# Toward the intrinsic architecture of planetary systems

### Wei Zhu (祝伟)

Cornell Planetary Lunch

2019-11-18

## K2: 2014-2019)

Images credit: NASA/Kepler

## Kepler's limitations

• Kepler detects transiting planets with $$P\lesssim1$$ yr and $$R_{\rm p} \gtrsim R_\oplus$$

Image credit: iLectureOnline

Figure adapted from Penny et al. (2019)

## Kepler's limitations & consequences

• Kepler only detects transiting planets

• Multi-planet systems are co-planar?

• Kepler only detects relatively large planets ($$R_{\rm p}\gtrsim R_\oplus$$)

• Planets around the same host have similar sizes/masses, regular spacings?

• Kepler only detects short-period planets ($$P\lesssim1$$ yr)

• Kepler systems formed in a different way?

## Constraining mutual inclinations with transit method

$$T_{\rm dur} \propto \frac{R_*}{v_{\rm orb}} \propto a^{1/2} \propto P^{1/3}$$ $$\longrightarrow$$ $$\xi \equiv \frac{T_{\rm dur, in}/P_{\rm in}^{1/3}}{T_{\rm dur, out}/P_{\rm out}^{1/3}} = \frac{\sqrt{1-b_{\rm in}^2}}{\sqrt{1-b_{\rm out}^2}}$$

If exactly coplanar: $$\frac{b_{\rm in}}{b_{\rm out}} = \frac{a_{\rm in}}{a_{\rm out}} = \left(\frac{P_{\rm in}}{P_{\rm out}}\right)^{2/3}$$

• However, this is more like a lower bound on mutual inclination.

## Coplanarity $$\rightarrow$$ "Kepler dichotomy"

• Half of transiting planets (tranets) are found in single-tranet systems.
• If assuming small (and fixed) mutual inclinations, two populations are needed:
• Compact systems
• 1-planet systems (or 2-planet with large mutual inclinations)

## No sign for Kepler dichotomy

Zhu, Petrovich, Wu et al. (2018)

(See also Xie et al. 2016, Munoz Romero et al. 2018, Weiss et al. 2018)

One-tranet hosts

Multi-tranet hosts

## Transit Timing Variation (TTV)

• No interaction

## Transit + TTV

• # of systems with k transiting planets
• ~50% in transit singles
• # of systems with k transiting planets AND at least one showing TTV signals (Holczer++16)
• ~50% in transit singles
• Majority of transit singles are actually multi-planet systems with large mutual inclinations

Transit singles

Transit multis

## Dynamics coupled with multiplicity

{\rm inclination~dispersion~of~} k{\rm -planet~system} \propto k^\alpha
• Low-multiples: dynamically hot.
• High-multiples: dynamically cold.

## Eccentricity also shows multiplicity dependence

\frac{P^2}{4\pi^2} = \frac{a^3}{GM_\star} = \frac{3(a/R_\star)^3}{4\pi G\rho_\star}
\rightarrow T_0 \propto P^{1/3} \rho_\star^{-1/3}
T_0 = \frac{2R_\star}{2\pi a /P}

Xie, Dong, et al. (2016)

• Fewer-planet systems are dynamically hotter.
• 30% of Sun-like stars have Kepler planets.
• Each system has on average 3 planets.
• Our solar systems fits "well" in this picture.
i,~e \propto k^\alpha

Zhu, Petrovich, Wu et al., 2018

(See Xie et al. 2016, Van Eylen et al. 2018, Mills et al. 2019 for eccentricity constraints)

Orbital eccentricity

## Fraction of Sun-like stars with Kepler-like planets

• >50% (Fressin et al. 2013; Petigura et al. 2013; Winn & Fabrycky 2015)

• 30% (Zhu, Petrovich, Wu et al. 2018)
{\rm \#~of~planetary~systems} = \frac{\rm \#~of~detected~systems}{\rm detectability~of~individual~system}

## Planet-planet mutual inclinations affect the occurrence rate of planetary systems

mutual inclination

## Planet-planet mutual inclinations affect the occurrence rate of planetary systems

{\rm \#~of~planetary~systems} = \frac{\rm \#~of~detected~systems}{\rm detectability~of~individual~system}
• Transit probability: $$g_{\red{j}\blue{k}}$$
• Intrinsic multiplicity distribution: $$f_{\blue{k}}$$
• % of stars with planets: $$F_{\rm p} = \sum_\blue{k} f_{\blue{k}}$$
• Total # of surveyed stars: $$N_*$$
• # of systems with $$\red{j}$$ tranets:

$$N_{\red{j}} = N_* \sum_{\blue{k}} f_{\blue{k}} \cdot g_{\red{j}\blue{k}}$$

• # of single-tranet systems:

$$N_{\red{1}} = N_* \sum_{\blue{k}} f_{\blue{k}} \cdot g_{\red{1}\blue{k}}$$

• Total # of transiting systems:

$$\sum_{\red{j}} N_{\red{j}} = N_* \sum_{\blue{k}} f_{\blue{k}} \cdot\sum_{\red{j}} g_{\red{j}\blue{k}}$$

