Lecture 4Moving Atmospheres

Formal Solution Along Characteristics1D Gustafsson et al.

Formal Solution Along Characteristics3D Cartoon

Mihalas Method

Want to work in the “Co-moving Frame”The emissivity η and opacity χ depend upon the angle as well as frequencyin the inertial frame because of Doppler shifts, aberration, and advectioninduced by the motion of the material in the frame.The goal of this section is to rewrite equation (2.1) with all material andradiation-field quantities measured in the comoving frame; in that frame boththe opacity and emissivity are isotropic, and can be related directly to propervariables that specify the thermodynamic state of the material. Futhermore,in that frame both the scattering properties of the material and the rateequations describing the mechanisms populating and depopulating itsinternal energy states are most easily defined. . .In our analysis we shall, however, leave both the space and time variables inthe inertial frame, as this is the only frame in which synchronism of clockscan be effected, and further this choice obviates the need to develop a metricfor accelerated fluid frames (Castor 1972) which in general can only be doneapproximately. With this choice of frame we can write exact Lorentztransformations for all the material and radiation-field quantities and usethese to develop a transfer equation that will remain valid for relativistic flowin the limit as v/c −→ 1. . .

Original Approach

Characteristic Equations

drdsM

= γ(µ+ β),

dµdsM

= γ(1− µ2)

[1 + µ

β

r− γ2(µ+ β)

dβdr

].

Characteristics are Curved

Tetrad FormalismCastor (2009)

Tetrad Formalism (cont’d)Castor (2009)

Formal Solution Along CharacteristicsAffine Method (Chen et al. 2007)

sb n

6 61 in out

in out in out1 = onstantR

m1mm+1

The Affine Approach IThe Boltzmann Equation is an integro-differential equation for theinvariant photon distribution function F (x ,p) on the photon’s7-dimensional phase-space (x ,p). This can be thought of as thephoton“on-shell" subspace of a full 8 dimensional particlephase-space.The number of photons ∆N found by observer u(x) in a small6-element ∆Vx ∆P of phase-space at (x ,p) is measured by the 6-formδN, i.e., ∆N = δN(∆Vx ,∆P) where

δN ≡ F (x ,p) δV6, (1)

δ V6 ≡ −(u(x) · p) δ Vx δP. (2)

In the above, u(x) is an arbitrary observer’s unit 4-velocity atspacetime point x ,

−(u(x) · p) = h/λ (3)

is the magnitude of the photon’s 3-momentum as seen by observeru(x), δ Vx is the observer dependent 3-dimensional volume element at

The Affine Approach IIx , and δP is the covariant volume element on the photons’3-dimensional momentum space at (x ,p).The collisionless Boltzmann equation simply states that F [x(ξ),p(ξ)]remains invariant (constant) along the Lagrangian flow of photons inphase-space generated by their geodesic motion in spacetime.Constancy of F [x(ξ),p(ξ)] is a natural consequence of Liouville’stheorem, i.e., δV6 is invariant under this flow, and the constancy of ∆Ndue to the absence of non-gravitational interactions of the photons.Any lack of constancy of ∆N in a finite volume ∆V6 is accounted for bya collision term. To exhibit covariance, the Boltzmann equation withcollisions, is often written as a differential equation

dFdξ

=dxα

dξ∂F∂xα

+dpα

dξ∂F∂pα

=

(dFdξ

)coll

, (4)

with F (x ,p) explicitly given as a function of 8 variables (all 4components of momentum are included but constrained by p · p = 0).The collision term on the R.H.S. is a measure of the rate of change of

The Affine Approach III

the number of photons ∆N in a ∆V6 transported along the would-bepaths of non-interacting photons in phase-space.According to the geometrical optics approximation, photons travel onnull spacetime geodesics independently of their wavelengths. Affineparameters, ξ, unique to each wavelength, can be chosen whichgenerate the following orbits on phase-space:

dxα

dξ= pα, (5)

dpα

dξ= −Γα

βγpβpγ , (6)

which reduces (4) to

pα ∂F∂xα

− Γαβγpβpγ ∂F

∂pα=

(dFdξ

)coll

. (7)

