# THE PANCHROMATIC HUBBLE ANDROMEDA TREASURY. IV. A PROBABILISTIC APPROACH TO INFERRING THE HIGH-MASS STELLAR INITIAL MASS FUNCTION AND OTHER POWER-LAW FUNCTIONS

## Abstract

We present a probabilistic approach for inferring the parameters of the present-day power-law stellar mass function (MF) of a resolved young star cluster. This technique (1) fully exploits the information content of a given data set; (2) can account for observational uncertainties in a straightforward way; (3) assigns meaningful uncertainties to the inferred parameters; (4) avoids the pitfalls associated with binning data; and (5) can be applied to virtually any resolved young cluster, laying the groundwork for a systematic study of the high-mass stellar MF (M {approx}> 1 M {sub Sun }). Using simulated clusters and Markov Chain Monte Carlo sampling of the probability distribution functions, we show that estimates of the MF slope, {alpha}, are unbiased and that the uncertainty, {Delta}{alpha}, depends primarily on the number of observed stars and on the range of stellar masses they span, assuming that the uncertainties on individual masses and the completeness are both well characterized. Using idealized mock data, we compute the theoretical precision, i.e., lower limits, on {alpha}, and provide an analytic approximation for {Delta}{alpha} as a function of the observed number of stars and mass range. Comparison with literature studies shows that {approx}3/4 of quoted uncertainties are smaller than themore »

- Authors:

- Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195 (United States)
- Center for Cosmology and Particle Physics, New York University, 4 Washington Place, New York, NY 10003 (United States)
- Max Planck Institute for Astronomy, Koenigstuhl 17, D-69117 Heidelberg (Germany)
- Raytheon Company, 1151 East Hermans Road, Tucson, AZ 85756 (United States)
- Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 (United States)
- Department of Astronomy, University of Michigan, 500 Church Street, Ann Arbor, MI 48109 (United States)
- Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218 (United States)
- Minnesota Institute for Astrophysics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455 (United States)

- Publication Date:

- OSTI Identifier:
- 22167243

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Astrophysical Journal; Journal Volume: 762; Journal Issue: 2; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ACCURACY; APPROXIMATIONS; ASTRONOMY; ASTROPHYSICS; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; DISTRIBUTION FUNCTIONS; GALAXIES; GIANT STARS; LUMINOSITY; MARKOV PROCESS; MASS; MONTE CARLO METHOD; PROBABILISTIC ESTIMATION; PROBABILITY; STAR CLUSTERS

### Citation Formats

```
Weisz, Daniel R., Fouesneau, Morgan, Dalcanton, Julianne J., Clifton Johnson, L., Beerman, Lori C., Williams, Benjamin F., Hogg, David W., Foreman-Mackey, Daniel T., Rix, Hans-Walter, Gouliermis, Dimitrios, Dolphin, Andrew E., Lang, Dustin, Bell, Eric F., Gordon, Karl D., Kalirai, Jason S., and Skillman, Evan D., E-mail: dweisz@astro.washington.edu.
```*THE PANCHROMATIC HUBBLE ANDROMEDA TREASURY. IV. A PROBABILISTIC APPROACH TO INFERRING THE HIGH-MASS STELLAR INITIAL MASS FUNCTION AND OTHER POWER-LAW FUNCTIONS*. United States: N. p., 2013.
Web. doi:10.1088/0004-637X/762/2/123.

```
Weisz, Daniel R., Fouesneau, Morgan, Dalcanton, Julianne J., Clifton Johnson, L., Beerman, Lori C., Williams, Benjamin F., Hogg, David W., Foreman-Mackey, Daniel T., Rix, Hans-Walter, Gouliermis, Dimitrios, Dolphin, Andrew E., Lang, Dustin, Bell, Eric F., Gordon, Karl D., Kalirai, Jason S., & Skillman, Evan D., E-mail: dweisz@astro.washington.edu.
```*THE PANCHROMATIC HUBBLE ANDROMEDA TREASURY. IV. A PROBABILISTIC APPROACH TO INFERRING THE HIGH-MASS STELLAR INITIAL MASS FUNCTION AND OTHER POWER-LAW FUNCTIONS*. United States. doi:10.1088/0004-637X/762/2/123.

```
Weisz, Daniel R., Fouesneau, Morgan, Dalcanton, Julianne J., Clifton Johnson, L., Beerman, Lori C., Williams, Benjamin F., Hogg, David W., Foreman-Mackey, Daniel T., Rix, Hans-Walter, Gouliermis, Dimitrios, Dolphin, Andrew E., Lang, Dustin, Bell, Eric F., Gordon, Karl D., Kalirai, Jason S., and Skillman, Evan D., E-mail: dweisz@astro.washington.edu. Thu .
"THE PANCHROMATIC HUBBLE ANDROMEDA TREASURY. IV. A PROBABILISTIC APPROACH TO INFERRING THE HIGH-MASS STELLAR INITIAL MASS FUNCTION AND OTHER POWER-LAW FUNCTIONS". United States. doi:10.1088/0004-637X/762/2/123.
```

```
@article{osti_22167243,
```

