Toward a Galactic Distribution of Planets. I. Methodology & Planet Sensitivities of the 2015 High-Cadence Spitzer Microlens Sample

TL;DR

This paper analyzes microlensing events from the 2015 Spitzer campaign to measure planet sensitivities and develop a methodology for studying the Galactic distribution of planets, finding most lenses are M dwarfs and setting upper limits on planet occurrence.

Contribution

It introduces a new methodology to statistically study the Galactic distribution of planets using microlensing parallax measurements from combined Spitzer and ground observations.

Findings

Majority of microlenses are M dwarfs.

95% upper limit of 49% on stars hosting typical microlensing planets.

Predicted that about one-third of planet detections are in the Galactic bulge.

Abstract

We analyze an ensemble of microlensing events from the 2015 Spitzer microlensing campaign, all of which were densely monitored by ground-based high-cadence survey teams. The simultaneous observations from Spitzer and the ground yield measurements of the microlensing parallax vector $\pi_{\rm E}$, from which compact constraints on the microlens properties are derived, including $\lesssim$25\% uncertainties on the lens mass and distance. With the current sample, we demonstrate that the majority of microlenses are indeed in the mass range of M dwarfs. The planet sensitivities of all 41 events in the sample are calculated, from which we provide constraints on the planet distribution function. In particular, assuming a planet distribution function that is uniform in $\log{q}$, where $q$ is the planet-to-star mass ratio, we find a $95\%$ upper limit on the fraction of stars that host typical microlensing planets of 49\%, which is consistent with previous studies. Based on this planet-free sample, we develop the methodology to statistically study the Galactic distribution of planets using microlensing parallax measurements. Under the assumption that the planet distributions are the same in the bulge as in the disk, we predict that $\sim$1/3 of all planet detections from the microlensing campaigns with Spitzer should be in the bulge. This prediction will be tested with a much larger sample, and deviations from it can be used to constrain the abundance of planets in the bulge relative to the disk.

Tables

Table 1. Table 1: Summary of the 50 events in our sample. Here (RA,Dec) are the equatorial coordinates, and ( l 𝑙 l , b 𝑏 b ) are the Galactic coordinates. We also include the subjective selection dates and objective selection dates (if objective criteria are met), OGLE-IV Bulge fields and cadences. In the last column, we present the HJD dates of the first and last Spitzer observation, as well as the total number of observations from Spitzer .

OGLE #	RA (deg)	Dec (deg)	$l$ (deg)	$b$ (deg)	Subjective	Objective	OGLE-IV fields,	Spitzer observations
					selection	selection	cadences (per day)	start, stop, #
0011	269.217833	$-29.283250$	$0.957550$	$-2.289872$	5-30-11:59	—	BLG505, 30	7184.96, 7222.58, 53
0029	269.944167	$-28.644944$	$1.828415$	$-2.522683$	5-10-14:33	6-01	BLG505, 30	7185.31, 7222.89, 52
0034	270.580333	$-27.516083$	$3.088357$	$-2.452817$	4-28-17:01	6-08	BLG511, 10	7186.01, 7222.92, 62
0081	268.653000	$-28.996278$	$0.957513$	$-1.719135$	6-01-14:25	—	BLG505, 30	7184.10, 7221.81, 57
0350	268.248583	$-31.820278$	$-1.657210$	$-2.846465$	5-19-20:45	6-01	BLG535, 3	7183.95, 7221.76, 61
0379	269.104292	$-29.574056$	$0.656089$	$-2.350021$	5-19-20:45	6-01	BLG505, 30	7184.61, 7222.58, 54
0388	268.468917	$-28.534028$	$1.274824$	$-1.346174$	5-10-14:33	6-01	BLG500, 10	7183.98, 7221.81, 66
0461	270.043208	$-28.156944$	$2.295828$	$-2.356385$	5-19-20:45	—	BLG504, 10	7185.79, 7222.90, 58
0529	270.264667	$-29.922917$	$0.854524$	$-3.397576$	5-16-22:18	6-08	BLG513, 3	7185.79, 7222.92, 51
0565	269.153708	$-29.128056$	$1.063892$	$-2.163672$	5-16-22:18	6-01	BLG505, 30	7184.62, 7222.59, 53

