CHAP 6 Energy Transfer

Introductory Remarks We will consider following a beam of light along some path from the source to our eyes (or a detector). How do we quantify the net result, depending on the nature of the intervening material?

Sidebar: Transport of Energy This can happen in four ways: Thermal conduction Convection Radiation Electrical conduction

What is Astronomically Important?

Are Planets ‘Astronomical’? Consider heat flow in the Earth. Conduction and convection dominate.

How About in Stars? The important processes are energy transport by convection and by radiation. The relative importance depends on (a) the temperature gradient and (b) the opacity. When it is hard for energy to flow purely by radiation, either because of a steep gradient or high opacity, convection begins.

Different Regimes

Things to Note Low-mass stars are fully convective. Note the implications for potential stellar lifetimes: fresh fuel gets cycled into the core. Sun-like stars are convective in the outer parts, thanks to the opacity. This is why we see granulation on the solar surface. The core is radiative: remember the discussion of photons ‘random walking’ through that material?

Solar Granulation

High-Mass Stars Have convective cores (very high temperature gradients)

General Remark The treatment of convection in stellar structure calculations is very complex and uncertain. This is yet another great uncertainty in the difficult art of studying stellar evolution. Outside of stars (and planets!), however, convection is not important in astrophysics and we can focus our attention on radiative transfer -- how energy gets from point A to point B along a path we will follow through the intervening medium.

…and Also Within Stars We will apply the same techniques to the outer parts of stars, the ‘reversing layers’ where the absorption lines are formed.

The Equation of Transfer

Consider a signal of specific intensity I ν passing through a ‘cloud’ (not necessarily a discrete region; could be part of a bigger complex or a zone within a star) of total thickness l. The intensity will (or at least may) change with distance depending on whether energy is scattered out of the beam, absorbed, scattered into the beam, or generated by some mechanism.

The Properties May Vary Along the Path - so we do this incrementally

Why an Intensity Change? The beam enters the incremental step with specific intensity I ν but emerges with I ν + d I ν, which may be greater or smaller --- energy can be removed from the beam (scattered, absorbed) or added (emission).

Note the distinction between the (incremental) geometrical path length dl and the (incremental) optical thickness dτ ν. If the material is transparent and non-emissive, dτ ν = 0 even though there is a real geometrical step being taken: the intensity will not change over that step of size dl.

Gains and Losses d I ν = d I ν loss + d I ν gain But d I ν loss = - α ν I ν dr (negative sign since these are losses; α ν is the absorption coefficient from before) The proportionality to I ν is because the more photons there are in the beam, the greater the total energy loss – i.e. some fraction of the photons get scattered or absorbed

Gains Next We write the gains simply in terms of a new emission coefficient j ν (in units of ergs sec -1 cm -3 Hz -1 sr -1 ) So d I ν gain = j ν dr Note that the emission depends only on the material along the path, not on the strength of the incoming beam, so there is no I ν factor.

Two Ways of Expressing This 1. How does the specific intensity vary as we move along a given geometrical path? (steps in dr) or 2. How does the specific intensity vary as we move along a path of growing optical depth (e.g. looking deeper into a stellar atmosphere)?

These Differ Any given geometrical step dr may not involve any change in optical depth dτ. How they scale depends on the distribution of the material, the physical conditions, state of ionization, etc. The convention is that r increases in the direction of the beam (coming in on one side of the cloud, passing through the layer of interest, and out the other, towards our eyes) but the optical depth τ increases as we look towards the source, in the opposite direction.

So We Can Write the Equation of Radiative Transfer in either form

The ‘Source Function’ S ν = j ν / α ν encapsulates the absorption and emission properties of the material in the cloud.

Note the Frequency Dependence! Dust can absorb (obscure) visible light [top panel] but emit in the infrared [bottom panel]

Notation The radiation has specific intensity I ν0 as it enters a cloud /layer, then passes through some geometrical thickness l and/or optical depth τ ν, and exits with final specific intensity I ν.

Specific Cases, to Aid Understanding Case A: if there is no ‘cloud’ / intervening material Then α ν = 0 (no absorption) j ν = 0 (no emission along the line of sight) Here we use the functional form with respect to r Clearly dI ν / dr = 0  I ν = constant.

