Capture of free-floating planets by stellar systems

TL;DR

This study uses 3D simulations to analyze how free-floating planets can be captured by stars, estimating a 1% capture rate and proposing factors that could increase this likelihood.

Contribution

The paper introduces a new simulation framework and analytical model to estimate free-floating planet capture rates by stars, highlighting the potential for higher capture probabilities.

Findings

Approximately 1% of stars may temporarily capture free-floating planets.

The capture cross section depends on mass, impact parameters, and velocities.

Additional physical processes could further increase capture likelihood.

Abstract

Evidence of exoplanets with orbits that are misaligned with the spin of the host star may suggest that not all bound planets were born in the protoplanetary disk of their current planetary system. Observations have shown that free-floating Jupiter-mass objects can exceed the number of stars in our galaxy, implying that capture scenarios may not be so rare. To address this issue, we construct a three-dimensional simulation of a three-body scattering between a free-floating planet and a star accompanied by a Jupiter-mass bound planet. We distinguish between three different possible scattering outcomes, where the free-floating planet may get captured after the interaction with the binary, remain unbound, or "kick-out" the bound planet and replace it. The simulation was performed for different masses of the free-floating planets and stars, as well as different impact parameters, inclination angles and approach velocities. The outcome statistics are used to construct an analytical approximation of the cross section for capturing a free-floating planet by fitting their dependence on the tested variables. The analytically approximated cross section is used to predict the capture rate for these kinds of objects, and to estimate that about 1\% of all stars are expected to experience a temporary capture of a free-floating planet during their lifetime. Finally, we propose additional physical processes that may increase the capture statistics and whose contribution should be considered in future simulations.

Figures

Equations

$$\mathcal{B}$$

$$r_{min}=r_B\frac{{\left({\left(1+\rho \right)}^{\text{nicefrac}32}-1\right)}^2}{2{\rho }^2},$$

$$\mathrm{d}\sigma =P\cdot 2\pi b\mathrm{d}b.$$

$$\mathrm{R}\left(M,r_{min}\right)$$

$$⟨\mathrm{R}⟩=N_{stars}\int \mathrm{R}\left(M,r_{min}\right)f\left(M,r_{min}\right)\mathrm{d}Md{r}_{min}.$$

$$\frac{1}{2}{m}_f{v}_{\infty }^2=\frac{1}{2}{m}_f{v}^2-\frac{G{m}_f(M+m_B)}{\sqrt{b^2+{\left(40r_B\right)}^2}}.$$

$${\mathcal{L}}_{kinetic}$$

$${\mathcal{L}}_{interaction}$$

$$\delta E_f=\sqrt{\sum _i{\left(\frac{\partial E_f}{\partial y_i}\left|e_i\right|\right)}^2},$$

$$\tau \left(M\right)=\frac{fM{c}^2}{L}\sim {\tau }_{\odot }{\left(\frac{M}{M_{\odot }}\right)}^{-2.5},$$

Full text

have shown that free-floating Jupiter-mass objects can exceed the number of stars in our galaxy, implying that capture scenarios may not be so rare. To address this issue, we construct a three-dimensional simulation of a three-body scattering between a free-floating planet and a star accompanied by a Jupiter-mass bound planet. We distinguish between three different possible scattering outcomes, where the free-floating planet may get captured after the interaction with the binary, remain unbound, or “kick-out” the bound planet and replace it. The simulation was performed for different masses of the free-floating planets and stars, as well as different impact parameters, inclination angles and approach velocities. The outcome statistics are used to construct an analytical approximation of the cross section for capturing a free-floating planet by fitting their dependence on the tested variables. The analytically approximated cross section is used to predict the capture rate for these kinds of objects, and to estimate that about 1% of all stars are expected to experience a temporary capture of a free-floating planet during their lifetime. Finally, we propose additional physical processes that may increase the capture statistics and whose contribution should be considered in future simulations.

Free-floating planets: capture rate - planetary systems and satellites: temporary capture, planet exchange - techniques: scattering simulations

1 Introduction

The canonical planet formation mechanism is an in-situ planet formation theory, dictating that planets form inside a protoplanetary disk. If so, planetary systems would have to be the only source of free-floating planets. An alternative suggestion adopts an ex-situ approach, where these planetary-mass free-floating objects (hereafter, referred as "free-floaters") may form by gravitational collapse of interstellar gas blobs.

