---
date: '2024-01-08'
description: solar system structure, exoplanet detection techniques including transit photometry, radial velocity, astrometry, and gravitational microlensing.
id: W1
modified: 2026-06-05 15:08:34 GMT-04:00
tags:
  - astron2e03
title: Solar systems in the context of exoplanets
created: '2024-01-08'
published: '2024-01-08'
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slug: thoughts/university/twenty-three-twenty-four/astron-2e03/W1
permalink: https://aarnphm.xyz/thoughts/university/twenty-three-twenty-four/astron-2e03/W1.md
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---
Ref: [[thoughts/university/twenty-three-twenty-four/astron-2e03/Solar System Exoplanets 2024.pdf|Solar System Exoplanets 2024]]

## Obj.

- content of solar system and orbital properties
- Compare properties of Solar System to known exoplanetary
- _six_ techniques for exoplanet detection & limitation.

---

## How people learn?

> Student enter the classroom with preconceptions about how the world works. If their _initial understanding is not fully engaged, they may fail to grasp new concepts_

_develop competence_

1. foundation knowledge
2. interrelationships among facts and concepts
3. retrieval and application.

## Solar system

Sun <span>&rarr;</span> terrestrial planets <span>&rarr;</span> asteroid belt <span>&rarr;</span> Jovian (gas giants) \~ Ice giant planets <span>&rarr;</span> Trans-Neptunian objects (TNOs) (Dwarf planets <span>&rarr;</span> Kuiper belt <span>&rarr;</span> Oort cloud)

> `1 au` (astronomical unit): average distance between Earth and Sun

> Planetary orbits are (nearly) _co-planar_

- Dispersion in mutual inclinations: $\Delta{i} \approx 2\text{ deg}$
- Pluto and many other TNOs are \_more highly inclined

## Consequence of **Protoplanetary disks**

_from Alma telescope_

- radio images of _warm dust continuum_ ($\leq 10^6\text{ Myrs}$)
- Disk sizes $\approx 100\text{ au}$
- Variety of morphologies

> \[!question\] Question
>
> Concentric gaps opened by _protoplanets_?

- Due to active construction of _two protoplanets?_

> \[!question\] Question
>
> What other _dynamical properties_ do you expect for planets formed from a disk?

- **Keplerian Motion**: Planets formed from a disk are expected to exhibit Keplerian motion <span>&rarr;</span> direct consequence rather than properties

## Regular vs. Irregular Satellites (aka, moons)

| Regular Satellites                                        | Irregular                     |
| --------------------------------------------------------- | ----------------------------- |
| Resemble mini planetary systems                           | Irregular orbits              |
| prograde                                                  | prograde or retrograde orbits |
| low mutual inclinations, e.g: 4 Galilean moons of Jupyter | highly elliptical             |
| nearly circular orbits                                    | highly inclined               |

![[thoughts/university/twenty-three-twenty-four/astron-2e03/exoplanets-discovery-technique.webp|Exoplanets discovery technique]]

> Most exoplanetary systems are compact

Kepler-11 System

## Transit

![[thoughts/university/twenty-three-twenty-four/astron-2e03/transit.webp|Trasit]]

- Time-resolved photometry (i.e. stellar brightness) = “light curve”
  Can measure:
- Orbital period
- Orbital inclination
  - Has to be edged-on
  - **relative to telescope**, not to the _star_
  - Reference is line-of-sight to exoplanetary system.
- Planet radius

### transit depth.

$$
\begin{aligned}
\mathbf{Z} &= \frac{\text{Area}_{pl}}{\text{Area}_{*}} = (\frac{R_{pl}}{R_*})^2 \\\
&\\\
Z&: \text{transit depth} \\\
R_{pl}&: \text{planet radius} \\\
R_{*}&: \text{stellar radius} \\\
\end{aligned}
$$

### limb-darkening

- appears fainter at their edges compared to centres
- depends on the **star’s temperature structure** and the **wavelength of the observations**

![[thoughts/university/twenty-three-twenty-four/astron-2e03/transit-graph.webp| Example transit graph]]

> The higher the depth, the larger the planet

> Limb-darkening only depends on the stars, and wavelength observing at

> Depth **doesn’t depends** on how far away the planets is away from the star (depends on the durations, orbiting more slowly)

> Duration is impacted by _period_ and _inclination_

### known transiting expolanets

![[thoughts/university/twenty-three-twenty-four/astron-2e03/radius-period-diagram.webp|Radius Period diagram]]

Geometric transit probability:

$$
\begin{align*}
P_{tr} &\approx \frac{R_{*}}{a} \\
&= 0.5\% \left( \frac{R_{*}}{R_{\odot}} \right) \left( \frac{a}{a_{\oplus}} \right)^{-1}
\end{align*}
$$

where $\odot$ and $\oplus$ is the _sun_ and _earth_ respectively

## Transit Timing Variations

_oscillating orbits_

![[thoughts/university/twenty-three-twenty-four/astron-2e03/transit-timing-variation.webp|Transit timing variation example]]

> B exhibits larger TTV

> A is more massive, since B is influenced by A (pulled by gravitational effect)

## Radial velocity

Only sees the bigger stars
chemical abundances in star atmosphere <span>&rarr;</span> graphs (dotted vertical lines)

Time-resolved spectroscopy to measure _Doppler-shifted spectral features_

> Radial velocity shift translates into wavelength shift

$$
\frac{\lambda_{obs}}{\lambda_{ref}} = \sqrt{\frac{1+v_{rad}/c}{1-v_{rad}/c}}
$$

Can measure

- Orbital period
- Orbital eccentricity
- Planet’s minimum mass

semi-amplitude of RV signal _K_

> K depends on the orbital inclination _i_ such that RV method is _sensitive an upper limit on planetary mass_

$$
\begin{align}
K &= M_p(\frac{2\pi{G}}{PM_{*}^{2}})^{1/3} \\\
K &= M_p \sin{i} (\frac{2\pi{G}}{PM_{*}^{2}})^{1/3}
\end{align}
$$

_Derivation_

$$
\begin{align}
a_sM_s &= a_pM_p \\\
P^2 &= \frac{4\pi^2}{GM_{*}}a_p^3
\end{align}
$$

$M_p$: planet mass, $i$: orbital inclination, $P$: orbital period, $M_{*}$: stellar mass

> - Insensitive to face-on, maximally sensitive to edge-on
> - Easier to detect big planets

Transits + Radial Velocity (Radius + mass) <span>&rarr;</span> planet bulk density

## Astrometry

> proper motions

![[thoughts/university/twenty-three-twenty-four/astron-2e03/astrometry.webp]]

### Aside - Parallax

`1 pc = 1 AU / 1"`
1” arcsec = 1/60 arcminutes = (1/60)/60 degrees

parsec is the distance from two planets

Consider a star-planet system located at _d_ from us

$x=d\theta = 1{AU}(\frac{d}{1pc})(\frac{\theta}{1"})$

$$
\triangle{\theta} = \frac{M_p}{d}(\frac{GP^2}{4\pi^2M^2_{*}})^{1/3}
$$

biased on long period

### Gravitational Microlensing

> Mass bends spacetime <span>&rarr;</span> light ray are bent by a curved spacetime <span>&rarr;</span> massive object act as _gravitational lens_

![[thoughts/university/twenty-three-twenty-four/astron-2e03/gravitational-microlensing.webp]]