Intra-system uniformity

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

Image from Lissauer et al. (2011)

Outer-to-inner radius ratio

Outer planet larger

Inner planet larger

0.4 $R_\oplus$

0.9 $R_\oplus$

1.0 $R_\oplus$

0.5 $R_\oplus$

11 $R_\oplus$

9 $R_\oplus$

4.0 $R_\oplus$

3.9 $R_\oplus$

Do Solar system planets show

size (or mass) similarity?

No, if knowing all planets

Giant planets are not observable: orbital period too long
Mercury & Mars undetectable: too small
Yes, if only Venus & Earth are seen

Intra-system uniformity:

intrinsic or observation bias?

Do they have outer ( $\gtrsim 1$ au) giant planet companions?

Are there smaller undetectable inner ( $\lesssim 1$ au) planets?

Cold Jupiters

Super Earths

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

Zhu & Wu, 2018, AJ, 156, 92

(see also Bryan et al. 2019, Herman, Zhu, & Wu 2019)

Super Earth-cold Jupiter correlations

Cold Jupiters

Super Earths

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

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

(see also Bryan et al. 2019, Herman, Zhu, & Wu 2019)

Super Earth-cold Jupiter correlations

Cold Jupiters

Super Earths

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

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}) = 30\%

P({\rm SE}|{\rm CJ}) = \frac{P({\rm SE})}{P({\rm CJ})} P({\rm CJ}|{\rm SE}) \approx 100\%

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

(see also Bryan et al. 2019, Herman, Zhu, & Wu 2019)

Super Earth-cold Jupiter correlations

Miranda Herman

(Zhu et al. 2018)

HATNet

Keck

Cold Jupiters

(~10%)

Cold Neptunes

Most Kepler-like planetary systems have outer giant planets.

Almost all cold giant planets have inner small planets.

Intra-system size and mass similarities no longer hold.

Kepler-like systems frequently have

outer giant companions

Kepler

planets

(30%)

(Zhu et al. 2018)

Zhu & Wu (2018)

Herman, Zhu, & Wu (2019)

Do they have outer ( $\gtrsim 1$ au) giant planet companions?
- Most of them do.

Are there smaller undetectable inner ( $\lesssim 1$ au) planets?
- Does dynamical stability allow for undetectable small planets?
- Is there sign for such planets?

Intra-system uniformity:

intrinsic or observation bias?

Does dynamical stability allow for undetectable small planets?

Yes, even for high-multiples ( $\geq 4$ ).

Even less an issue for 2-tranet & 3-tranet systems, which contribute >80% of Kepler multi systems.

From Weiss et al. (2018)

{\rm Transit ~ S/N} = \frac{(R_{\rm p}/R_\star)^2 \cdot \sqrt{\rm \# ~ of ~ transit ~ events}}{\rm photometric ~ noise ~ per ~ transit}

{\rm Transit ~ S/N} = \frac{(R_{\rm p}/R_\star)^2 \cdot \sqrt{\rm \# ~ of ~ transit ~ events}}{\rm photometric ~ noise ~ per ~ transit}

Is there a sign for smaller undetectable planets?

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.

CDPP CDF

From Ciardi et al. (2013)

Kepler detections pile up toward the

detection threshold.

{\rm S/N} = \frac{(R_{\rm p}/R_\star)^2 \cdot \sqrt{\rm \# ~ of ~ transit ~ events}}{\rm photometric ~ noise ~ per ~ transit}

{\rm S/N} = \frac{(R_{\rm p}/R_\star)^2 \cdot \sqrt{\rm \# ~ of ~ transit ~ events}}{\rm photometric ~ noise ~ per ~ transit}

On the apparently similar sizes

Zhu, arXiv:1907.02074

Correlation strength $r$

Forward modeling

A more proper way is fully forward modeling the detection and selection processes from the intrinsic planetary (radius or mass) distribution. Instead of randomly drawing parameters from the observed distribution, one should draw from the intrinsic distribution and then apply the same detection criteria (e.g., S/N cut) on these simulated planets. This process requires knowing the intrinsic planet distribution function and having access to the automated Kepler detection pipeline. It is further complicated by the fact that the planet distribution function is period dependent (e.g., Dong, & Zhu 2013; Hsu et al. 2019) and possibly multiplicity dependent, and that the Kepler detection efficiency is weakly multiplicity dependent (Zink et al. 2019).

There is a shortcut that circumvents these problems. As it is the transit S/N that determines whether or not a planet is detected, we can simply start from the observed S/N distribution.

Why resampling S/N makes sense?

Page 2 of Zhu, arXiv:1907.02074

Period ratio of inner pair

Period ratio of outer pair

Stability

boundary

Kepler multi-planet systems are dynamically compact (Pu & Wu 2015).

Stability boundary (measured in period ratio) shifts to smaller value as planet radius decreases.

Systems around

noisy stars
intermediate stars
quiet stars.

On the apparently regular spacings

Zhu, arXiv:1907.02074

Period ratio of inner pair

Period ratio of outer pair

Kepler multi-planet systems are dynamically compact (Pu & Wu 2015).

Stability boundary (measured in period ratio) shifts to smaller value as planet radius decreases.

Systems around

noisy stars
intermediate stars
quiet stars.

On the apparently regular spacings

Zhu, arXiv:1907.02074

Weiss et al. (2018) cut at Period ratio=4.
Detection bias & dynamical stability.

Data used in Weiss et al. (2018)

All planet triplets

Zhu, arXiv:1907.02074

On the apparently regular spacings

Summary

In (super) Kepler's eye, Solar system planets would show size/mass similarity, even though they do not intrinsically.

Do Kepler planets have outer giant planet companions?
- Most of them do (Zhu & Wu, 2018; Herman, Zhu, & Wu, 2019).

Are there smaller undetectable inner planets?
- Does dynamical stability allow for undetectable small planets?
  - Yes, even for high-multiples.
- Is there sign for such planets?
  - Yes. Kepler detections pile up toward detection threshold (set by S/N).
The sample S/N cut corresponds to different radius thresholds for different stars (Zhu, arXiv:1907.02074).

Q: Do planets show intra-system uniformity intrinsically?

back up slides

Period ratio of inner pair

Period ratio of outer pair

Stability boundary

Noisy stars

Intermediate stars

Quiet stars

Kepler systems do not show intrinsic intra-system uniformity

Wei Zhu (祝伟)

Intra-system uniformity

Do Solar system planets show

size (or mass) similarity?

Intra-system uniformity:

intrinsic or observation bias?

Super Earth-cold Jupiter correlations

Super Earth-cold Jupiter correlations

Super Earth-cold Jupiter correlations

Kepler-like systems frequently have

outer giant companions

Intra-system uniformity:

intrinsic or observation bias?

Does dynamical stability allow for undetectable small planets?

Is there a sign for smaller undetectable planets?

On the apparently similar sizes

Why resampling S/N makes sense?

On the apparently regular spacings

On the apparently regular spacings

On the apparently regular spacings

Summary

back up slides