What we learn from multi-planet systems

Wei Zhu (祝伟)

Planet Group Meeting

Tsinghua University

2021 March 5th

Zhu & Dong 2021, ARAA in press (arXiv:2103.02127)

Solar system as an example

Planets have diverse properties ($\sim10^2$ in separation and $\sim10^4$ in mass).
Planets have small but substantial orbital eccentricities ($\lesssim0.2$) and mutual inclinations ($\lesssim 6^\circ$).
Solar system was once dynamically active, and may become chaotic again in $\sim$ Gyr timescale.

Image credit: iLectureOnline

Transit (ground)

Transit (space)

Microlensing

Imaging

Based on data from NASA Exoplanet Archive.

Hot Jupiters

Cold Jupiters

Super Earths

Cold Neptunes

Sub-Earths

What You See Is Not What You Get!

w/ known companions

w/o known companions

Based on data from NASA Exoplanet Archive.

Hot Jupiters

Cold Jupiters

Super Earths

Cold Neptunes

Sub-Earths

Exoplanets frequently reside in multi-planet systems.

Planets inside the same system may have formed and evolved in a correlated way?

1. Orbital properties of planets in the same system are correlated.

No strong preference for mean-motion resonances (e.g., Lissauer et al. 2011, Fabrycky et al. 2014).

Spacing depends on multiplicity.
~80-90% of Kepler planet pairs do not allow intermediate planets (e.g., Fang & Margot 2013).
Kepler systems are dynamically packed (e.g., Pu & Wu 2015). However, interior/exterior planets remain allowed.

Spacing in mutual Hill radii

Orbital eccentricity depends on multiplicity

The transit duration method $$ \frac{T}{T_0} = \sqrt{1-b^2} \frac{\sqrt{1-e^2}}{1+e\sin{\omega}} $$ where $ T_0 \propto P^{1/3} \rho_\star^{-1/3} $.
- Transit singles have $\sigma_e\approx0.3$, whereas transit multis have $\sigma_e\approx0.05$ (Van Eylen et al. 2015, 2019, Xie et al. 2016, Mills et al. 2019).

Transit duration $T$

Orbital eccentricity depends on multiplicity

The transit duration method $$ \frac{T}{T_0} = \sqrt{1-b^2} \frac{\sqrt{1-e^2}}{1+e\sin{\omega}} $$ where $ T_0 \propto P^{1/3} \rho_\star^{-1/3} $.
- Transit singles have $\sigma_e\approx0.3$, whereas transit multis have $\sigma_e\approx0.05$ (Van Eylen et al. 2015, 2019, Xie et al. 2016, Mills et al. 2019).
Systems with more planets have smaller eccentricities and are dynamically colder.
- RV planets show a similar behaviour (e.g., Limbach & Turner 2015, Zinzi & Turrini 2017).

Transit singles

Transit doubles

Transit triples

Transit quadruples

Based on DR25 MCMC parameters and Berger et al. (2020) stellar parameters.

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}

Seager & Mallen-Ornellas (2003)

Transit singles have $\sigma_e\approx0.3$, whereas transit multis have $\sigma_e\approx0.05$ (Van Eylen et al. 2015, 2019, Xie et al. 2016, Mills et al. 2019).

Transit duration $T$

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}

Based on DR25 MCMC parameters and Berger et al. (2020) stellar parameters.

Systems with more planets have smaller eccentricities and are dynamically colder.

RV planets show a similar behaviour (e.g., Limbach & Turner 2015, Zinzi & Turrini 2017).

Transit singles

Transit doubles

Transit triples

Transit quadruples

Measure mutual inclinations in transit

The weighted transit duration ratio (Steffen et al. 2010).

$ \xi \equiv \frac{\rm (Transit~chord~length)_{in}}{\rm (Transit~chord~length)_{\rm out}} = \frac{T_{\rm in} P_{\rm in}^{-1/3}}{T_{\rm out} P_{\rm out}^{-1/3}} = \sqrt{\frac{1-b_{\rm in}^2}{1-b_{\rm out}^2}} $

Transit chord length = $ 2\sqrt{1-b^2} = v \cdot T $

$b$

Measure mutual inclinations in transit

The weighted transit duration ratio (Steffen et al. 2010).