Equations

$$t_{\mathrm{E}}\equiv \frac{{\theta }_{\mathrm{E}}}{{\mu }_{\mathrm{rel}}}.$$

$${\theta }_{\mathrm{E}}\equiv \sqrt{\kappa M_{\mathrm{L}}{\pi }_{\mathrm{rel}}};\kappa \equiv \frac{4G}{c^2\mathrm{AU}}\simeq 8.14\frac{\mathrm{mas}}{M_{\odot }},$$

$$\mbox{\boldmath$\pi_{\rm E}$}\equiv\frac{\pi_{\rm rel}}{\theta_{\rm E}}\frac{\mbox{\boldmath$\mu_{\rm rel}$}}{\mu_{\rm rel}}\ ,$$

$${\tilde{r}}_{\mathrm{E}}=\frac{\mathrm{AU}}{{\pi }_{\mathrm{E}}},$$

$$M_{\mathrm{L}}=\frac{{\theta }_{\mathrm{E}}}{\kappa {\pi }_{\mathrm{E}}};{\pi }_{\mathrm{rel}}={\theta }_{\mathrm{E}}{\pi }_{\mathrm{E}},$$

$$\mbox{\boldmath$\pi_{\rm E}$}\approx\frac{{\rm AU}}{D_{\perp}}\left(\Delta\tau,\Delta\beta\right)\ ,$$

$$\Delta \tau \equiv \frac{t_{0,\mathrm{sat}}-t_{0,\oplus }}{t_{\mathrm{E}}};\Delta \beta \equiv u_{0,\mathrm{sat}}-u_{0,\oplus }.$$

$$\mbox{\boldmath$\pi_{\rm E}$}\approx\frac{{\rm AU}}{D_{\perp}}\left(\Delta\tau,\Delta\beta_{\pm,\pm}\right)\ ,$$

$$\left\{\begin{array}[]{ll}\Delta\beta_{+,+}\equiv+u_{0,{\rm sat}}-u_{0,\oplus}&,(+,+){\rm\ solution}\cr\Delta\beta_{+,-}\equiv-u_{0,{\rm sat}}-u_{0,\oplus}&,(+,-){\rm\ solution}\cr\Delta\beta_{-,-}\equiv-u_{0,{\rm sat}}+u_{0,\oplus}&,(-,-){\rm\ solution}\cr\Delta\beta_{-,+}\equiv+u_{0,{\rm sat}}-u_{0,\oplus}&,(-,+){\rm\ solution}\cr\end{array}\right.\ .$$

$$F^{j}(t_{i})=F_{\rm s}^{j}\cdot A^{j}(t_{i};t_{0},u_{0},t_{\rm E},\rho,\mbox{\boldmath$\pi_{\rm E}$})+F_{\rm b}^{j}\ .$$

$$\mbox{\boldmath$\tilde{v}_{\rm hel}$}=\mbox{\boldmath$\tilde{v}_{\rm geo}$}+\mbox{\boldmath$v$}_{\oplus,\perp};\quad\mbox{\boldmath$\tilde{v}_{\rm geo}$}=\frac{{\rm AU}}{t_{\rm E}}\frac{\mbox{\boldmath$\pi_{\rm E}$}}{\pi_{\rm E}^{2}}\ ,$$

$$t_{\mathrm{E}}^{\prime }\equiv \frac{{\tilde{r}}_{\mathrm{E}}}{{\tilde{v}}_{\mathrm{hel}}};{\tilde{r}}_{\mathrm{E}}\equiv \frac{\mathrm{AU}}{{\pi }_{\mathrm{E}}}.$$

$$n_{\star }(x,y,z)=n_{\mathrm{B}}(x^{\prime },y^{\prime },z^{\prime })+n_{\mathrm{D}}(R,z).$$

$$n_{\mathrm{B}}=n_{\mathrm{B},0}{e}^{-r_s^2/2};r_s\equiv {\left\{{\left[{\left(\frac{x^{\prime }}{x_0}\right)}^2+{\left(\frac{y^{\prime }}{y_0}\right)}^2\right]}^2+{\left(\frac{z^{\prime }}{z_0}\right)}^4\right\}}^{1/4},$$

$$n_{\mathrm{D}}=n_{\mathrm{D},0}\exp \left[-\left(\frac{R-R_{\mathrm{GC}}}{R_0}+\frac{|z|}{z_{\mathrm{D},0}}\right)\right].$$

$$\mbox{\boldmath$\mu_{v}$}(x,y,z)=\frac{n_{\rm B}}{n_{\star}}\mbox{\boldmath$\mu$}_{v,\rm B}+\frac{n_{\rm D}}{n_{\star}}\mbox{\boldmath$\mu$}_{v,\rm D}\ ,$$

