微引力透镜 (Gravitational microlensing)

Paczynski (1986); Mao & Paczynski (1991)

$t_{\rm E} \sim 30{\rm days} \left(\frac{M_{\rm L}}{M_\odot}\right)^{1/2}$

Nature of microlenses

Milky Way

(not to scale)

Halo

Magellanic Clouds

Sun

Bulge

Disk

"I discovered this phenomenon, but I don't think it can ever be observed."

Microlensing probes cold planets

Animations from B. Scott Gaudi

Transit (ground)

Transit (space)

Radial velocity

Microlensing

Imaging

Figure adapted from Zhu & Dong (2021)

Transit (ground)

Transit (space)

Radial velocity

Microlensing

Imaging

Figure from Zhu & Dong (2021)

• 最小质量星云模型: $\Sigma(r) \propto r^{-3/2}$ $\longrightarrow$ 质量(1-10 AU) > 质量(0.1-1 AU).

冷行星的重要性

Mimimum mass extra-solar nebula

Mimimum mass solar nebula

Weidenschilling (1977); Hayashi (1981); Chiang & Laughlin (2013)

• 成熟行星系统:

Microlensing from ground: KMTNet

• 3x1.6m telescopes with wide FoV;
• ~10$^8$ stars observed once every >20 min;
• Every year ~3000 microlensing events discovered, with ~30 planet detections；
• Current limit: planet-to-star mass ratio $q\sim10^{-5}$.
  • ref: Earth/Sun $q=3\times10^{-6}$.

Credit to: KASI, Gould et al. (2020)

Exoplanet statistics from microlensing

• Cold Neptunes are more abundant than cold Jupiters.
• Cold Neptunes are perhaps the most abundant?

Figures from Suzuki et al. (2016)

(see also Gould et al. 2010, Cassan et al. 2012, Clanton & Gaudi 2014, Udalski et al. 2018, Jung et al. 2019)

Microlensing@Tsinghua

• Real-time alert & follow-up observations (e.g. TAP), data reduction;
• Planet search, light curve modeling;

Yee, Zang, et al. (2021)

Cold Earths are abundant

• 14 planets from KMTNet 2019 prime fields
• 23 planets from MOA-II 5 yr survey.

Zang et al. (in prep)

1. Cold lower-mass planets are more abundant than cold higher-mass ones.

   The occurrence rate is integrated to ~1planet/star.

2. Multi-planet systems from microlensing: what can we learn?

Two-planet system OGLE-2012-BLG-0026L

Han et al. (2013), Beaulieu et al. (2016)

Microlensing probing projected configuration

Animations from B. Scott Gaudi

Planetary signals (~days) vs. orbital period (~years).

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

Effect of orbital orientations

Madsen & Zhu, 2019, ApJL, 878, 29

Dynamical stability

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

Madsen & Zhu, 2019, ApJL, 878, 29

Pluto & Neptune

Eccentric orbits

Eccentric orbits & in MMRs

Nearly circular orbits & out of MMRs

Madsen & Zhu, 2019, ApJL, 878, 29

Compare with similar planet pairs from radial velocity (RV)

MMR

non-MMR (probably)

Madsen & Zhu, 2019, ApJL, 878, 29

Orbital evolution prefers

mean-motion resonances

Lee & Peale (2002)

Evolution of GJ 876 system

MMR

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

Madsen & Zhu, 2019, ApJL, 878, 29

Cold planet multiplicity

• 3 two-planet detections out of ~110 microlensing planetary systems.
• Low detection efficiency of multi-planet systems via microlensing (~5%, Zhu et al. 2014).

Figure from Weicheng Zang

Solar system-like architecture may be common?

Figure adapted from Zhu & Dong (2021)

Multi-planet systems from microlensing

• OGLE-2006-BLG-109L (Gaudi et al. 2008, Bennett et al. 2010);
• OGLE-2012-BLG-0026L (Han et al. 2013, Beaulieu et al. 2016);
• OGLE-2018-BLG-1011L (Han et al. 2019);
• OGLE-2019-BLG-0468L (Han et al. 2022);
• KMT-2020-BLG-0414L (Zang et al. 2011);
• KMT-2021-BLG-1077L (Han et al. 2022);
• ...

1. Cold lower-mass planets are more abundant than cold higher-mass ones.

   The occurrence rate is integrated to ~1planet/star.