Mutual inclination disperson $ \sigma_i\sim 3^\circ $ (e.g., Fang & Margot 2012, Fabrycky et al. 2014).
- A lower limit.

Weighted transit duration ratio $\xi$

Observed $\xi$ distribution

$\sigma_i=0$

$\sigma_i=1.8^\circ$

$\sigma_i=6^\circ$

Coplanarity $\longrightarrow$ Kepler dichotomy?

Figure from Ballard & Johnson (2016)

With a fixed mutual inclination distribution, at least two different populations of planetary systems are needed.
- Regardless of the intrinsic multiplicity distribution (e.g., Lissauer et al. 2011, Johansen et al. 2012, He et al. 2019).
Transit method cannot constrain the mutual inclination because of the degeneracy with intrinsic multiplicity.
- Transit + RV (Tremaine & Dong 2012, Figueira et al. 2012).
- Transit + TTV (Zhu et al. 2018).

Mutual inclination also depends on multiplicity

Transit doubles

Transit triples

Transit quadruples

$ \xi = \frac{T_{\rm in} P_{\rm in}^{-1/3}}{T_{\rm out} P_{\rm out}^{-1/3}} = \sqrt{\frac{1-b_{\rm in}^2}{1-b_{\rm out}^2}} $

Weighted transit duration ratio

Systems with more planets have smaller mutual inclinations and are dynamically colder.
- E.g., Zhu et al. (2018), He et al. (2020).

Fewer-planet systems are dynamically hotter.

Each system has on average 3 planets (average multiplicity).

Dynamical evolution may have reshaped the architecture.

Zhu et al., 2018

(see also He et al. 2020)

Intrinsic architecture of the inner ($<1$ AU) system

$ \sigma_i,~\sigma_e \propto k^\zeta $

{\rm \#~of~planetary~systems} = \frac{\rm \#~of~detected~systems}{\rm detectability~of~individual~system}

Planet-planet mutual inclinations affect the frequency of planetary systems

Coplanarity $\longrightarrow$>50% of Sun-like stars have Kepler-like planets (e.g., Fressin et al. 2013, Petigura et al. 2013).

mutual inclination

{\rm \#~of~planetary~systems} = \frac{\rm \#~of~detected~systems}{\rm detectability~of~individual~system}

Planet-planet mutual inclinations affect the frequency of planetary systems

Fraction of Sun-like stars with Kepler-like planets is ~30% (Zhu et al. 2018).
Fraction of Sun-like stars with planets in an extended parameter space may be higher (e.g., Mulders et al. 2018, He et al. 2020).
- Or not, if planet occurrences are strongly correlated.

(Colors mean different multiplicities.)

1. Orbital properties of planets in the same system are correlated $\longrightarrow$ Dynamical evolution may have largely reshaped the system architecture.

2. The existences of planets in the same system are correlated.

Cold Jupiters

Super Earths

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

Zhu & Wu, 2018, AJ, 156, 92

Inner-outer correlation

Kepler-48 as an example

$M_{\rm b}=3.9\pm2.1\,M_\oplus$

$P_{\rm b}=4.78$ d

$M_{\rm c}=14.6\pm2.3\,M_\oplus$

$P_{\rm c}=9.67$ d

$M_{\rm d}=7.9\pm4.6\,M_\oplus$

$P_{\rm d}=42.90$ d

$M_{\rm e}=2.1\pm0.1\,M_{\rm J}$

$P_{\rm d}=982$ d

Marcy et al. (2014)

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 (almost) always have inner super Earth companions!

$ P({\rm SE}|{\rm CJ}) \cdot P({\rm CJ}) = P({\rm CJ}|{\rm SE}) \cdot P({\rm SE}) $

$ \rightarrow P({\rm SE}|{\rm CJ})=100\% $

Cold Jupiters

Super Earths

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

Super Earths preferentially have cold Jupiter companions $$ P({\rm CJ}|{\rm SE}) \approx33\% $$
- $ P({\rm CJ})=10\%$ (Cumming et al. 2008).
- $P({\rm CJ}|{\rm SE})>50\%$, if [Fe/H]>0.