It was suggested that high-speed gas blobs from the explosive death of stars may form free-floaters by accretion of interstellar ambient matter as they slow down, cool by radiation and collapse into a hot Jupiter once their mass exceeds the Jeans mass (Dado et al., 2011). Such blobs and other filamentary structures are observed in large numbers in nearby supernova remnants (Fesen et al., 2006), planetary nebulae (O’Dell et al., 2002; Matsuura et al., 2009) or star formation regions, and are considered to be common in these stellar stages. Some of these blobs were observed as dense conglomerations of gas within H II regions, were they are referred to as globulettes (Haworth et al., 2015).

Measurements of the angle between the planetary orbital axis and the stellar spin axis (spin-orbit angle) reveal that a considerable fraction of the hot Jupiters have misaligned spin-orbit. This misalignment does not settle with the expectation of a close alignment between the spin of the star and the orbital motion of the planets, as they all should inherit their angular momentum from the protostellar disk. As for now, the spin-orbit angle of 87 planets was calculated from light curves that exhibit anomalies due to the Rossiter-McLaughlin effect. About 40% of them show significant spin-orbit misalignment, and nine of them are retrograde planets (Campante et al., 2016).

Observations of supernova remnants and planetary nebulae each show thousands of blobs that can lead to considerable number densities of free-floaters, which may be large enough for captures to be common. A wide-field image of the Helix Nebula in the $2.12\mathrm{\,\mu m}$ molecular hydrogen line shows more then 40,000 blobs that constitute the only source of the $\mathrm{H}_{2}$ surface brightness (Matsuura et al., 2009). If we assume that the only source of free-floaters are planetary nebulae and that the $\sim 40,000$ blobs found in the Helix Nebula are a typical number for stars at this evolutionary stage, then a planetary nebulae number density of $\sim 5\times 10^{-3}\mathrm{\,pc^{-3}}$ (Holberg et al., 2002) predicts a blobs number density of $n_{f}\approx 200\mathrm{\,pc^{-3}}$. This tells us that there may be one thousand Jupiter mass objects for every star in the Galaxy. In order to predict the capture rate of this kind of free-floaters by planetary systems, we evaluate the capture cross section by simulating three-dimensional scatterings between a planetary-mass free-floater and a star-planet binary.

2 Approach

Significant orbit perturbations of a bound planet due to interactions with free-floaters are expected to occur for impact parameters of $-b_{max}\leq b\leq b_{max}$, where the value of $b_{max}$ is determined by evaluating the maximal closest approach distance $r_{min}$ from the host star, for which significant orbital perturbations are still possible. Let us assume that a free-floater with a relative velocity $v_{\infty}$ at infinity is approaching a star-planet binary with an impact parameter $b$. Assuming that the mass of the star $M$ is much larger then the mass of the bound planet $m_{B}$ and the free-floater $m_{f}$, the trajectory of the free-floater is the one of a test-particle, and determined by the mass of the star. The cross section for significant perturbations induced by this flux is the product of the geometrical area times the gravitational focusing factor

[TABLE]

We use an analytical expression for the closest approach $r_{min}$, that holds for $m_{f},m_{B}\ll M$ (Donnison, 1984)

[TABLE]

where $r_{B}$ is the semi-major axis of the bound planet, and $\rho$ is the the mass ratio between the free-floater and the bound planet (hereafter, the planetary-mass ratio).

We may write the differential size of the capture cross section $\mathrm{d}\sigma$ as a product of the differential cross section for a significant interactions $2\pi b\mathrm{d}b$, times the fraction of interactions that resulted in a capture $P$

[TABLE]

The interaction cross section depends on the relative velocity $v$, the mass of the host star $M$, the planetary-mass ratio $\rho$, and the semi-major axis of the bound planet $r_{B}$; the last two determine the closest approach radius. The capture probability may also depend on similar variables, but dependence on relative velocity and the inclination angle is expected, since the perturbing impulse is affected.

