Astro 497: Week 2, Friday

Exoplanet Detection: Transits

Logistics

• How was length of lab 2?

Overview of Today

• Transit Method

• Observables

• Transit Probability

• Transit Surveys

• Multiple Transiting Planet Systems

• Strengths & Weaknesses

• Multiple Transiting Plantes

Transit Method

Credit: NASA

Credit: Joshua Winn (2010)

Normalized Flux versus time

$$\begin{equation} f(t) = 1 + k^2 \frac{I_p(t)}{I_\star} - \begin{cases} k^2 \alpha_{\rm tra}(t) & \text{transits,} \\ 0 & \text{outside eclipses,} \\ k^2 \frac{I_p(t)}{I_\star} \alpha_{\rm occ}(t) & \text{occultations.} \end{cases} \end{equation}$$

• Disk-averaged intensity of the star: $I_\star$

• Disk-averaged intensity of the planet: $I_p$

• Planet-star radius ratio: $k = R_p/R_\star$

Keplerian Orbit

Motion in the Orbital plane:

$$r = \frac{a(1-e^2)}{1+e\cos f}$$

• Star-planet sepration: $r$

• Semimajor axis: $a$

• Eccentricity: $e$

• True anomaly: $f$

Orbit projected onto the sky

$$\begin{eqnarray} X & = & -r \cos(\omega+f), \\ Y & = & -r \sin(\omega+f)\cos i,\\ Z & = & r \sin(\omega+f)\sin i \end{eqnarray}$$

• Inclination: $i$

• Arguement of pericenter: $\omega$

• Longitude of ascending node: $\Omega$ (arbitrarily set to $180\degree$)

When do transits occur?

$$X^2+Y^2 \le R_\star + R_p$$

$$f_{{\rm tra}} = +\frac{\pi}{2} - \omega$$

$$f_{{\rm occ}} = -\frac{\pi}{2} - \omega$$

Transit Observables

Credit: Joshua Winn (2010)

One transit

• Transit Depth: $\delta$ (dimensionless)

• Impact Parameter: $b$ (units of stellar radii)

• Ingress Duration: $\tau$

• Transit Durations:

  • Total duration: $t_{IV}-t_{I}$

  • Full-transit duration: $t_{III}-t_{II}$

  • Mathematically-convenient duration: $T$

  • Best-measured duration: $(t_1+t_2-t_3-t_4)/2$

Impact parameter

$$b = \frac{a \cos i}{R_\star} \left(\frac{1-e^2}{1 + e\sin\omega}\right)$$

Transit depth

$$\delta \approx \left(\frac{R_p}{R_\star}\right)^2~\left[1 - \frac{I_p(t_{\rm tra})}{I_\star}\right]$$

Limb Darkening

Credit: Figure 3 of Knutson et al. (2007)

See Mandell & Agol (2002) and numerous implementations (e.g., from Eric Agol and Transits.jl)

Multiple Transits

• Orbital period: $P$

• Epoch of $n$th transit: $t_n$

• Deviations from mean values

  • Transit Timing Variations (TTVs): $\delta t_i$

  • Transit Duration Variations (TDVs): $\delta T_i$

  • Transit Depth Variations (TdVs): $\delta d_i$

Transit Probability

Credit: Joshua Winn (2010)

$$p_{\rm tra} = \left(\frac{R_\star \pm R_p}{a}\right) \left(\frac{1 + e\sin\omega}{1 - e^2} \right)$$

If marginalize over $\omega$:

$$p_{\rm tra} = \left(\frac{R_\star \pm R_p}{a}\right) \left(\frac{1}{1 - e^2} \right)$$

For a small planet on a circular orbit:

$$p_{\rm tra} = \frac{R_\star}{a}~\approx 0.005~\left( \frac{R_\star}{R_{\odot}} \right) \left( \frac{a}{{\rm AU}} \right)^{-1}$$

Q: Could you please explain how to derive the transit and occupation probability equations, i.e. (9) and (10), from equations (7), (8)? Do we need the assumption that a >> R, where a is the orbital semi-major axis and R is star/planet radii?

Q: Is there a limit to how far out a planet can be from its star and we still detect it by transit?

Transit Surveys

Ground-based Transit Surveys

Early Surveys

Credit: Horne 2002

Notable ground-based transit surveys

Dozens of large planets:

• SuperWASP

• HATNet

Small number of small planets around nearby stars:

• MEarth

• TRAPPIST

Credit: David Anderson (CC-BY-SA-3.0 license)

Space-based Transit Surveys

• CoRoT

• Kepler/K2

• TESS

• Plato

Kepler/K2

Credit: NASA

TESS

Credit: NASA

Plato

Copyright: Copyright: ESA/ATG medialab

Source: Exoplanet Archive 9/2/2022

Credit: Lissauer et al. 2022, submitted to AAS Journals

Stregnths & Weaknesses

Strengths

• Small telescope can do high-quality science

• CCDs make relative photometry (relatively) easy

• Can observe many stars at once

• Not restricted to specific spectral types.

• Transit signal-to-noise $\sim R_p/R_\star$

• Transit probability high for short-period planets.

Weaknesses

• Transits only provide information about a planet during a very small fraction of it's orbit.

  • Scheduling observations of a full transit can be difficult .

  • If observing from the ground, then day light or weather can lead to missing a transit.

• Transit probability decreases with increasing orbital separation

  • Most planets won't transit (as seen from Earth)

  • Unlikely that transits can be used to study any particular planet

• Other astrophysical/atmospheric objects/effects cause similar photometric effects.

  • Follow-up observations are often needed to validate or confirm transiting planet candidates.

• Need to observe for multiple orbital periods to measure period robustly.

Reading Questions

Multiple Transiting Planets

Kepler-11

Credit: NASA/Tim Pyle

TRAPPIST-1

Credit: NASA

Helper Code