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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
http://slidepdf.com/reader/full/the-simple-model-fitting-analytic-solutions-of-the-equation-of-transfer-to 1/44
8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds
http://slidepdf.com/reader/full/the-simple-model-fitting-analytic-solutions-of-the-equation-of-transfer-to 2/44
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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8/9/2019 The Simple Model: Fitting analytic solutions of the equation of transfer to observations reveals infall rates for star-forming molecular clouds
http://slidepdf.com/reader/full/the-simple-model-fitting-analytic-solutions-of-the-equation-of-transfer-to 4/44
Astronomical Observations
Astronomers explore the Universe by making three types of
observations:
1. Robotic Probes.
2. Neutrino Detectors.
3. Electromagnetic 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,
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
http://slidepdf.com/reader/full/the-simple-model-fitting-analytic-solutions-of-the-equation-of-transfer-to 6/44
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
http://slidepdf.com/reader/full/the-simple-model-fitting-analytic-solutions-of-the-equation-of-transfer-to 7/44
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.
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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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 .
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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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
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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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.
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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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 .
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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Infalling Molecular Cloud
Hot
Radially Infalling Cloud
Observer
Cool
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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Infalling Cloud: Doppler Shifts
Observer
Radially Infalling Cloud
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
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.
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.
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
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.
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.
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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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)
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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Gradient Methods
Parameter
E r r o r
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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Gradient Methods
Parameter
E r r o r
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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Gradient Methods
Parameter
E r r o r
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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Gradient Methods
Parameter
E r r o r
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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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.
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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Simulated Annealing
Parameter
E r r o r
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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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.
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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Differential Evolution
E r r o r
Parameter
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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Differential Evolution
E r r o r
Parameter
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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Differential Evolution
E r r o r
Parameter
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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Differential Evolution
E r r o r
Parameter
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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Differential Evolution
E r r o r
Parameter
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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A Fit to Data: L1544
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 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?
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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Simulated Fits
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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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
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 Moral of the Story
1. Know your problem.
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 Moral of the Story
1. Know your problem.
2. Look for reasonable simplifications.
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 Moral of the Story
1. Know your problem.
2. Look for reasonable simplifications.
3. Know where these simplifications apply.
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 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.