Mean Intensity. Same units as I ν : J/m 2 /s/hz/sr (ergs/cm 2 /s/hz/sr) Function of position (and time), but not direction

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1 MCRT: L5 MCRT estimators for mean intensity, absorbed radiation, radiation pressure, etc Path length sampling: mean intensity, fluence rate, number of absorptions Random number generators

2 J Mean Intensity I dω 2π 4π 4π 0 0 π I sinθ dθ dφ Same units as I : J/m 2 /s/hz/sr (ergs/cm 2 /s/hz/sr) Function of position (and time), but not direction Determines heating, ionization, level populations, etc s I (r, n) dω θ da

3 R * θ * I * r What is J at r from a star with uniform specific intensity I * across its surface? I I * for 0 < θ < θ * (µ * < µ < ); µ cos θ I 0 for θ > θ * (µ < µ * ) J 2 µ * I dµ 2 I * ( µ * ) J I ( * 2 R 2 * / r 2 ) w I * w dilution factor Large r, w R 2 /4r 2

4 Monochromatic Flux Energy passing through a surface. Units: J/s/m 2 /Hz df I cos θ dω F I cosθ dω 2π 0 π 0 I cosθ sinθ dθ dφ If I is isotropic thermal radiation, get: F π I π B

5 Momentum Flux The momentum of a photon is E/c Momentum flux p in the direction of n photon flux times momentum per photon: p (n) / c I cos 2 θ dω One factor of cos θ comes from foreshortening of da Only the normal component of the momentum acts on the surface, hence the second factor of cos θ Units: N/m 2 /Hz

6 Energy density of radiation Consider a cylinder along a ray of length c dt. Define: u (Ω) energy per unit solid angle per unit volume per unit frequency in the cylinder: de u (Ω) dv dω dv u (Ω) (da c dt) dω dv All this radiation will exit the cylinder through da in time dt, so: de I da dω dt dv Equating gives: u (Ω) I / c Integrating over angles, we obtain the specific energy density, u (units J/m 3 /Hz). This is the energy per unit volume per unit frequency interval, u u (Ω) d Ω ( / c) I dω ( 4π/ c) J The total energy density of radiation requires one more integration over frequencies (this has dimensions of Energy / Volume): u u dv (4 π / c) J dv

7 Moments of the Radiation Field First three moments of specific intensity are named J (zeroth moment), H (first), and K (second): Ω Ω Ω d cos 4 d cos 4 d 4 2 θ π θ π π I K I H I J Physically: J mean intensity; H F / 4π K related to momentum flux: π K c p 4

8 Intensity Moments The moments of the radiation field are: J I dω dω 2 H I µ K I µ dω 4π 4π 4π J mean intensity; H flux; K momentum flux Compute these moments throughout the slab. First split the slab into layers, then tally number of packets, weighted by powers of their direction cosines to obtain J, H, K. Contribution to specific intensity from a single packet is: ΔI ΔE µ ΔAΔt Δ ΔΩ F µ N ΔΩ π B µ N ΔΩ

9 Substitute into intensity moment equations and convert the integral to a summation to get: J B 4 N i µ i H B 4 N i µ i µ i K B 4 N i µ i 2 µ i Note the mean flux, H, is just the net energy passing each level: number of packets traveling up minus number traveling down. Pathlength formula (Lucy 999) Long history of use in neutronics J i L 4π N ΔV i l

10 J i L 4π N ΔV i l Some Monte Carlo photon packets may pass through a cell without interacting (scatter or absorbed), but the path length estimator ensures they still contribute to the estimates for mean intensity, absorbed energy, radiation pressure, etc

$More packets pass through a cell than interact with a cell Mean intensity, J, related to photon energy density, u, via Packet contributes a fraction ε t/δt to the energy density of a cell where t l/c$

11 Summing path lengths gives better estimates for intensities, absorbed energy, radiation pressure, etc. More packets pass through a cell than interact with a cell Mean intensity, J, related to photon energy density, u, via Packet contributes a fraction ε t/δt to the energy density of a cell where t l/c is time the packet spends in a cell, so can form Monte Carlo estimator: u Where ε MC packet energy L Δt / N. Hence, get estimator for J which will be accurate in optically thin regions: J i u 4 π J /c c Δt ΔV i L 4π N ΔV i ε l l

12 How much energy absorbed in a cell? Could count number of absorption events in each cell, but this is inaccurate for optically thin systems. We know the change in intensity for radiation passing through a medium with absorbing particles is di - I n σ abs dl - I dτ abs Hence, a Monte Carlo estimator for absorbed energy: E i abs L 4π N ΔV i nσ abs l

13 Random Number Generators Want random numbers in range 0 < ξ < Generate sequence of numbers rapidly No patterns or correlations Pass statistical tests for randomness Because using computer algorithms the sequence will be periodic, so period should be as long as possible pseudo random numbers Anyone who considers arithmetical methods for producing random digits is, of course, in a state of sin. John von Neumann

14 Random Number Generators Pseudo random number generators PRNGs ξ ran(seed), seed usually an integer, updated in each call to ran Can easily de-bug Monte Carlo codes: if use same seed for PRNG will get same random sequence Use uniformly distributed random numbers from (0 < ξ < ) to sample from complex functions Much research devoted to PRNGs Computer modular arithmetic, remainders and bits

15 Middle Square Method Used by von Neumann in 940s Recursive relation: square a n-digit number and take the middle n digits (add zeroes to make a 2n digit number if necessary), repeat process e.g., n 4: gives 5946, ξ gives 3549, ξ gives 5954, ξ etc, etc Problems: short period, can get stuck in very short loops, or crash (e.g., 0000) Von Neumann acknowledged problems, but found this fast (940s), adequate for problems, and crashes were obvious

16 Linear Congruential Generators Based on integer recurrence relation: y i+ mod(a y i + c, M) mod(a,b) gives remainder when A divided by B Careful choice of a, c, M gives periods around 2 32 (0 9 ) Random number in range (0,) from: ξ i y i M ran2.f from Numerical Recipes is more complex, involving shuffling of sequences. Authors offered prize of $000 if anyone finds a statistical test that ran2 fails.

17 Good and Bad Random Number Generators y i+ mod(a y i + c, M ) a 366, c 50889, M a 37, c 87, M 256 ξ i y i M First return maps plot successive pairs of (ξ i, ξ i+ ) ξ i+ ξ i xn+ Xn x n ξ i X n (a) (b) Figure 4.4. First return maps for (a) Random Number Generator #; (b) Random Number Generator #2. ξ i

18 Should we worry? Maybe but if you read you ll get scared! OK for our needs, need better for cryptography Mersenne Twister: period ~ 4.3 E600

Monte Carlo Radiation Transfer I

Monte Carlo Radiation Transfer I Monte Carlo Photons and interactions Sampling from probability distributions Optical depths, isotropic emission, scattering Monte Carlo Basics Emit energy packet, hereafter