Let us assume that a homogeneous flux of free-floaters with a given mass $m_{f}$, relative velocity $v$ and a number density $n_{f}$ is approaching with a given inclination angle $\theta_{i}$ at a planetary system. This flux will produce $n_{f}v\mathrm{d}\sigma\left(v,\theta,M,r_{min}\right)/\mathrm{d}\Omega$ captures per second per solid angle. For a given relative velocity dispersion $\mathcal{S}$, the capture rate for a specific stellar mass and $r_{min}$ is obtained by integrating with respect to velocity and over the full solid angle

[TABLE]

where $f_{\mathcal{S}}$ is the relative velocity distribution. The expected capture rate is obtained after integrating with respect to the $M$ and $r_{min}$ according to their corresponding distribution function, and multiplying by the total number of stars $N_{stars}$

[TABLE]

3 Numerical Method

We follow the work of Varvoglis et al. (2012) where they simulated scattering events of a Jupiter-mass free-floater by a binary of Sun-mass star and a Jupiter-mass planet. The free-floater was placed on a parabolic coplanar orbit and sufficiently far away to be considered as an unbound object. They mapped the resulting outcomes of the simulation for a grid of impact parameters $b$ and initial orbital phases $\phi_{B}$ of the planetary system, and calculated the fraction of captures $P$. This is because different initial orbital phases lead to different impulse durations and strengths. It turned out that about $50\%$ of the initial condition grid led to a so-called “temporary capture”, where the fraction of captures that end up with moderate values of semi-major axis and eccentricity is in the order of $1\%$. These captures are considered as temporary because simulating this isolated system for a large number of revolutions of the captured free-floater shows that most of them will eventually gain energy and escape. They also performed simulations for different free-floater masses, showing that the fraction of captures grows linearly with the mass, but not significantly.

The free-floater, as well as the host star and the bound planet, are set as point-mass objects. The mass of the star and the free-floater are simulation variables, while the mass of the bound planet is of one Jupiter-mass. The bound planet is set to revolve around the host star on simple circular orbit. We define the inclination angle to be the one between the initial velocity vector of the free-floater $v_{f}$ and the orbital axis of the bound planet, (see Fig. 1). The initial distance of the free-floater from the system is set to be $r_{0}=\sqrt{40m_{f}/m_{B}r_{B}+b^{2}}$, so that the initial binding energy between the free-floater and the binary is negligible. The corresponding initial velocity is obtained through conservation of energy

[TABLE]

The simulations are carried for different masses of the free-floater and star, velocities and inclination angles. Each simulation run preforms multiple scattering that corresponds to a grid of impact parameters $b$ and initial orbital phases of the bound planets $\phi{}_{B}$. The array of impact parameters $b$ spans up to some maximal value $b_{max}$, for which significant perturbation may still occur (Eq. 2).

One may notice from Fig. 1 that all values of $b$ lead for the path of the free-floater to intersect with the y-axis if no interaction with the bound planet took place. This means that the capture statistics that obtained from a simulation for some inclination angle $\theta_{i}$ do not necessarily predict the percentage of captures produced by a flux of free-floaters that passes through an area of $\pi b^{2}$. To account for this, the effect of a non-zero second component of the impact parameter, orthogonal to $b$ and the velocity vector $v_{0}$, will be be tested separately.

The dynamical system is described by a simple Lagrangian. It includes the kinetic term of the two planets, the interaction term between them, and their interaction terms with the star

[TABLE]

We do not add a kinetic term for the star since we assume standard planetary masses, for which the displacement of the star due to interaction is negligible down to red dwarf masses.

We preform the simulation for 175 values of $-b_{max}<b<b_{max}$ and 125 values of $0<\phi<2\pi$ for each inclination angle, velocity at infinity, mass of the free-floater and mass of the host star. The equations of motion are integrated until the free-floater travels out approximately twice its initial distance from the star. At this time, we calculate the sum of the kinetic and potential energy for two pairs: $E_{f}$ - star and free-floater, and $E_{B}$ - star and bound planet. The sign of the energy distinguishes between three outcomes, visualized in Fig. 2:

• •

Flyby: $E_{f}\geq 0,\leavevmode\nobreak\ E_{B}<0$

• •

Capture: $E_{f}<0,\leavevmode\nobreak\ E_{B}<0$

• •

Exchange: $E_{f}<0,\leavevmode\nobreak\ E_{B}\geq 0$

The outcomes are saved together with their corresponding grid parameters $\left(b,\phi_{B}\right)$, for which the fraction of captures $P$ is calculated.

