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8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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November 17, 2006Mathematics and Computer Science Seminar

The Simple Model:

Fitting analytic solutions of the equation of transferto observations reveals infall rates

for star-forming molecular clouds

Christopher H. De Vries

Philip C. Myers

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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Astronomical Observations

Astronomers explore the Universe by making three types of

observations:

1. Robotic Probes.

2. Neutrino Detectors.

3. Electromagnetic Radiation.

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Electromagnetic Radiation

This talk will focus on analysis of radiation from astronomical

objects. Specifically the millimeter and sub-millimeter emissionfrom molecular gas clouds.In order the interpret what we see, we need to understand

1. what causes the radiation,

2. what happens to the radiation as it travels,

3. and what effect our detectors have on that radiation.

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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Electromagnetic Radiation

This talk will focus on analysis of radiation from astronomical

objects. Specifically the millimeter and sub-millimeter emissionfrom molecular gas clouds.In order the interpret what we see, we need to understand

1. what causes the radiation, (Signal)

2. what happens to the radiation as it travels, (Transfer)

3. and what effect our detectors have on that radiation.

(Noise)

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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Electromagnetic Radiation

This talk will focus on analysis of radiation from astronomical

objects. Specifically the millimeter and sub-millimeter emissionfrom molecular gas clouds.In order the interpret what we see, we need to understand

1. what causes the radiation, (Signal)

2. what happens to the radiation as it travels, (Trans-fer)

3. and what effect our detectors have on that radiation.(Noise)

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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Specific Intensity

In radiative transfer we measure the change in specific intensity

or brightness of radiation. Specific intensity (I ν) is defined asthe amount of energy passing through a small area in a smallrange of directions at a small range of frequencies in a smalltime.

I ν =dE

dAdtdΩ dν

The units of specific intensity (in cgs) are therefore erg cm−2

s−1 ster−1 Hz−1.

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Radiative Transfer

Radiative transfer is huge topic which I cannot cover in detail,

but it is described by a very modest differential equation calledThe Equation of Transfer .

dI ν

ds = −ανI ν + jν

s Distance (cm)

αν Absorption Coefficient (cm−1)

jν Emission Coefficient (erg cm−3 s−1 ster−1 Hz−1)

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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Optical Depth

In a purely absorbing medium with the equation of transfer is

easy to solve, and depends only on the integral of the absorptioncoefficient along the radiation’s path.

dI nu

ds

= −ανI ν

I ν(s) = I ν(s0)exp

−

s

s0

aν(s)ds

The intensity decays exponentially as it travels through an ab-sorbing medium. We define the optical depth (τ ) as the integralof aν along the path. Using τ we can restate the equation of transfer as

dI ν

dτ = −I ν + S ν .

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Radio Astronomers and Temperature

Radio astronomers are peculiar in that we assign a temperature

to everything (even when that temperature has no real thermalmeaning).

• Brightness Temperature (T B) — Proportional to specificintensity.

• Excitation Temperature (T ex)— Proportional to thesource function.

I will slow a lot of observations where “temperature” is the unitof the observation, but these will be intensities or brightness of radiation.

dT B

dτ = −T B + T ex

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Molecular Emission and Absorption

What do we look at to observe “dark” molecular clouds? Emis-

sion from molecules within those clouds.

• At narrow bandwidths (spectral lines)

• At radio (millimeter and submillimeter) wavelengths

• Caused by quantum mechanical processes (rotationaltransitions).

• Specific lines caused by specific molecules.

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Star Formation and Infalling Clouds

Molecular clouds are huge, stretching up to 100 parsecs in size.

They are also incredibly diffuse with a density lower than thebest vacuum achievable on Earth.

Stars have an average density greater than water, with ex-

tremely high density in the core. They are also more thanone million times smaller than a small molecular cloud core.

In order to form stars clouds must undergo a phase of massive

collapse .

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Infalling Molecular Cloud

Hot

Radially Infalling Cloud

Observer

Cool

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Infalling Cloud: Doppler Shifts

Observer

Radially Infalling Cloud

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The Asymmetric Infall Profile

Observer

Radially Infalling Cloud

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The Asymmetric Infall Profile

Observer

Radially Infalling Cloud

1. There must be a rising ex-citation temperature gradi-

ent along the line of sight.