The Affine Approach IVThe R.H.S. is typically separated into absorption and emission terms(

dFdξ

)coll

= −fF + g, (8)

where f (x ,p) and g(x ,p) are identified respectively with the invariantabsorptivity and emissivityThe radiation transport equation is an integro-differential equation forthe specific intensity I(x ,p) which is equivalent to the Boltzmannequation (4) or (7) for F (x ,p). Both are functions on the photon’s7-dimensional phase-space (x ,p); however, I(x ,p) depends on achoice of an observer at each point of spacetime through

Iλ(x ,p) = −c2

h(u(x) · p)5F (x ,p). (9)

We have chosen to follow our group’s convention and use λ ratherthan ν as often appears in the literature; however, it is straightforward

The Affine Approach V

to change between Iλ and Iν using λIλ = νIν . Once the observers arechosen, Iλ(x ,p) like F (x ,p), is a scalar. Defining a set of observers isequivalent to giving a unit time-like vector field on spacetime, u(x),which appears in Eq. (9). Just as in equation (3), −u(x) · p is equal tothe the photon’s momentum as seen by observer u(x). If u(x)describes the material fluid with which the photons interact, Iλ is calledthe comoving specific intensity and λ the comoving wavelength.The transport equation for Iλ(x ,p) is obtained from equations (4) and(8) by substituting Iλ(x ,p) for F (x ,p) using Eq. (9)

dIλdξ

= −(χλhλ

+5λ

dλdξ

)Iλ + ηλhλ, (10)

The Affine Approach VIwhere the observer dependent absorptivity χλ and emissivity ηλ arerelated to f and g by

χλ = − f(u(x) · p)

,

ηλ =

(c2

h

)(u(x) · p)4 g. (11)

The other term (∝ dλ/dξ) on the right in Eq. (10) is present becausethe definition of Iλ, Eq. (9), explicitly depends on comoving λ. If as iscustomary we divide the extinction into two parts: “true absorption” κλ

and “scattering” σλ, then χλ = κλ + σλ. For a comoving observer wewill also assume the emissivity is given by thermal emission (trueabsorption opacity κλ times the Planck function Bλ) and that scatteringis elastic and isotropic. For a comoving observer, χλ depends only onthe magnitude of the momentum and not its direction (given isotropicsources), and consequently is a function of only x and u · p in anarbitrary coordinate system.

The Affine Approach VII

If the energy density in the radiation field is written as ελ = 4πJλ/c theemissivity becomes

ηλ = κλBλ + σλJλ. (12)

Applied Affine Method IUsing the general formalism developed in Chen et al. we can derivethe transfer equation in flat spacetime with arbitrary flows. We chooseto work in spherical coordinates without loss of generality.The photon’s four-momentum can be written

pa ≡ dxa

dξ=

hλ∞

(1, n), (13)

where h is the Planck constant, ξ is the affine parameter, λ∞ is the restframe wavelength, and n is the 3-D direction of the photon as seen bya distant stationary observer. The four-velocity of the co-movingobserver in an arbitrary flow can be written

ua = γ(r, t)[1, ~β(r, t)], (14)

and the co-moving wavelength λ can be obtained using

hλ

= −(u · p). (15)

Applied Affine Method IIThe 3-D geodesic in the flat spacetime can be parametrized as

r(s) = r0 + n s, (16)

where r0 is the starting point of the characteristics, and s is the restframe physical distance related to the affine parameter ξ by

s ≡ hλ∞

ξ. (17)

The radiative transfer equation can be written in terms of the affineparameter ξ as (see Eq. (10) of Chen et al.):

∂Iλ∂ξ

∣∣∣∣λ

+dλdξ

∂Iλ∂λ

= −(χλhλ

+5λ

dλdξ

)Iλ + ηλhλ, (18)

where Iλ(r, t ; n) is the specific intensity measured in a frame wherewavelengths are measured by a co-moving observer and all otherquantities are measured by an observer at infinity.

