Solar Physics Course Lecture Art Poland Modeling MHD equations And Spectroscopy

Why What I am going to talk about should lead to our understanding of physical processes in outer solar atmosphere: –Heating –Energy transport –Solar wind acceleration –Magnetic field evolution

Overview Modeling features in the solar atmosphere involves solving the full MHD equations. The solution of these equations needs initial conditions and boundary conditions. To get realistic values, you need observations. In this lecture I will first talk about how to get the observations, and then how they are used in the solution of the equations.

What Features Closed loops Open magnetic field

What Quantities Do We Need? The equations tell us we need to observe –Temperature –Density –Velocity –Magnetic field To do time dependence we need each as a function of time.

Questions How can we measure these quantities at the Sun? Spectroscopic observations What causes an absorbtion line? How is one formed? Why are some lines in emission?

Atomic Structure

Comments Equations derived from observations in the lab. of atomic spectra. Quantum mechanics gives more precise description. What I am showing helps visualize the structure.

Level Splitting, Momentum When there are multiple electrons in an atom, the n levels are split. The split levels are referred to as s,p,d,f, –For n=1 there are only s levels –For n=2 there are s and p levels –For n=3 there are s,p,and d levels –etc

Spin The other quantity is electron spin. –He for example has 2 electrons, both can occupy the n=1 level because one has a spin of +1/2 and the other a spin of -1/2 –Transitions between spin level have a very low probability, and are referred to as forbidden transitions. –When they are opposite to each other they are referred to as singlets –When they are the same, triplets –Spin combined with momentum can also give doublets.

Sample Energy Diagram Allowed transitions Forbidden transitions Magnetic fields can split sub- levels (ie 2 2 s into 2 levels).

Summary Atomic structure - Spectral lines Electron transitions –N levels –Momentum s,p,d,f –Spin Momentum and spin splitting occurs in magnetic field – can use splitting to measure strength of field.

How to Use Spectra Velocity? –Doppler shift –Sometimes just an asymmetry in profile What else? Temperature – next topic Density – next topic

Gas State 1) The basic observed equation is P=NkT, N=ρ/uM H –a. This is important: if you know two, you know the third. –b. Can make observations that yield T, and ρ so you can get P. 2) Temperature and mean velocity – visualization again –a. Perfect box with perfect collisions: collision momentum with wall is 2mv x –b. Number of collisions v x /2L L is size of box –c. Total of all momentum is ΣΜv x 2 /L –d. Momentum is pressure so P=ML -3 Σv x 2 –e. Define mean v x 2 =n -1 Σv x 2 –f. v x 2 =v y 2 =v z 2 =1/3v 2 –g. P=n/3L 3 (Mv 2 )=1/3(NMv 2 ) –h. So average energy ½ Mv 2 =3/2 kT This is important because it relates energy, velocity, to temperature. (not bulk velocity)

Get The T Velocity of atoms and electrons related to T was just shown. What do we need to get the temperature of the gas? First assumption? –Assume a Maxwellian velocity distribution What must be assumed for this to be valid? –Collisions (not so good at very low  f(v,T)= (M/2πkT) 3/2 v 2 e- (Mv2/2kT) Maxwellian tail of distribution

Line Profile Emission –Where to measure for T? –Can get T from line width.

Boltzmann Distribution a. N i /N=g i /ue -ε/kT ε=hν b. Can use the ratio for 2 energy levels to get relative populations between two energy levels. c. Measure two lines from same atom to get T d. N i /N j =g i /g j e -δε/kT

Saha Equation a. equation of ionization state: N i /N j (Ne)=(2πmkT) 3/2 /h 3 (2(u i (T)/u j (T))e -ε/kT b. Used to determine gas temperature N T FeIII FeIV

How to Get Lines –I is the intensity you observe –S is emission/cm 3 –  is optical depth  is frequency –How do you get an absorption line from this? –How do you get an emission line?

Other issues Non-LTE A or f values –Line brightness –Collision prob. Plank function Differential emission measure Log Ne 2

Summary Spectra can give us: –T via line width, line ratios –Density via line ratios or diff. emission measure –Velocity via line shift

MHD Equations +H-C

–Force Balance or –Conservation of Momentum

Energy Conservation and Maxwell’s equation

How to Solve Depends on the problem –Near the Sun small area, cartesian –Whole Sun or Heliosphere, spherical Coupled differential equations. The big problem is steep gradients. –Transition region (T gradient) –Flares (P gradients, shocks) The boundary conditions you choose almost always dictate the solution.

Results The output of these programs are table of numbers, T(x,y,z,t), P(x,y,z,t),etc. Need to visualize the results Need to make visualizations something that you can compare with observations.

Variable Grid Mesh Major Breakthrough Paramesh Steep T gradient in transition region. Almost no gradient in corona.

Gridding Changes as Calculation progresses

Other Issues Conduction –Isotropic –Anisotropic –along B field Radiative losses –Optically thin - collisions –Non-LTE T(collision) T(radiation) Heating –Constant –Alfven waves – a function of B Make each of these a replaceable module in your program.

Conduction Conduction only along B field –How the grid is oriented with respect to B Excess heating low down Numerical diffusion makes it wider.

Energy Balance

Model Output All profiles from same T different v

Differential Emission Measure