• Adding up the two equations:

$$N_{\red{1}} + \sum_{\red{j}} N_{\red{j}} = N_* \sum_{\blue{k}} f_{\blue{k}} \cdot \left( g_{\red{1}\blue{k}} + \sum_{\red{j}} g_{\red{j}\blue{k}} \right)$$

$$\red{j}$$ ($$\geq1$$): # of tranets

$$\blue{k}$$ ($$\geq1$$): # of planets

With very few assumptions, we have:

• $$F_{\rm p}=30\%$$ for Sun-like hosts
• $$F_{\rm p}=50\%$$ for M-dwarf hosts.

## Planetary system intrinsic architecture

• Multi-planet systems are not always coplanar

• Kepler only detects relatively large planets ($$R_{\rm p}\gtrsim R_\oplus$$)

• Planets around the same host have similar sizes/masses, regular spacings?

• Kepler only detects short-period planets ($$P\lesssim1$$ yr)

• Kepler systems formed in a different way?

## Intra-system uniformity?

1. Lissauer et al. (2011): Adjacent Kepler planets tend to have similar sizes.
2. Ciardi et al. (2012): The outer planet is preferentially larger than the inner one.
3. Weiss et al. (2018): Kepler multi-planet systems are like "peas in a pod."
4. Millholland et al. (2017): Planets in the same system should also have similar masses.

Image from Lissauer et al. (2011)

Outer planet larger

Inner planet larger

Data

## Detection threshold variation

Noisy sample

Quiet sample

Zhu, arXiv:1907.02074

Data from Weiss et al. (2018)

• The sample S/N cut corresponds to different radius thresholds for different stars.

Stellar noise (ppm)

• Kepler detections pile up toward the

detection threshold (defined by signal-to-noise ratio, S/N).

CDF

Detection threshold

• Detection threshold (in radius) varies for different stars.

• There are more small planets than large ones (e.g., Hsu et al. 2019).

Systems around

• noisy stars
• intermediate stars
• quiet stars.

## Detection threshold variation naturally leads to the size correlation

• Detection threshold (in radius) varies for different stars.

• There are more small planets than large ones (e.g., Hsu et al. 2019).

Systems around

• noisy stars
• intermediate stars
• quiet stars.

## What we learn about planet formation and evolution from no intra-system similarity?

• (Unlikely) Planet formation does not care about initial conditions (e.g., stellar properties).

• Dynamical evolution has erased most of the planet's memory of initial conditions:
• Large eccentricities and mutual inclinations;
• Dynamically compact (Pu & Wu 2015);
• No (strong) preference for mean-motion resonances (e.g., Fabrycky et al. 2014);
• Compositional diversity.

## Planetary system intrinsic architecture

• Multi-planet systems are not always coplanar

• Kepler multi-planet systems do not show intra-system uniformity

• Kepler only detects short-period planets ($$P\lesssim1$$ yr)

• Kepler systems formed in a different way?

Figure adapted from Penny et al. (2019)

## Kepler systems vs. Solar system

• Typically super Earths
• ~3 per system
• Gas fraction: a few %.

Cold Jupiters

Super Earths

22 from Kepler (triangles) + 39 from RV (squares)

Zhu & Wu, 2018, AJ, 156, 92

## Inner-outer correlation

Cold Jupiters

Super Earths

P({\rm CJ}|{\rm SE}) \approx 33\% {\rm ~vs.~} P({\rm CJ})=10\%

22 from Kepler (triangles) + 39 from RV (squares)

• 1/3 of Kepler systems have cold Jupiter companions.
• >50%, if [Fe/H]>0.

Zhu & Wu, 2018, AJ, 156, 92

## Inner-outer correlation

Cold Jupiters

Super Earths

P({\rm CJ}|{\rm SE}) \approx 33\% {\rm ~vs.~} P({\rm CJ})=10\%

22 from Kepler (triangles) + 39 from RV (squares)

• 1/3 of Kepler systems have cold Jupiter companions.
• >50%, if [Fe/H]>0.

