12. Circumstellar Matter Monte Carlo Radiation Transfer I

1. 12. Circumstellar MatterMonte Carlo Radiation Transfer I • Circumstellar material • Monte Carlo “Photons” and interactions • Sampling from probability distributions • Optical depths, isotropic emission, scattering

2. Circumstellar Matter • Show stellar spectrum: detailed RT discussed above • Show images of circumstellar environments • Star formation, reflection nebulae, PNe, HII regions • Typical optical depths ~ 10s • Need 2D and 3D radiation transfer

3. Monte Carlo Basics • Emit energy packet, hereafter a “photon” • Photon travels a distance L • Something happens… • Scattering, absorption, re-emission

4. Photon Packets If the total input luminosity is L, then each photon packet carries energy Ei / Dt = L / N, where N is the number of Monte Carlo photons. A Monte Carlo photon represents Ng real photons, where Ng = Ei / hni. So, a Monte Carlo photon packet’s contribution to the Specific intensity is: Energy packet

5. N • (spectrum) q (phase function) Note, In is a distribution function. We will be working with discrete energies. Binning the photon packets into directions, frequencies, etc, enables us to simulate a distribution function. e.g., spectrum: bin in frequency; scattering phase function: bin in angle

6. Volume = A dl Number density n Cross section s A dl Photon Interactions Energy removed from beam per t / n / dW = Ins Number of photons absorbed/scattered from beam per sec = Ins n A dl Number of photons absorbed/scattered from beam per sec per area = Ins n dl

7. L N segments of length L / N Intensity differential over dl is dIn = - Inns dl. Therefore In (l) = In (0) exp(-nsl) Fraction scattered or absorbed / length = ns ns = volume absorption coefficient = rk Mean free path = 1 / n s = average distance between interactions Probability of interaction over dl is ns dl Probability of traveling dl without interaction is 1 – ns dl Probability of traveling L before interacting is P(L) = (1 – nsl / N) (1 – nsl / N) (1 – nsl / N) … = (1 – nsl / N)N = exp(-nsL) P(L) = exp(-t) t = number of mean free paths over distance L.

8. N exp(-t). t Probability Distribution Function The probability distribution function (PDF) for photons to travel optical depth t before an interaction is exp(-t). If we pick t uniformly over the range 0 to infinity we will not reproduce exp(-t). We want to pick lots of small ts and fewer large ts. Same with a scattering phase function: want to get the correct number of photons scattered into different directions, forward and back scattering, etc.

9. Cumulative Distribution Function Want to randomly choose t, q, l, … so that PDF is reproduced x is a random number uniformly chosen in range [0,1] The above equation is the fundamental principle behind Monte Carlo techniques and is used to sample randomly from PDFs.

10. P(q) F(x) qi xi q 1 x A e.g., P(q) = cos q and we want to map x to q. Choose random qs to “fill in” P(q) If we sample many random qi in this way and “bin” them, we will reproduce the curve P(q) = cos q.

11. Choosing a Random Optical Depth P(t) = exp(-t), i.e., photon travels t before interaction Since x is in range [0,1], then (1-x) is also in range [0,1], so we may write: We find the physical distance, L, that the photon has traveled from:

12. Random Isotropic Direction Solid angle is dW = sin q dq df, we want to choose (q, f) so they fill in PDFs for q and f. P(q) normalized over [0, p], P(f) normalized over [0, 2p]: P(q) = ½ sin qP(f) = 1 / 2p Using fundamental principle from above: Use this formula for emitting photons isotropically from a point source, or for choosing a scattering direction for isotropic scattering.