$${\sigma }_{v,i}^2(x,y,z)={\left(\frac{n_{\mathrm{B}}}{n_{\star }}\right)}^2{\sigma }_{v,i,\mathrm{B}}^2+{\left(\frac{n_{\mathrm{D}}}{n_{\star }}\right)}^2{\sigma }_{v,i,\mathrm{D}}^2(i=x,y,z).$$

$$\frac{{\rm d}\xi(M_{\rm L})}{{\rm d}\log{M_{\rm L}}}\propto\left\{\begin{array}[]{ll}M_{\rm L}^{0.7}&,\ 0.013<M_{\rm L}/M_{\odot}<0.08\cr M_{\rm L}^{-0.3}&,\ 0.08<M_{\rm L}/M_{\odot}<0.5\cr M_{\rm L}^{-1.3}&,\ 0.5<M_{\rm L}/M_{\odot}<1.3\end{array}\right.\ .$$

$$D_{8.3}\equiv \frac{\mathrm{kpc}}{\left({\pi }_{\mathrm{rel}}/\mathrm{mas}\right)+1/8.3}.$$

$$D_{8.3}\to D_{\mathrm{L}}(D_{\mathrm{L}}\ll D_{\mathrm{S}});$$

$$(8.3\mathrm{kpc}-D_{8.3})\to D_{\mathrm{L}\mathrm{S}}(D_{\mathrm{L}\mathrm{S}}\ll D_{\mathrm{S}}),$$

$${\pi }_{\mathrm{rel}}\to \frac{{\tilde{v}}_{\mathrm{hel}}}{{\mu }_{\mathrm{sgrA}\star }},$$

$$\mbox{\boldmath$\tilde{v}_{\rm hel}$}=\frac{D_{\rm S}}{D_{{\rm L}{\rm S}}}\mbox{\boldmath$v$}_{{\rm L},{\rm gc}}-\frac{D_{\rm L}}{D_{{\rm L}{\rm S}}}\mbox{\boldmath$v$}_{{\rm S},{\rm gc}}-\mbox{\boldmath$v$}_{\odot,{\rm gc}}\ ,$$

$$\mbox{\boldmath$\mu_{\rm rel}$}=\frac{\mbox{\boldmath$v$}_{{\rm L},{\rm gc}}-\mbox{\boldmath$v$}_{\odot,{\rm gc}}}{D_{\rm L}}-\frac{\mbox{\boldmath$v$}_{{\rm S},{\rm gc}}-\mbox{\boldmath$v$}_{\odot,{\rm gc}}}{D_{\rm S}}\ .$$

$$\frac{{\rm d}^{4}\Gamma}{{\rm d}D_{\rm L}{\rm d}\log M_{\rm L}{\rm d}^{2}\mbox{\boldmath$\mu_{\rm rel}$}}=n_{{\rm L},\star}D_{\rm L}^{2}(2\theta_{\rm E})\mu_{\rm rel}f_{\mu}(\mbox{\boldmath$\mu_{\rm rel}$})\frac{{\rm d}\xi(M_{\rm L})}{{\rm d}\log M_{\rm L}}\ .$$

$$\begin{aligned} \frac{{\rm d}^{4}\Gamma}{{\rm d}D_{8.3}{\rm d}t_{\rm E}^{\prime}{\rm d}^{2}\mbox{\boldmath$\tilde{v}_{\rm hel}$}}&=\frac{{\rm d}^{4}\Gamma}{{\rm d}D_{\rm L}{\rm d}\log M_{\rm L}{\rm d}^{2}\mbox{\boldmath$\mu_{\rm rel}$}}\left|\frac{\partial(D_{\rm L},\log M_{\rm L},\mbox{\boldmath$\mu_{\rm rel}$})}{\partial(D_{8.3},t_{\rm E}^{\prime},\mbox{\boldmath$\tilde{v}_{\rm hel}$})}\right|\cr&=4n_{{\rm L},\star}\frac{D_{\rm L}^{4}}{D_{8.3}^{2}}\mu_{\rm rel}^{2}f_{\tilde{v}}(\mbox{\boldmath$\tilde{v}_{\rm hel}$})\frac{{\rm d}\xi(M_{\rm L})}{{\rm d}\log M_{\rm L}}\end{aligned}\ .$$