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The Asymmetric Infall Profile

Observer

Radially Infalling Cloud

1. There must be a rising ex-citation temperature gradi-

ent along the line of sight.

2. There must be a velocitygradient along the line of sight.

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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The Asymmetric Infall Profile

Observer

Radially Infalling Cloud

1. There must be a rising ex-citation temperature gradi-

ent along the line of sight.

2. There must be a velocitygradient along the line of sight.

3. The line must be opticallythick.

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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Traditional Modeling of Infall

1. Choose a hydrodynamic simulation which includes rele-

vant physical processes.

2. Simulate the radiative processes and thermodynamicswithin the cloud.

3. Assume a chemistry model for the cloud.

4. Model the radiative emissions of the cloud.

Remarkably, nearly all these models predict the excitation tem-perature depends linearly on optical depth!

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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The Simple Model

We exploit this relationship by building simple models of col-

lapsing clouds.

1. The excitation temperature rises linearly to a peak andthen falls with the same slope as a function of opticaldepth.

2. We assume some uniform rate of infall over the entirecloud.

8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds

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The Simple Model

Τex

Tpk

τ

f

τ

r

Tbg

τ

cvc v

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The Simple Model

We exploit this relationship by building simple models of col-

lapsing clouds.

1. The excitation temperature rises linearly to a peak andthen falls with the same slope as a function of opticaldepth.

2. We assume some uniform rate of infall over the entirecloud.

The equation of transfer is integrable in this case and simula-tions can by calculated in second by a computer rather thanhours or weeks taking the traditional approach. This allows usto consider fitting our models to real observations.

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Fitting a Model to Observations

The key of fitting is to numerically minimize the difference

between the model predictions and observations by changingparameters of the models. This is often referred to as χ2-minimization.

Minimization is an iterative process. You must

1. choose parameters,

2. calculate a result,

3. and compare that result with observations.

4. (repeat as necessary)

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Gradient Methods

Parameter

E r r o r

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Gradient Methods

Parameter

E r r o r

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Gradient Methods

Parameter

E r r o r

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Gradient Methods

Parameter

E r r o r

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Simulated Annealing

The standard solution to ending up in a local minimum is to

use a process called simulated annealing.

• Mirrors process of annealing or controlled cooling to cre-ate crystals.

• Although you tend to follow the gradient down and re-duce error, you allow a probability of going upwards andincreasing error.

•

As time goes on the probability of taking a step upwardsdecreases.

• If probability is reduced at the right rate, you will end upin the global minimum.

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Simulated Annealing

Parameter

E r r o r

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Differential Evolution

Simulated annealing requires careful control of the probability

of an upward step during the simulation. Differential evolution(Storn & Price 1997) is self-regulating with fewer free param-eters.

• Start with a population of solutions.

• Allow the solutions to vary by the differences betweenthem (self-scaling).

• Keep good solutions and throw out the bad solutions ineach generation.

• Repeat until you believe convergence is reached.

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Differential Evolution

E r r o r

Parameter

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Differential Evolution

E r r o r

Parameter

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Differential Evolution

E r r o r

Parameter

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Differential Evolution

E r r o r

Parameter

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Differential Evolution

E r r o r

Parameter

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A Fit to Data: L1544

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The Correct Solution

So far we have:

• A model that is easy to calculate

• A method for fitting that model to data

How do we know if our best fit model is right? Is a modelcorrect merely because it explains our observations? Not nec-essarily. We must continue to scrutinize it.

Does our model reproduce the features of hydrodynamicallysimulated clouds?

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Simulated Fits

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Actual Data

Since this model is easy to

use and runs quickly (and Iput the code online) observa-tional astronomers have be-gun using this model to fit

their asymmetric line profilesand derive infall velocities andother physical parameters fortheir observations.

The fits are good, but do theyreally tell us anything aboutthe physics of the cloud?

Williams, Lee, & Myers 2006

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The Moral of the Story

1. Know your problem.

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The Moral of the Story

1. Know your problem.

2. Look for reasonable simplifications.

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The Moral of the Story

1. Know your problem.

2. Look for reasonable simplifications.

3. Know where these simplifications apply.

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The Moral of the Story

1. Know your problem.

2. Look for reasonable simplifications.

3. Know where these simplifications apply.

4. Convince people you are right.