Cold Jupiters almost always have inner super Earth companions $$ P({\rm SE}|{\rm CJ})=\frac{P({\rm CJ}|{\rm SE}) \cdot P({\rm SE})}{P({\rm CJ})} \approx100\% $$

Zhu & Wu (2018), Bryan et al. (2019)

Inner-outer correlation

Independent confirmations

Long-period ($>2\,$yr) transiting planets frequently have inner transiting companions.
- Herman, Zhu & Wu (2019); Masuda, Winn, & Kawahara (2020).
TESS detections of super Earths in RV cold Jupiter systems.
- pi Mensae (Huang et al. 2018, Gandolfi et al. 2018), HD 86226 (Teske et al. 2020)
TESS + Gaia.

2 yr

Figure from Herman, Zhu & Wu (2019)

Strong inner-outer correlation challenging formation models of Kepler planets

Migration models typically expect anti- or no correlations:
- Formation-and-migration (e.g., Bern model, Schlecker et al. 2020).
- Migration-then-assembly (e.g., pebble accretion, Izidoro et al. 2015, Bitsch et al. 2019).
In situ?

Pebbles ($\sim$ cm)

Pebble isolation mass ($\sim10\,M_\oplus$)

See reviews by Ormel (2017), Liu & Ji (2020)

On the Poisson process assumption

Fraction of stars with Kepler-like planets ~30%.
Fraction of stars with cold giants ~10%.
Fraction of stars with planets in the joint parameter space?
- 37% [$=1-(1-30\%)\times(1-10\%)$] under the Poisson assumption.
- ~30%, because of the strong correlation.

Figure adapted from Penny et al. (2019)

2. The existences of planets in the same system are correlated $\longrightarrow$ Planet formation is a global behaviour.

3. Physical properties of planets in the same system are (largely) not correlated.

Peas in a pod?

Weiss et al. (2018)

(See also Lissauer et al. 2011, Ciardi et al. 2013, Millholland et al. 2017)

Radius of the inner planet $R_\oplus$

Radius of the outer planet $R_\oplus$

Image from Chatterjee & Tan (2014)

(See also Ormel et al. 2017)

Mass/Radius similarities expected from some formation models

Pebble isolation mass (Lambrechits et al. 2014, Liu et al. 2019)

$$M_{\rm iso} \approx 10 \left(\frac{h/r}{0.04}\right)^3 \left(\frac{M_\star}{M_\odot}\right) M_\oplus $$

Pebble

($\sim$ cm)

Detection threshold

Systems around

noisy stars
intermediate stars
quiet stars.

Detection threshold variation naturally leads to the size correlation

Radius of inner Kepler planet

Radius of outer Kepler planet

Planet size

Stellar noise

Numbers

Planet size

Zhu (2020)

(see also Murchikova & Tremaine 2020)

Systems around

noisy stars
intermediate stars
quiet stars.

Detection threshold variation naturally leads to the size correlation

Radius of inner Kepler planet

Radius of outer Kepler planet

Planet size

Stellar noise

Numbers

Planet size

Zhu (2020)

(see also Murchikova & Tremaine 2020)

After the detection threshold variation is taken into account, the "peas in a pod" pattern is (largely) removed.

Future super-Kepler mission (e.g., Earth 2.0) should find sub-Earths in known systems.

Zhu (2020)

(see also Murchikova & Tremaine 2020)

Solar system in the context

See Zhu (2020) and Murchikova & Tremaine (2020).

Figure adapted from Penny et al. (2019)

Solar system planets have very diverse properties.
In a (super-)Kepler's view, Solar system planets (Venus & Earth) have very similar properties.

Kepler system vs. Solar system

Figure adapted from Penny et al. (2019)

Kepler systems are massive.
- Inside 1 AU: 3 super Earths (~5 $M_\oplus$) vs. ~2 $M_\oplus$.

(Un)Popularity of Solar system-like architecture

Figure from Zhu & Wu (2018)

Systems with inner small planets and outer giant planets are a popular kind.
Systems very similar to ours are very rare.
- Solar system has no super Earth (~70%).
- Solar system has a cold Jupiter (~10%).