$$\begin{aligned} \left|\frac{\partial(D_{\rm L},\log M_{\rm L},\mbox{\boldmath$\mu_{\rm rel}$})}{\partial(D_{8.3},t_{\rm E}^{\prime},\mbox{\boldmath$\tilde{v}_{\rm hel}$})}\right|&=\left(\frac{D_{\rm L}}{D_{8.3}}\right)^{2}\frac{\mu_{\rm rel}}{\tilde{v}_{\rm hel}}\frac{1}{M_{\rm L}}\left|\frac{\partial(M_{\rm L},\mu_{\rm rel})}{\partial(t_{\rm E}^{\prime},\tilde{v}_{\rm hel})}\right|\cr&=\left(\frac{D_{\rm L}}{D_{8.3}}\right)^{2}\frac{2\pi_{\rm rel}^{2}}{{\rm AU}^{2}t_{\rm E}^{\prime}}\end{aligned}\ .$$

$$P_{\mathrm{raw}}(D_{8.3})=\frac{{\int }_{D_{\mathrm{S},\min }}^{D_{\mathrm{S},\max }}\mathcal{P}(D_{8.3}|D_{\mathrm{S}})n_{\mathrm{S},\star }{D}_{\mathrm{S}}^{2-\gamma }\mathrm{d}{D}_{\mathrm{S}}}{{\int }_{D_{\mathrm{S},\min }}^{D_{\mathrm{S},\max }}{n}_{\mathrm{S},\star }{D}_{\mathrm{S}}^{2-\gamma }\mathrm{d}{D}_{\mathrm{S}}},$$

$$\small\mathcal{P}(D_{8.3}|D_{\rm S})\equiv\int\frac{{\rm d}^{4}\Gamma}{{\rm d}D_{8.3}{\rm d}t_{\rm E}^{\prime}{\rm d}^{2}\mbox{\boldmath$\tilde{v}_{\rm hel}$}}P(t_{\rm E}^{\prime}|{\rm Data})P(\mbox{\boldmath$\tilde{v}_{\rm hel}$}|{\rm Data}){\rm d}t_{\rm E}^{\prime}{\rm d}^{2}\mbox{\boldmath$\tilde{v}_{\rm hel}$}\ .$$

$$P_i(D_{8.3})=\frac{e^{-\Delta {\chi }_i^2/2}{\pi }_{\mathrm{E},i}^{-2}{P}_{i,\mathrm{raw}}(D_{8.3})}{\mathcal{N}},$$

Full text

gravitational lensing: micro — planetary systems — planets and satellites: dynamical evolution and stability — methods: statistical

1 Introduction

The distribution of planets in different environments is of great interest. Studies have shown that the planet frequency may be correlated with the host star metallicity (e.g., Santos et al., 2001, 2003; Fischer & Valenti, 2005; Wang & Fischer, 2015; Zhu et al., 2016b), the stellar mass (e.g., Johnson et al., 2010), stellar multiplicity (e.g., Eggenberger et al., 2007; Wang et al., 2014), and exterior stellar environment (e.g., Thompson, 2013). For this purpose, probing the planet distribution outside the Solar Neighborhood is important. In particular, the planet distribution in the Galactic bulge, given its unique environment, can provide an extra dimension to test and further develop our theories of planet formation.

Probing the distribution of planets in the Galactic bulge, or more generally, at all Galactic scales, is a unique application of Galactic microlensing, because of its independence on the flux from the planet host (Mao & Paczynski, 1991; Gould & Loeb, 1992). For example, Penny et al. (2016) used an ensemble of 31 microlensing planets and found tentative evidence that the bulge might be deficient of planets compared to the disk.

While microlensing is in principle sensitive to planets at various Galactic distances, the distance determination of any given microlensing event is nontrivial. This is because, in the majority of cases, the only relevant observable from the microlensing light curve is the Einstein timescale

[TABLE]

Here $\mu_{\rm rel}$ is lens-source relative proper motion, and $\theta_{\rm E}$ is the angular Einstein radius,

[TABLE]

where $M_{\rm L}$ is the lens mass, $\pi_{\rm rel}\equiv{\rm AU}(D_{\rm L}^{-1}-D_{\rm S}^{-1})$ is the lens-source relative parallax, and $D_{\rm L}$ and $D_{\rm S}$ are distances to the lens and the source (i.e., the star being lensed), respectively. In planetary events, $\theta_{\rm E}$ is usually also measurable through the so-called finite-source effect (Yoo et al., 2004), in addition to two parameters that characterize the planet itself: the planet/star mass ratio $q$ and the planet/star separation $s$ in units of $\theta_{\rm E}$ (Gaudi & Gould, 1997b). There nevertheless remains a degeneracy between the lens mass and lens distance (assuming the source is in the bulge, which is almost always the case). The difficulty in precisely determining the lens distance is a significant weakness of ground-based microlensing in determining the Galactic distribution of planets, as has been demonstrated by Penny et al. (2016).