Applied Affine Method III

We can rewrite Eq. 18 as

d(ct)dξ

1c∂Iλ∂t

∣∣∣∣λ

+d~rdξ· ∇Iλ +

dλdξ

∂Iλ∂λ

= −(χλhλ

+5λ

dλdξ

)Iλ + ηλhλ, (19)

The radiative transfer equation can be written in terms of physicaldistance parameter s instead of the affine parameter ξ as (see Eq. (18)of Chen et al.):

d(ct)dξ

1c∂Iλ∂t

∣∣∣∣λ

+dsdξ

d~rds· ∇Iλ +

dsdξ

dλds

∂Iλ∂λ

= −(χλhλ

+5λ

dsdξ

dλds

)Iλ + ηλhλ,

(20)Then using the definition of s from Eq. 17 and the fact that

d(ct)dξ

= cpt = h/λ∞

Applied Affine Method IV

we find

1c∂Iλ∂t

∣∣∣∣λ

+∂Iλ∂s

∣∣∣∣λ

+dλds

∂Iλ∂λ

= −(χλλ∞λ

+5λ

dλds

)Iλ + ηλλ∞λ. (21)

Eq. 21 can be put into our standard form:

1c∂Iλ∂t

∣∣∣∣λ

+∂Iλ∂s

+ a(s)∂

∂λ(λIλ) + 4a(s)Iλ = −χλf (s)Iλ + ηλf (s), (22)

wheref (s) ≡ λ∞

λ= γ(r, t)[1− n · ~β(r, t)] (23)

is simply the Doppler factor, and

a(s) ≡ 1λ

dλds. (24)

Applied Affine Method VUsing Eqs. (16) and (23), a(s) is found to be

a(s) =1

1− n · ~β

[dds

(n · ~β)− γ2β(1− n · ~β)dβds

](25)

where β is the magnitude of ~β, and

dds

=1c∂

∂t+ n · ∇ =

1c∂

∂t+

∂

∂s(26)

When we numerically integrate the radiation transfer equation, a(s)can be approximated as

a(s) ≈ δ(n · ~β)− γ2β(1− n · ~β) δβ

δs(1− n · ~β), (27)

where δs is the differential step size (physical distance) along thecharacteristics, δ(n · ~β) and δβ are respectively the changes of n · ~β

Applied Affine Method VIand β when we move one step forward which includes the changesinduced by both time and spatial advances, say

δβ = β(si+1, ti+1)− β(si , ti). (28)

Since few numerical schemes will be able to provide the fully implicitderivative δβ will often be obtained for example by the backwarddifference

δβ =1c

(β(si , ti)− β(si , ti−1) + (β(si , ti)− β(si−1, ti). (29)

√

In the stationary case, both β and f (s) are independent of time andspecializing Eqs. (16) and (23) to that case, a(s) becomes:

a(s) =1

1− n · ~β

[dds

(n · ~β)− γ2β(1− n · ~β)dβds

]

Applied Affine Method VII

=n · ∇(n · ~β)− γ2β(1− n · ~β)(n · ∇β)

1− n · ~β= −(n · ∇) ln(1− n · ~β)− γ2β(n · ∇β), (30)

where β is the magnitude of ~β, and we have used the fact that alongthe characteristics, d/ds no longer contains the time derivative and isthus directional derivative operator, i.e., , d/ds = n · ∇. When wenumerically integrate the radiation transfer equation, a(s) can beapproximated as

a(s) ≈ δ(n · ~β)− γ2β(1− n · ~β) δβ

δs(1− n · ~β), (31)

where δs is the differential step size along the characteristics, δ(n · ~β)and δβ are respectively the changes of n · ~β and β when we move onestep forward.

Applied Affine Method VIIIIn terms of its spherical components, ~β can be written

~β = βr er + βθeθ + βφeφ, (32)

where er , eθ, eφ are the spherical orthonormal basis vectors at pointr(r , θ, φ), i.e.,

er = (sin θ cosφ, sin θ sinφ, cos θ),eθ = (cos θ cosφ, cos θ sinφ,− sin θ),eφ = (− sinφ, cosφ,0), (33)

and consequently

n · ~β = βr n · er + βθn · eθ + βφn · eφ

≡ βr nr + βθnθ + βφnφ (34)

(note that along the characteristics, n has constant Cartesiancomponents, nx , ny , nz , but changing spherical components,

Applied Affine Method IXnr , nθ, nφ). Writing n = (nx ,ny ,nz), the spherical componentsnr , nθ,nφ can be easily computed using Eq. (33).At first glance comparing Eq 22 with Eq. 2.12 of (Mihalas 1980)something seems amiss. Both we and Mihalas (1980) work in theframe where spatial coordinates and clocks are measured by anobserver at rest. However, Mihalas’ time derivative contains a Dopplerfactor, whereas ours does not. Also, our terms on the right hand sidecontain Doppler factors, f (s), whereas those of Mihalas do not. Thediscrepency has been noted in passing by us and is due to the factthat our s is a true distance measured in the Observer’s frame,whereas that of Mihalas, sM , contains an extra Doppler factor:

sM =λ∞λ

s.

