Inferring Galactic Structure from a Billion Stellar Colors

Pan-STARRS 1

• $grizy$ photometry
• Covers entire sky north of $\delta = -30^{\circ}$

2MASS

• $J H K_s$ photometry
• Full-sky

~800 million stars with high-quality, multi-band photometry

Pixelization

Assume all stars in one pixel trace the same reddening column

Model Parameters

 $\hphantom{\mu_{k} ,\,} \Delta E_{j} ,$ $\ \ j = 1, \ldots , n_{\mathrm{bins}}$ $\mu_{k} ,\,$ $\delta_{k} ,\,$ $\Theta_{k} ,$ $\ \ k = 1, \ldots , n_{\mathrm{stars}}$
\left. \vphantom{ \begin{align} \Delta E_{j} \\ \Theta_{k} \\ \Theta_{k} \end{align} } \right\} \Rightarrow \vec{m}_{k}

Factorizing the Problem

$p \left( \left\{ \Delta E \right\} | \left\{ \vec{m} \right\} \right)$ $\propto p \left( \left\{ \Delta E \right\} \right)$ $\prod_{k}$ $\int$ $\mathrm{d} \mu_{k}$ $\mathrm{d} \Theta_{k}$ $\mathrm{d} \delta_{k}$ $p \left( \vec{m}_{k} \, | \, \mu_{k} , \Theta_{k} , \left\{ \Delta E \right\} , \delta_{k} \right)$
$\times \ p \left( \mu_{k} , \Theta_{k}, \delta_{k} \, | \, \left\{ \Delta E \right\} \right)$

For each star, pre-compute

$\tilde{p} \left( \mu_{k} , E \, | \, \vec{m}_{k} \right) =$ $\int \mathrm{d} \delta_{k}$ $\int \mathrm{d} \Theta_{k}$ $p \left( \mu_{k} , \Theta_{k} , \vphantom{\left( 1 + \delta_{k} \right)} \right.$ $\left( 1 + \delta_{k} \right)$ $\left. E \, | \, \vec{m}_{k} \right)$ $p \left( \delta_{k} \, | \, E \right)$

Then sample the line of sight

$p \left( \left\{ \Delta E \right\} | \left\{ \vec{m} \right\} \right)$ $\propto p \left( \left\{ \Delta E \right\} \right)$ $\prod_{k}$ $\int \mathrm{d} \mu_{k}$ $\tilde{p} \left( \mu_{k} , E \left( \mu_{k} , \left\{ \Delta E \right\} \right) | \, \vec{m}_{k} \right)$
$\hphantom{p \left( \left\{ \Delta E \right\} | \left\{ \vec{m} \right\} \right) \propto p \left( \left\{ \Delta E \right\} \right) \prod_{k} \int \mathrm{d} \mu_{k} \ } \underbrace{\hphantom{ \tilde{p} \left( \mu_{k} , E \left( \mu_{k} , \left\{ \Delta E \right\} \right) | \, \vec{m}_{k} \right) }}_{\mathrm{pre-computed}}$

Line integral through stellar probability densities

Mathematically equivalent to parameterization shown earlier.

Each star puts a constraint on the line-of-sight reddening.

Hundreds of stars per sightline.

50 pc orbit about the Sun

Cross-Sightline Correlations

The cross-sightline differences in the above map are not a priori likely.

Cross-Sightline Correlations

• Should include correlations between dust in neighboring sightlines, assuming some dust density power spectrum.
• Couples stellar reddenings across the sky.
• Since information is shared naturally across sightlines, we could theoretically shrink sightlines arbitrarily without loss of information.
• Computational difficulty of large, coupled problem.