ExoStat
Wednesday June 18th, 2014
Ian Czekala
Models make predictions of stellar properties as a function of mass and age. This relationship is rather uncertain for young stars.
Determined relative to host star properties, and biases or uncertainties strongly affect habitable zone and $\eta_\bigoplus$ (Kane11)
Submm interferometric observations of stars with circumstellar disks. Use Keplerian rotation to derive a very precise and accurate stellar mass ($< 5$%) (Rosenfeld+12)
There are many ways to determine
$\{ T_{\rm eff}, \log g, [{\rm Fe}/{\rm H}] \}$
Depending on your desired precision and the nature of your target, you can use photometric colors, astroseismology, optical/IR interferometry, or spectroscopy.
By far, spectroscopic inference is the most prevalent and general purpose (e.g., spectral typing).
To leverage the high quality synthetic spectra that now exist, we develop a forward model of the observed stellar spectrum.
Desire 

Problem 

Solution 

Spectroscopic fits are different from "normal" $ \chi^2$ fits
Existing stellar synthesis is amazing, but not perfect.
How can you fit ~1,000 or more lines
with only a few parameters?
or at least very difficult to attain in practicefailureperfection is not an option
the residuals are a spectrum $\textbf{R}$
the pixel "variances" are now a covariance matrix $C$
$$p(D  \theta) = \frac{1}{\sqrt{(2 \pi)^N \det(C)}} \exp\left ( \frac{1}{2} \textbf{R}^T C^{1} \textbf{R} \right ) $$
$$ \ln[p(D  \theta)] = \frac{1}{2} \textbf{R}^T C^{1} \textbf{R}  \frac{1}{2} \ln \det C  \frac{N}{2} \ln 2 \pi $$
the structure of $C$ is set by a kernel function analogous, say, to the twopoint correlation function
Matérn $\nu = 3/2$
$k(\lambda_i, \lambda_j  a, l) = a \left (1 + \frac{\sqrt{3} r_{ij}}{l} \right ) \exp \left (  \frac{\sqrt{3} r_{ij}}{l} \right )$
This covariance matrix accounts for the "global" correlated structure of the residuals
$\sim 5$ pixels correlated
A strong outlier generally looks like a Gaussian $\textbf{R}_\lambda(a, \mu, \sigma) = \frac{a}{\sqrt{2 \pi} \sigma} \exp \left (  \frac{r^2(\lambda,\, \mu)}{2 \sigma^2} \right ) $
covariance kernel is
$k(\lambda_i, \lambda_j) = \langle \textbf{R}_i \textbf{R}_j \rangle$
$k(\lambda_i, \lambda_j  a, \mu, \sigma) = \frac{1}{2 \pi} \left ( \frac{a}{\sigma} \right)^2 \exp \left (  \frac{r^2(\lambda_i,\, \mu) + r^2(\lambda_j,\, \mu)}{2 \sigma^2} \right ) $
(courtesy dfm/triangle.py and dfm/emcee)
M dwarf molecular bands are hard to model
...but, some bands tell you a lot about stellar properties
We want to include these areas but weight them appropriately
Workshop topic: custom design of covariance kernels?
This is not unlike how $UBVRI$ photometric passbands
are regressed out (Bessell & Simon 2012)
Pass this along to the stellar modellers.
If you're impatient/a DIYer, you can create your own selfcalibrated semiempirical libraries of pseudosynthetic spectra.
Slides at iancze.github.io/exostat
Email: iczekala@cfa.harvard.edu