Thus, we can transform from s to sM in Eq. 21 to find

λ

λ∞

1c∂Iλ∂t

∣∣∣∣λ

+∂Iλ∂sM

∣∣∣∣λ

+dλdsM

∂Iλ∂λ

= −(χλ +5λ

dλdsM

)Iλ + ηλ, (35)

Applied Affine Method X

which is very similar to the equation of Mihalas, except that thecoefficient multiplying the time derivative term is the inverse Dopplerfactor f (s)−1, since we are working with Iλ instead of Iν as doesMihalas.

Comparison with Mihalas ITo derive Mihalas Eq. 2.12, starting from the Initial frame transferequation 2.1

1c∂I(µ, ν)

∂t+ µ

∂I(µ, ν)

∂r+

1− µ2

r∂I(µ, ν)

∂µ= η − χI(µ, ν). (36)

Note this equation can be easily derived using

drds

= µ,dµds

=dds

(n · r

r

)=

1− µ2

r. (37)

Applying Eq. 2.2,2.3, 2.4 of Mihalas to change initial I, η, χ tocomoving I0, η0, χ0. Then change phase space coordinates from(t , r , µ, ν) to (t0 = t , r0 = r , µ0, ν0).

t0 = tr0 = r

µ0 =µ− β

1− µβ

Comparison with Mihalas IIν0 = νγ(1− βµ) (38)

note that β, and γ are functions of r , t .Work out the Jacobian Matrix

J =∂(t0, r0, µ0, ν0)

∂(t , r , µ, ν), (39)

and change the coordinate vectors using(∂

∂xi

)= JT

(∂

∂x0i

). (40)

The derivations are just straightforward algebra.To derive Milalas’ result using our affine method, starting from (ourinertial/comoving notation is exactly the opposite of that of Mihalas)

dFdξ

= −fF + g, (41)

Comparison with Mihalas III

using

χλ =fν, ηλ = ν4g, χλ = χν , ηλ = ν2ην . (42)

we obtain transfer equation in affine parameter

dIνdξ− 3ν

dνdξ

Iν = ν[−χν Iν + ην ] (43)

Changing from affine parameter to physical distance

ν0

ν

(dIνds− 3ν

dνds

Iν

)= −χν Iν + ην (44)

or

ν0

ν

(1c∂Iν∂t

+ µ0∂Iν∂r

+dµds

∂Iν∂µ

+dνds

∂Iν∂ν− 3ν

dνds

Iν

)= −χν Iν + ην (45)

Comparison with Mihalas IV

Using

µ(s) =µ0 − β1− βµ0

, ν(s) = ν0γ(1− βµ0) (46)

and

dµds

=∂µ

∂µ0

dµ0

ds+∂µ

∂β

dβds, µ0 = n·r, dµ0

ds=

1− µ20

r,

dβds

=1c∂β

∂t+µ0

∂β

∂r, µ0 =

µ+ β

1 + βµ,

(47)we can easily work out the same transfer equation (2.12) in Mihalas.

Formal Solution Along CharacteristicsAffine Method (Baron, Hauschildt, Chen 2009)

∂Iλ∂s|λ + a(s)λ

∂Iλ∂λ

= −[χλf (s) + 5a(s)]Iλ + ηλf (s). (48)

Funny Framer and n in Observer’s Frameλ is measured by a comoving observer

Scattering Problem in Co-moving Frame

Need Jλ in co-moving frame (in momentum space)So just integrate RTE equation over dΩ (where dΩ is measured by acomoving observer), but since Iλ is known in the funny frame:

Use Chain Rule

dΩ = (γ[1− β · n])−2dΩ0

= f (s)−2dΩ0.

Jλ = ΛSλ