In order to maximize the simulation speed, we used a grid resolution that is four times sparser then the one used by Varvoglis et al. (2012), but still high enough for statistical purposes. The numerical precision, however, must be sufficient enough so that the error in energy will be at least one order of magnitude smaller than the value itself. The numerical precision is set by two parameters: the first is the relative tolerance $r_{T}$, which determines the maximal relative error of the solution, and the second is the absolute tolerance $a_{T}$, which determines the last important digit of the solution. These parameters were set to $r_{T}=a_{T}=10^{-7}$, for which the accuracy and the simulation speed are sufficient. We assume that the maximal error in every solution component $y_{i}$, taken as $\left|e_{i}\right|=r_{T}\left|y_{i}\right|+a_{T}$, has been achieved, so that the maximal error in $E_{f}$ is

[TABLE]

where $E_{f}=\nicefrac{{1}}{{2}}m_{f}\dot{\mathbf{r}}^{2}-m_{f}/r$. We calculate $\delta E_{f}$ for every run of the simulation to assure that the number of runs that resulted with $\delta E_{f}/E_{f}>0.1$ is less then $1\%$.

4 Results

We start with a Jupiter-mass free-floater that approaches the binary on a parabolic and coplanar orbit $\left(v_{\infty}=0,\theta_{i}=90^{\circ}\right)$. The resulting outcome map is displayed in Fig. 3, where we use gray levels to distinguish between captures, flybys, and exchanges. The gray area, which represents regions of $b,\phi{}_{B}$ values that led to captures, covers almost $50\%$ of grid. The upper an lower impact parameter values that were tested are those beyond which exchanges do not occur, indicating that significant orbital perturbations are no longer possible or too rare. Although statistically negligible, exchange events may also produce misaligned orbits, and they are considered as captures.

This outcome map agrees with the one presented by Varvoglis et al. (2012), implying the consistency of the method. Additionally, a variety of $b,\phi{}_{B}$ grid values with specific outcomes were selected from the map and tested individually to assure that the final energy values do not vary, and that the trajectories of the bodies agree with the outcomes.

The majority of the captures, however, end up with energies that are $\sim 10^{-3}$ time smaller then the initial energy of the binary, and are very eccentric and elongated. The outcome of the simulations is also plotted in Fig. 4 in terms of the ratio between the final energy of the free-floater and the initial energy of the binary. One can see that the maximal energy transfer takes place around the boundary between capture and flyby regions, at which exchange events are prone to happen. For most $b,\phi{}_{B}$ values, the total energy of the free-floater is almost unchanged. There is “line” of significant positive energies around $b=0$, where the minimal approach of the free-floater from the star is comparable to the numerical precision, and deviation from conservation of energy becomes substantial. This small region is disregarded in the statistics.

The eccentricity as a function of the semi-major axis for the captured Jupiter-mass free-floaters is given in Fig. 5.

Only $\sim 0.9$% of the simulated scattering events ended up with a semi-major axis of $a_{sm}<50\mathrm{\,AU}$, which is the distance of the Kuiper belt from the sun. According to our result, almost all free-floaters will reach their aphelion beyond the Kuiper belt if they would be captured by our Sun, but the number of $a_{sm}<50\mathrm{\,AU}$ captures drops rapidly for slight deviations from coplanarity.

We cover a $v_{\infty}$ initial velocity range for which the capture probability experiences significant variations. To evaluate the effect of the lateral impact parameter $a\neq 0$ on the statistics (Fig. 1), we ran additional simulations for a Jupiter mass free-floater with a fixed maximal value of $a=b_{max}$. The resulting statistics do not show dramatic variations in terms of capture probability, and they are averaged with the ones for $a=b_{max}$. Fig. 6 shows the dependence of the averaged capture probability on the velocity of the free-floater for all different inclination angels that were tested.

The capture percentage is lower for free-floaters with higher initial velocity and lower inclination angle (towards a face-on inclination), which is expected since the orbital perturbation of the bound planet gets shorter. The resulting functionality is best fitted with a Gaussian $P\left(v\right)=P_{0}\exp\left(-v^{2}/2\Sigma^{2}\right)$, where the parameter $P_{0}$ is capture fraction for the case of $v_{\infty}=0$ and $\Sigma$ is the standard deviation of the Gaussian profile.