Zhu & Wu, 2018, AJ, 156, 92

## Inner-outer correlation

Cold Jupiters

Super Earths

P({\rm CJ}|{\rm SE}) \approx 33\% {\rm ~vs.~} P({\rm CJ})=10\%

22 from Kepler (triangles) + 39 from RV (squares)

P({\rm SE}) = 30\%
P({\rm SE}|{\rm CJ}) = \frac{P({\rm SE})}{P({\rm CJ})} P({\rm CJ}|{\rm SE}) \approx 100\%
• 1/3 of Kepler systems have cold Jupiter companions.
• >50%, if [Fe/H]>0.
• Cold Jupiters (almost) always have inner super Earth companions!

Zhu & Wu, 2018, AJ, 156, 92

## (Un)Popularity of Solar system-like architecture

P({\rm no~SE,~CJ}) = [1-P({\rm SE}|{\rm CJ})] \times P({\rm CJ}) \approx 1\%

Zhu & Wu, 2018, AJ, 156, 92

• Solar system has no super Earth (70%).
• Solar system has a cold Jupiter (10%).

## Early Jupiter formation?

• Can potentially explain why Solar system has no super Earths (see also Batygin & Laughlin 2015).
• If Jupiter is not atypical $$\rightarrow$$ Giant planet cores form early.

## Inner-outer correlations $$-$$ theoretical implications

• Formation
• Cold Jupiters do not prohibit super Earths' formation.
• Disfavoring pebble accretion.
• They do not compete for building blocks.
• Disfavoring disk migration.
• Cold Jupiters require more stringent formation conditions.
• In situ formation + massive disk?

## Inner-outer correlation

P({\rm SE}|{\rm CJ}) = 100\% \left( \frac{N_{\rm in,out}/N_{\rm out}}{5/12} \right) \left( \frac{R_\star/a_{\rm in}}{0.03} \right)^{-1} \left( \frac{i_0}{4^\circ} \right)
i_0 \propto k^{-2}

1 yr

1. 12 long-period giants in Kepler data
2. 5 have inner transiting companions
3. 1 has five inner planets:

Inner and outer regions correlate in occurrence rate & dynamical states

## $$\pi$$ Mensae system

10 M_{\rm J}\\ 3~{\rm AU}\\ e=0.6
5 M_\oplus\\ 6~{\rm d}

Huang et al., (2018)

• HD 86226 with a known RV cold Jupiter also has a transiting candidate (TOI-652.01).
• TESS + Gaia will discover 100s more such systems.
• Multi-planet systems can have large ($$\gtrsim 10^\circ$$) mutual inclinations & eccentricities.
• Intra-system diversity: planets had violent past.
• Inner-outer correlation: Solar-system like architecture may be common.
• We need more observational constraints on planet formation!
• Atmosphere characterization; mass distribution; orbital properties; icy companions...

## Kepler is both revolutionary & limited

Figure adapted from Penny et al. (2019)

Minimum-mass solar/extra-solar nebula:

Weidenschilling (1977); Hayashi (1981)

\Sigma(r) \propto r^{-3/2}

The outer region dominates the mass and angular momentum budget

## The importance of outer region

Mimimum mass extra-solar nebula

Mimimum mass solar nebula

Chiang & Laughlin (2013)

• Multi-planet systems can have large ($$\gtrsim 10^\circ$$) mutual inclinations & eccentricities.
• Intra-system diversity: planets had violent past.
• Inner-outer correlation: Solar-system like architecture may be common.
• We need more observational constraints on planet formation!
• Atmosphere characterization; mass distribution; orbital properties; icy companions...

## gravitational microlensing

Figure from Penny et al. (2019)

# Back-up slides

## Full forward modeling confirming

(Even though the authors stated the opposite)

Non-clustered model

Clustered periods & sizes model

Transit depth ratio

Transit depth ratio

Transit depth ratio

Transit depth ratio

Period ratio of inner pair

Period ratio of outer pair

Stability boundary

Sensitivity limit

Noisy stars

Intermediate stars

Quiet stars

## Gravitational Microlensing

Brightness

Pollack et al. (1996)

Suzuki et al. (2016)

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

## 30-100 $$M_\oplus$$ planets unpredicted by core accretion theory

OB120026L:

x

y

Planet 1

Planet 2

Star

Planet 2

Planet 1

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

Madsen & Zhu, ApJL, 2019, 878, 29

Light curve from Han et al. (2013)

## Free-floating planets from microlensing

t_{\rm E}=0.3 {\rm~d} \\ M \approx \left(\frac{0.3~\rm d}{1~\rm d}\right) M_{\rm J} = 30~M_{\rm E}
t_{\rm E}=0.16 {\rm~d} \\ M \approx 8~M_{\rm E}
t_{\rm E}=0.9 {\rm~d} \\ M \approx 0.8~M_{\rm J}

Image credit: KASI

By Wei Zhu（祝伟）

# Planet intrinsic architecture (Cornell)

Planet lunch talk at Cornell University

• 251