2. Cold planets frequently have companions.

   Microlensing allows to constrain the orbital configuration of multi-planet systems.

3. Detecting BHs via microlensing

Nature of microlenses

Microlensing toward the bulge:

White dwarf (WD): 17%

Neutron star (NS): 3%

Black hole (BH): ~1%

Gaudi (2012)

(see also Gould 2000)

微引力透镜 (Gravitational microlensing)

Paczynski (1986); Mao & Paczynski (1991)

$t_{\rm E} \sim 30{\rm days} \left(\frac{M_{\rm L}}{M_\odot}\right)^{1/2}$

背景恒星亮度

背景恒星

”透镜“星体

$t_{\rm E}$

First microlensing BH candidate?

Mao et al. (2002)

(see also Bennett+2002, Agol+2002)

Microlensing parallax

Microlensing parallax: an observable more sensitive to mass

Image credit: Bill Saxton, NRAO/AUI/NSF

• Parallax (absolute): $\pi \equiv \frac{\rm au}{d}$
• Microlensing parallax: $$\pi_{\rm E} \equiv \frac{\pi}{\theta_{\rm E}} \approx f(M_{\rm L}, D_{\rm L})$$

Parallax effect in BH event

Lam et al. (2020)

Typical BH events have

• long timescales ($t_{\rm E} \gtrsim 100$ d);
• small parallaxes ($\pi_{\rm E} \lesssim 0.05$).

Combining timescale & parallax

• Event timescale $$t_{\rm E} \propto \theta_{\rm E} \propto M^{1/2}$$
• Microlensing parallax $$\pi_{\rm E} \propto \theta_{\rm E}^{-1} \propto M^{-1/2}$$

3 unknowns: $M$, $D$, $\mu$

2 observables: $t_{\rm E}$, $\pi_{\rm E}$

Karolinski & Zhu (2020)

Finding BH events: previous attempts

(See also Mao et al. 2002; Bennett et al. 2002)

• Wyrzykowski et al. (2016) searched for BH events in >3600 events that were found by OGLE-III survey.
• Out of 59 events that showed parallax effect, they identified
  • 13 events with dark lenses (WD, NS, or BH), 2 of which are likely BHs.

Parallax effect in BH event is undetectable

(except for very rare cases)

Karolinski & Zhu (2020)

(see also Ma, Zhu, & Yang, 2022 submitted)

First isolated stellar-mass BH from microlensing?

First isolated stellar-mass BH from microlensing?

• Photometric & (HST) astrometric microlensing:
  • $M_{\rm L} = 7.1 \pm 1.3 M_\odot$ (Sahu et al. 2022)
  • $1.6<M_{\rm L}/M_\odot < 4.5$ (Lam et al. 2022)
• Difference coming from the parallax determination:

  • $\pi_{\rm E}=0.089 \pm0.014$ (Sahu et al.) or $0.13-0.3$ (Lam et al.)

1. Cold lower-mass planets are more abundant than cold higher-mass ones.

   The occurrence rate is integrated to ~1planet/star.

2. Cold planets frequently have companions.

   Microlensing allows to constrain the orbital configuration of multi-planet systems.

3. Microlensing can detect isolated BHs & BH binaries with wide separations.

   Constraining the parallax effect to a good precision is the key.

Summary

Future prospect: Microlensing from space

• Higher resolution：Individual stars resolved.
• Deeper observations：
  • More stars $\rightarrow$ More microlensing events
  • Smaller stars $\rightarrow$ Lower-mass planets

• Expected yield (3 months, 3.6 deg$^2$):
  • ~100 cold exoplanets (1-10 AU);
  • ~40 free-floating planets (FFPs);
  • Transiting planets with $P\lesssim30$ days.

Future prospect: Disk microlensing

• E.g., ASASSN microlensing, ZTF microlensing, Gaia microlensing, etc.

1. Cold lower-mass planets are more abundant than cold higher-mass ones.

   The occurrence rate is integrated to ~1planet/star.

2. Cold planets frequently have companions.

   Microlensing allows to constrain the orbital configuration of multi-planet systems.

3. Microlensing can detect isolated BHs & BH binaries with wide separations.

   Constraining the parallax effect to a good precision is the key.

4. Future prospects

   • Space-based microlensing, Disk microlensing

Summary