A pair of planets likely in mean-motion resonance from gravitational microlensing

Wei Zhu (祝伟)

Astrophysical Dynamics, Shanghai

(Based on Madsen & Zhu, 2019, ApJL, 878, 29)

Sabrina Madsen

Microlensing probes the cold planet population

Mao & Paczynski (1991); Gould & Loeb (1992); Gaudi (2012)

For typical disk surface density profile (\( \Sigma \propto r^{-1} \) or \( \propto r^{-3/2} \) ), there is more mass in the outer region

Microlensing commonly probes projected configuration

Orbital period: years

planetary perturbation: days

Mao & Paczynski (1991); Gould & Loeb (1992); Gaudi (2012)

How to probe orbital configuration of microlensing planets

  • Light curve modeling (e.g., Gaudi++2008; Ryu++2018)
  • Radial velocity follow-up (e.g., Yee++2016 for a binary)
  • Dynamical stability


  • Direct imaging system HR 8799:
    • Likely double 2:1 mean-motion resonances (b & c, c & d)

Fabrycky & Murray-Clay (2010)

(see also Wang et al. 2018)




Long-term stability constrains orbital configuration of multi-planet systems

Marois et al. (2008)

Two-planet system OB120026L

planet 2

planet 1

Han et al. (2013)

(see also Beaulieu et al. 2016)

Both planets inside Einstein ring

Both planets outside Einstein ring

\red{q_1\sim 1\times 10^{-4},~s_1=0.96 {~\rm or~} 1.03} \\ \blue{q_2\sim 8\times 10^{-4},~s_2=0.81{~\rm or~} 1.25}

Madsen & Zhu, 2019, ApJL, 878, 29

Effect of orbital orientations

planet 2

planet 1

Dynamical stability

planet 2

planet 1

  • Randomize e vector
  • N-body integration
  • Reject unstable orbits

Madsen & Zhu, 2019, ApJL, 878, 29

Hadden & Lithwick (2018)

Pluto & Neptune

Mean-motion resonances

Eccentric orbits

Eccentric orbits & in MMRs




Nearly circular orbits & out of MMRs

Madsen & Zhu, 2019, ApJL, 878, 29

Compare with similar planet pairs

from RV


non-MMR (probably)

Madsen & Zhu, 2019, ApJL, 878, 29

Orbital evolution prefers

mean-motion resonances

Lee & Peale (2002)

Evolution of GJ 876 system


non-MMR (probably)

A pair of microlensing planets likely in

mean-motion resonance

  • Microlensing can also probe the detailed dynamical state of multi-planet systems
    1. Two planets close to each other;
    2. Azimuthal offset;
    3. Stability & evolution history --> MMRs
  • More similar systems from microlensing

planet 2

planet 1

Madsen & Zhu, 2019, ApJL, 878, 29

Pollack et al. (1996)

Suzuki et al. (2016)

(see Herman, Zhu, & Wu 2019 for the radius distribtuion)

30-100 \(M_\oplus\) planets not predicted by core accretion theory


How often do cold Jupiters have cold Neptune companions?

  • A single detection out of ~20 microlensing systems with Sun-like hosts
  • Low detection efficiency for Neptunes (~5%, Zhu et al. 2014).
  • Perhaps all cold Jupiter systems also have cold Neptunes: P(CN|CJ)~100%.
    • Mean-motion resonances may also be common.



Multi-planet systems from microlensing: Connection to the overall exoplanet demographics

Hot Jupiters


Cold Jupiters


Cold Neptunes


Super Earths


(Zhu et al. 2018)

Hot Jupiters have distant companions

(Knutson et al. 2014)

Hot Jupiters are lonely (Steffen et al. 2010)

Super Earths & cold Jupiters tend to co-exist

(Zhu & Wu 2018)

  • Why is the mass function so smooth? Is it related to multiplicity?
  • How many Neptunes per system?
  • Do cold Jupiters frequently have cold Neptune companions?
  • ...

Data from NASA Exoplanet Archive


Orbital motion in microlensing

& RV follow-ups

Skowron et al. (2011); Yee et al. (2016)

(OGLE-2006-BLG-109, Gaudi et al. 2008, Bennett et al. 2010;

OGLE-2016-BLG-1190, Ryu et al. 2018;

Gaia16aye, Wyrzykowski et al. 2019)

OB120026: Sun-like host with a cold Jupiter & a cold Neptune

Beaulieu et al. (2016)

Orbital solution of microlensing system

OGLE-2006-BLG-109: A Jupiter/Saturn analog

Gaudi et al. (2008); Bennett et al. (2010)

MMR in Microlensing (Shanghai)

By Wei Zhu

MMR in Microlensing (Shanghai)

Talk at the Astrophysical Dynamics Conference in Shanghai

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