Case B: if there is an absorbing cloud but it is not emitting. In this case, we use the relationship between I ν and τ ν (the optical depth) Heres, j ν = 0 = S ν so dI ν / dτ ν = I ν Rearrange ( dI ν / I ν = dτ ν ) and integrate through the cloud to get I ν = I ν0 exp (- τ ν )

The Implication “One optical depth” == “one e-folding distance” Since e -1 = 0.368, we lose ~63 percent of the photons in the first optical depth of the material. Two optical depths eliminate 86% of the photons; Three optical depths eliminate 95%.

Case C: the converse: a cloud that is emitting but not absorbing. (Example: a cool dust cloud that is transparent to infrared radiation.) We use the geometrical (dr) formulation rather than terms involving optical depth. In this case, α ν = 0. A simple integration yields I ν = I ν0 + j r dr In other words, integrate the total emission along the line of sight to see how it supplements the specific intensity.

Case D: A cloud in complete thermodynamic equilibrium (TE). In TE, the radiation temperature T equals the kinetic temperature T everywhere, and the specific intensity I ν is given by the Planck function B ν (T) everywhere. This means there is no intensity gradient, so d Iν / dr = 0 and 0 = - α ν B ν (T) + j ν Whence B ν (T) = j ν / α ν

Kirchoff’s Law Since B ν (T) is the same everywhere in the cloud, this means that j ν and α ν are in equilibrium: In other words, there are equivalent emissions and absorptions everywhere in the material if it is in complete TE (which makes intuitive sense)

Case E: A cloud that both emits and absorbs (the most general case). Simplification: take the emission and absorption properties to be constant along some line of sight (valid at least over small regions). Then

Consider the Terms Separately Rewrite this: I ν = I νο exp(-τ ν ) + S ν - S ν exp(-τ ν ) The first term describes the attenuation of the incoming beam because it is passing through absorbing/scattering material. The second term describes the contribution to the emission from the cloud itself. The third term describes the cloud’s absorption of its own emission.

A Couple of Limiting Cases If the cloud is opaque, τ ν >> 1 and I ν = S ν (where S ν is simply j ν / α ν ) If the cloud is ‘optically thin’ (nearly transparent), τ ν << 1 and exp (-τ ν ) ~ 1 – τ ν whence I ν = I νο (1 - τ ν ) + S ν τ ν = I νο (1 - τ ν ) + j ν l In the last of these expressions, note that I νο is slightly diminished (by a factor of 1 - τ ν ), and then we simply add on the emissivity through the entire pathlength of the cloud.

The Most Important Case Case F: an emitting and absorbing cloud that is in LTE. Example: a region in the interior of a star: not full LT, but the temperature gradients are very small. (We considered this earlier.) In this case, the source function is the Planck function (emissions and absorptions are effectively in balance).

But There is a Flow of Radiation So I ν is not constant, and we write with implications to follow

Some of Those Implications Consider as before the two extreme cases: (1) optically thick ( τ ν >> 1), whence I ν = B ν (T) and (2) optically thin (τ ν << 1), whence I ν = I νο (1 - τ ν ) + B ν (T) τ ν = I νο (1 - τ ν ) + j ν l

The First of These Reminds us that an optically thick cloud in LTE emits like a black body. Any background source contributes negligibly to the emission. Example: a star! All we learn from the emission (a Planck curve) is the temperature.

The Second of These I ν = I νο (1 - τ ν ) + B ν (T) τ ν = I νο (1 - τ ν ) + j ν l The observed intensity I ν depends on the background source I νο since it can be seen through the cloud. But the cloud itself also contributes – notice the second term! – in a way that depends both on the temperature (via B ν (T)) and the density (since it determines τ ν )

In Short By looking at the light of a background source, we can probe the temperature and density of the foreground ‘cloud’ (or layer). This is, of course, what facilitates stellar spectroscopy: the light from the stellar interior is transmitted through the low-density (‘transparent’) outer parts of the stellar atmosphere.

Suppose There is No Background Source The equations for the LTE case reduce to I ν = B ν (T) (optically thick, background source irrelevant) and I ν = B ν (T) τ ν (with τ ν << 1) This means that the observed brightnesss temperature places a lower limit on the temperature of the source. Remember that