To evaluate the dependence of the capture probability on the planetary-mass ratio, we preform a full simulation for two additional lighter free-floaters

• Earth mass and Mercury mass and two additional coplanar scatterings for $7m_{\jupiter}$ and $13m_{\jupiter}$ free-floaters. The statistics for the different inclination angles and velocities hardly differed from the ones for Jupiter-mass, and are effectively independent of the mass at this regime. The later two exhibited a slight increase in the capture percentage, as displayed in Fig. 7.

The dependence of capture probability on the stellar mass is also needed to be analytically approximated. However, since 85% of all stars are sub-solar, we performed additional coplanar scattering simulations with lower mass host-stars to evaluate their effect. The the resulting capture probabilities are shown in Fig. 8, and small differences of velocity dependence between the four tested masses can be noticed.

This trend can be attributed to the shorter impulse that is needed for the free-floater to loose its energy. Still, assuming that this velocity dependence is best fitted with a Gaussian, the resulting standard deviations differ only by $\sim 0.3\mathrm{\,km\cdot s^{-1}}$. Since the typical velocity dispersion in our galaxy is in the order of $\sim 10^{2}\mathrm{\,km\cdot s^{-1}}$, we do not expect the capture rate to be significantly affected.

5 The cross section construction

The fitted Gaussians profiles, shown in Fig. 6, predict the fraction of captures $P$ for a given velocity. To account for the dependence on the inclination angle $\theta_{i}$, we analyze the variations of the standard deviation $\Sigma$ and the $P_{0}$ with the inclination angle, which are shown in Fig. 9.

We fit a linear function for both $\Sigma$ and $P_{0}$, so that the capture probability, given in c.g.s units, is approximated analytically with $P\left(v,\theta_{i}\right)=\left(0.018\theta_{i}+0.451\right)\exp\left[-v^{2}/10^{8}\left(0.75\theta_{i}+3.5\right)^{2}\right]$. The orbital radius of the bound planet $r_{B}$ was constant throughout the simulations, so a possible dependence of the capture probability on $r_{b}$ is not accounted for; however, a Jupiter-Sun separation is indeed typical according to the NASA Exoplanet Archive111http://exoplanetarchive.ipac.caltech.edu/.

6 The capture rate

We evaluate the capture rate for the galactic thin disk, which contains most of the stars in our galaxy $\left(N_{s}\sim 10^{11}\right)$. With a moderate typical velocity dispersion of $\sigma_{v}=40\mathrm{\,km\cdot s^{-1}}$ and higher metallicity values, the thin disk may be considered as a plausible source of free-floaters and with higher capture rates (Murdin, 2001). With a relative velocity dispersion of $\sigma_{rv}=\sqrt{2}\sigma_{v}$, we use a Boltzmann velocity distribution function for the region in question. The interaction cross section that results for velocities lower than $v_{\mathrm{lim}}=0.07\mathrm{\,km\cdot s^{-1}}$ covers an area that is larger then the typical separation between the stars. To overcome this difficulty, we use a constant interaction cross section of $\mathcal{B}_{\mathrm{lim}}=n_{stars}^{-\nicefrac{{2}}{{3}}}\simeq 3.7\mathrm{\,pc^{2}}$ for velocities below $v_{\mathrm{lim}}$, where $n_{stars}=0.14\mathrm{\,pc^{-3}}$ is the number density of stars at the solar neighborhood. This accounts for slow free-floaters that enter the unit volume of the star, while any free-floater outside of this volume is dominated by the gravity of another star. After integrating with respect to velocity and solid angle (Eq. 4), we obtain the capture rate as a function of the stellar mass and the planetary-mass ratio $\mathrm{R}\left(M,\rho\right)$.

We assume that a Jupiter mass is a typical mass of a free-floater, as predicted through observations by Sumi et al. (2011), and use the probability distribution function given by Malhotra (2015) for planetary masses. This distribution function is based on masses of observed Kepler exoplanets for which a debiased distribution was constructed. We also use a present-day mass function for the galactic thin disk as presented by Chabrier (2003) and constructed from the observed luminosity function.

