation Discovery of solar rotation

„ The Sun rotates differentially, both in latitude „ Galileo Galilei and (equator faster than poles) and in depth (more Christoph Scheiner complex). noticed already that sunspots move across „ Standard value of solar rotation: Carrington rotation the solar disk in period: 27.2753 days (the time taken for the solar accordance with the coordinate system to rotate once). rotation of a round body „ Sun’s rotation axis is inclined by 7.1o relative to the ÎSun is a rotating Earth’s orbital axis (i.e. the Sun’s equator is inclined sphere by 7.1o relative to the ecliptic). „ Movie based on Galileo Galilei’s historical data

23 Surface differential rotation Surface differential rotation „ Poles rotate more slowly than equator. „ Description: „ Surface differential rotation from Ω = A + B sin2ψ + C sin4ψ measurements of: „ Tracers, such a where ψ is the latitude, A = Ω at the equator sunspots or magnetic field elements (always and A+B+C = Ω at the poles. indicators of the rotation rate of the „ Different tracers give different A, B, C values. magnetic field) Equator E.g. spots rotate faster than the surface gas. „ Doppler shifts of the gas „ Coronal holes (not plotted) rotate rigidly

How come different rotation laws? Internal differential rotation

„ Are different tracers anchored at different „ Method: Helioseismic inversions depths in the convection zone? „ In a non-rotating star the individual modes of oscillation, „ Evidence in support comes from sunspots: young described by “quantum numbers” n,l,m are degenerate in spots rotate faster than older spots ( → older that their frequency depends only on n and l, but not on m. spots are slowed down by the surrounding gas) „ Similar to Zeeman effect. Note that m distinguishes between the surface distribution of oscillation nodes. For „ How come coronal holes rotate rigidly, while a spherically symmetric star (no rotation) all these modes the underlying photospheric magnetic field must have same frequency. rotates differentially? „ In a rotating Sun the degeneracy is removed and modes ÎIndividual magnetic features must move in and with different m have slightly different frequency. out of coronal holes „ Since modes with different l sample the solar latitudes in ÎSupport: evidence for enhanced magnetic different ways, it is possible to determine not just vertical, reconnection at the edges of coronal holes but also latitudinal differential rotation by helioseismology.

Internal differential rotation II Internal differential rotation III : tachocline „ Structure of internal rotation deduced from MDI data Large radial „ Note: differential rotation in CZ, solid rotation below gradients in rotation rate at bottom of CZ (tachocline), but also just below solar surface (enigmatic). Note the slight missmatch of helio-seismic and Doppler measurements

24 What about rotation of solar core? What about rotation of solar core?

„ Rotation rate of solar core is not easy to determine, since p-modes are rather insensitive to the innermost part of the Sun. „ Different values in the literature for the core’s rotation rate: Ω(r=0) = Ω(r=R) … 2 Ω(r=R) „ One way to set limits on Ω(r=0): quadrupole moment of Sun. „ Solar rotation leads to oblateness, i.e. diameter is larger at equator than between the poles. „ If core rotates more rapidly than surface, then oblateness will be larger than expected due to surface rotation rate.

Solar oblateness Evolution of solar rotation

„ Oblateness = ∆R/R -5 „ „ Direct measurements: ∆R/R ≈ 10 „ Young stars are seen to rotate up to 100 times „ Very tricky, since oblateness 10-5 corresponds to ∆R = 14 faster than the Sun. km (best spatial resolution achievable: 100 km). „ Did the Sun also rotate faster when it was young? „ Systematic errors due to concentration of magnetic activity to low latitudes → affects measurements of solar „ Skumanich law: Ω ~ t-1/2, where t is the age of the diameter, since shape of limb is distorted. star (deduced from observing stars in clusters of „ Initial measurements due to Dicke & Goldenberg (1967) -5 different ages). gave ∆R/R ≈ 5x10 → required change of general relativity to explain motion of Mercury’s perihelion (but ÎSun also rotated faster as a young star. was consistent with Brans-Dicke gravitation theory) „ Helioseismic measurements give for the acoustic radius of the Sun (which is not the same as the „ Question: where did all the angular momentum go? -5 optical radius, but similar): ∆R/R ≈ 10 (Redouane Mecheri)

Evolution of solar rotation II Evolution of solar rotation III

„ Answer part 2: Magnetic field! „ Question: where did all the angular momentum go? „ Answer part 1: Solar wind! The solar wind carries „ Solar wind is channeled by magnetic field up away angular momentum with it. Torque j, i.e. rate to the Alfven radius RA, i.e. point where wind of change in angular momentum, exerted by solar speed > Alfven speed. Up to that radius, the wind (without magnetic field): wind rotates rigidly with the solar surface (forced to do so by rigid field lines), i.e. it only j = Ω Rdm/dt carries angular momentum away beyond RA. „ Here dm/dt is the solar mass-loss rate (mass carried Proper expression for Torque: away by solar wind) j = Ω RAdm/dt „ Problem: j is 2-3 orders of magnitude too small to cause a significant braking of solar rotation… „ RA typically is 10-20 times larger than R .

25 Evolution of rotation IV

„ j = Ω RAdm/dt ~ Ω ÎThe faster the star rotates, the quicker it spins down. „ Additional corrections: „ dm/dt depends on Ω (more rapidly rotating star, more magnetic field, hotter the corona, larger the dm/dt)

„ RA depends on Ω (although not in a straightforward manner: more rapidly rotating star, more magnetic field, but also larger the density and velocity of wind). „ In general j = kΩα , where α typically > 1, although there are signs that for very large Ω, the α value becomes very small (saturation).

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