title = {THE PANCHROMATIC HUBBLE ANDROMEDA TREASURY. IV. A PROBABILISTIC APPROACH TO INFERRING THE HIGH-MASS STELLAR INITIAL MASS FUNCTION AND OTHER POWER-LAW FUNCTIONS},

author = {Weisz, Daniel R. and Fouesneau, Morgan and Dalcanton, Julianne J. and Clifton Johnson, L. and Beerman, Lori C. and Williams, Benjamin F. and Hogg, David W. and Foreman-Mackey, Daniel T. and Rix, Hans-Walter and Gouliermis, Dimitrios and Dolphin, Andrew E. and Lang, Dustin and Bell, Eric F. and Gordon, Karl D. and Kalirai, Jason S. and Skillman, Evan D., E-mail: dweisz@astro.washington.edu},

abstractNote = {We present a probabilistic approach for inferring the parameters of the present-day power-law stellar mass function (MF) of a resolved young star cluster. This technique (1) fully exploits the information content of a given data set; (2) can account for observational uncertainties in a straightforward way; (3) assigns meaningful uncertainties to the inferred parameters; (4) avoids the pitfalls associated with binning data; and (5) can be applied to virtually any resolved young cluster, laying the groundwork for a systematic study of the high-mass stellar MF (M {approx}> 1 M {sub Sun }). Using simulated clusters and Markov Chain Monte Carlo sampling of the probability distribution functions, we show that estimates of the MF slope, {alpha}, are unbiased and that the uncertainty, {Delta}{alpha}, depends primarily on the number of observed stars and on the range of stellar masses they span, assuming that the uncertainties on individual masses and the completeness are both well characterized. Using idealized mock data, we compute the theoretical precision, i.e., lower limits, on {alpha}, and provide an analytic approximation for {Delta}{alpha} as a function of the observed number of stars and mass range. Comparison with literature studies shows that {approx}3/4 of quoted uncertainties are smaller than the theoretical lower limit. By correcting these uncertainties to the theoretical lower limits, we find that the literature studies yield ({alpha}) = 2.46, with a 1{sigma} dispersion of 0.35 dex. We verify that it is impossible for a power-law MF to obtain meaningful constraints on the upper mass limit of the initial mass function, beyond the lower bound of the most massive star actually observed. We show that avoiding substantial biases in the MF slope requires (1) including the MF as a prior when deriving individual stellar mass estimates, (2) modeling the uncertainties in the individual stellar masses, and (3) fully characterizing and then explicitly modeling the completeness for stars of a given mass. The precision on MF slope recovery in this paper are lower limits, as we do not explicitly consider all possible sources of uncertainty, including dynamical effects (e.g., mass segregation), unresolved binaries, and non-coeval populations. We briefly discuss how each of these effects can be incorporated into extensions of the present framework. Finally, we emphasize that the technique and lessons learned are applicable to more general problems involving power-law fitting.},

doi = {10.1088/0004-637X/762/2/123},

journal = {Astrophysical Journal},

number = 2,

volume = 762,

place = {United States},

year = {Thu Jan 10 00:00:00 EST 2013},

month = {Thu Jan 10 00:00:00 EST 2013}

}