The most efficient way to determine or better constrain the lens distance $D_{\rm L}$ is by measuring the so-called microlens parallax vector $\pi_{\rm E}$

[TABLE]

which can be effectively achieved by simultaneously observing the same event from at least two well-separated ($\mathcal{O}(1~{}{\rm AU})$) observatories (Refsdal, 1966; Gould, 1994). This is because, for typical Galactic microlensing events, the projected Einstein radius on the observer plane,

[TABLE]

is of order $\sim 10~{}{\rm AU}$, and thus observers separated by $\sim$1 AU would see considerably different light curves of the same microlensing event. For events with $\theta_{\rm E}$ measurements, including most planetary events, most binary events, and relatively rare single-lens events, the measurements of $\pi_{\rm E}$ directly yield the lens mass and lens-source relative parallax

[TABLE]

the latter being a good proxy for distinguishing disk and bulge lenses (see Section 4). For the great majority of single-lens events, $\theta_{\rm E}$ cannot be measured from the microlensing light curve, but the lens distribution ($M_{\rm L}$ and $\pi_{\rm rel}$) can be much more tightly constrained once $\pi_{\rm E}$ is measured, as first pointed out by Han & Gould (1995).

For this reason, the Spitzer Space Telescope has been employed for microlensing (Dong et al., 2007; Gould et al., 2013, 2014, 2015a, 2015b, 2016). The 2014 Spitzer microlensing experiment served as a pilot program that successfully demonstrated the ability to measure microlens parallax using Spitzer (Udalski et al., 2015a; Yee et al., 2015a; Calchi Novati et al., 2015; Zhu et al., 2015a). Starting in 2015, the main goal of Spitzer microlensing campaigns became measuring the Galactic distribution of planets (Calchi Novati et al., 2015; Yee et al., 2015b).

It is by no means trivial to organize Spitzer and ground observations to enable a measurement of the Galactic distribution of planets that is unbiased by observational decisions. On the one hand, microlensing events must be chosen for Spitzer observations very carefully in order to maximize both the sensitivity to planets of the whole sample and the probability that these observations will actually lead to a microlens parallax measurement. On the other hand, these observational decisions cannot in any way be influenced by whether planets have (or have not) been detected. The first objective requires that observational decisions make maximal use of available information, while the second means that a certain “blindness” to this information must be rigorously enforced. Yee et al. (2015b) discussed in great detail how to optimize observations while enforcing this blindness, and a short summary is given in Section 2.3. Interested readers are urged to consult Yee et al. (2015b) for more details.

Following the Yee et al. (2015b) protocol, the 2015 Spitzer microlensing campaign observed 170 microlensing events that were first found in the ground-based microlensing surveys, namely the Optical Gravitational Lensing Experiment (OGLE, Udalski, 2003; Udalski et al., 2015b) and the Microlensing Observations in Astrophysics (MOA, Bond et al., 2001; Sako et al., 2008). In this work, we present analysis of 50 of them that fall within the footprints of OGLE and the prime fields of the newly established KMTNet (Korean Microlensing Telescope Network, Kim et al., 2016).

The present work is not aimed at directly answering how planets are distributed within the Galaxy. Instead, we develop a framework within which the above question can be ultimately addressed. It is nevertheless true that the 50 events in our sample, observed at $\sim$10 min cadence nearly continuously throughout year 2015, are more sensitive to planets than the majority of the remaining events in the 2015 Spitzer sample. Another significant contributor to the overall planet sensitivity would be high-magnification events, which have nearly 100% sensitivity to planets (Griest & Safizadeh, 1998; Gould et al., 2010) but are considerably rarer. These high-magnification events will be analyzed separately.

This paper is organized as follows. Section 2 summarizes our observations and reduction methods for both ground-based and space-based data; Section 3 describes our selection of the raw sample; in Section 4 we provide the methodology for analyzing individual events, including four-fold solutions, distance and mass estimations, and planet sensitivity computation. This method is then applied to the current sample, and results are presented in Section 5. In Section 6 we discuss the implications of this work, as well as outline the path for future work.