The number density of free-floaters $n_{f}$ is assumed to be constant, as the density distribution of the free-floaters is yet unknown. Using a number density of $n_{f}=0.24\mathrm{\,pc^{-3}}$, as estimated by Sumi et al. (2011) from microlensing surveys, the capture rate in the galactic thin disk is $R\simeq 6.4\times 10^{-6}\mathrm{\,yr^{-1}}$ (Eq. 5), which is about one “temporary capture” every $1.5\times 10^{5}\mathrm{\,yr}$. One problem with this prediction is that the estimated number density is model dependent, since a delta-function mass distribution was assumed for the free-floaters population. A second problem is that the microlensing event detection was limited to free-floaters with masses of $m_{f}>0.1M_{J}$. Sumi et al. also tested a power-law distribution for the free-floaters and received a number density that is 11 times higher. If we adopt the ex-situ approach for the origin of free-floater as introduced by Dado et al. (2011), then a possible number density of $n_{f}\approx 200\mathrm{\,pc^{-3}}$ is predicted, for which the capture rate in the galactic thin disk is $R\approx 0.0046\mathrm{\,yr^{-1}}$; that is, we expect a “temporary capture” to occur about every 218 years.

The lifetime of stars varies according to their mass, due to fact that the mass is the energy source of the star and it dictates the energy output $L$ (luminosity). A rough relation between the lifetime and mass is derived by evaluating the time required for the star to consume itself,

[TABLE]

where $\tau_{\odot}=10^{10}\mathrm{\,yr}$ is the lifetime of the Sun. We use this estimation to evaluate the number of temporary captures that are expected for different stellar masses. As displayed in Fig. 10, it turns out that about one out of every $\sim 760$ solar-mass stars is expected to experience capture a free-floater during its lifetime, while for red-dwarf stars with a mass of $0.1M_{\odot}$ the expectancy is a capture by one out of every 25 stars.

As more then 85% of the stars are of sub-solar masses, they are expected to be the main contributors to the capture rate. We estimate the expected fraction of stars that will capture a free-floater during their lifetime by calculating the expected value of $\mathrm{R}\left(M\right)\tau\left(M\right)$, where we use the initial stellar mass-function. We predict that $\sim 1\%$ of all $0.1M_{\odot}<M<2M_{\odot}$ stars are expected to experience a capture during their lifetime.

The capture rate has a strong dependence on the velocity dispersion, as illustrated in Fig. 11, indicating that the majority of captures are expected to take place in very cold regions (i.e. of slow planets).

7 Discussion

The vast majority of the capture events ends up with a very small binding energy between the free-floater and the host star, resulting in very elongated and eccentric orbits. Captured free-floaters may stay bound for a finite period of time until they gain energy back from planet-planet perturbation and get “kicked out” back to the interstellar medium. The following perturbations can also cause the captured free-floater to lose energy and thus “tighten” its orbit. Longer two-dimensional simulation carried by Varvoglis et al. (2012) showed that 90% of the captured free-floaters eventually got ejected back to the interstellar medium. In most of our scattering simulations, the bound planet experienced only one perturbation, as the equations of motion were integrated up to the point where the free-floater travels about twice its initial distance.

The strong velocity dependence of the cross section results in a very low capture rate of one capture every 218 years in the galactic thin disk, implying that captures are expected to occur in low velocity dispersion regions. Moreover, we simulated low-velocity scatterings because captures are expected at this velocities range. Since typical velocities in our galaxy are much higher, scattering simulations at higher velocities are needed to avoid extrapolations.

Moreover, 42% of the detected planetary systems contain more then one planet, and about one third of the stellar systems contain at least two stars. Simulating a planetary system with multiple bound planets should produce more boundaries between flyby and capture regions (see Figs. 4 and 3), at which significant orbit perturbations occur and hence increase the fraction of strong captures.

One must remember that our simulation assumed point masses and only the gravitational forces that they produce. We did not include any additional effects that may subtract energy from the interacting free-floater, such as tidal force and collision with clouds of debris; these should increase the capture fraction and tighten the orbits of captured planets. Not all captures, however, would be subjected to dissipation mechanisms in the same manner. Treating the eccentricity and the semi-major axis as additional variables of the cross section would allow us to distinguish between captures that are too loose and inclined to remain bound, and those that might get tighter and circularized on reasonable time-scales.