---
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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---
Ref: [[thoughts/university/twenty-three-twenty-four/astron-2e03/Solar System Exoplanets 2024.pdf|Solar System Exoplanets 2024]]

## Obj.

- Describe the contents of the Solar System and relate orbital properties to formation.
- Compare the Solar System with observed exoplanetary systems, accounting for detection bias.
- Explain the physics and limitations of the six detection techniques in the lecture.

---

## 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

Moving outward from the Sun: terrestrial planets, asteroid belt, gas giants (Jupiter and Saturn), ice giants (Uranus and Neptune), then trans-Neptunian objects. Pluto is a dwarf planet in the Kuiper belt. The distant Oort cloud is approximately spherical; it does not continue the planets’ flat arrangement.

An astronomical unit, $1\,\mathrm{au}$, is approximately the mean Earth–Sun distance. Planetary orbits are nearly _coplanar_: the lecture gives a dispersion in mutual inclinations of $\Delta i\approx 2^\circ$. Pluto and many other trans-Neptunian objects have more inclined orbits.

## Consequence of **Protoplanetary disks**

ALMA images in slide 8 show thermal continuum emission from dust around young stars, with ages $\lesssim 10\,\mathrm{Myr}=10^7\,\mathrm{yr}$ and disk sizes around $100\,\mathrm{au}$. The images show rings, gaps and other structures; their spatial resolution limits which inner regions can be distinguished.

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

Possible, though a gap is insufficient evidence to count planets. ALMA’s [HD 169142 analysis](https://www.almaobservatory.org/en/audiences/new-alma-image-reveals-migrating-planet-in-protoplanetary-disk/) models several rings with a single migrating planet.

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

A rotating disk supplies a common direction of orbital angular momentum. Planets formed within it should initially have mostly prograde, nearly coplanar orbits; damping by gas also favours low eccentricities. Later encounters can change those orbits. Keplerian motion alone says little about a disk origin: any bound two-body orbit under Newtonian gravity follows Kepler’s laws.

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

| Regular satellites                     | Irregular satellites                                               |
| -------------------------------------- | ------------------------------------------------------------------ |
| Prograde orbits                        | Prograde or retrograde orbits                                      |
| Low mutual inclinations                | Often highly inclined                                              |
| Nearly circular orbits                 | Often highly eccentric                                             |
| Example: Jupiter’s four Galilean moons | Orbital arrangement differs from a small coplanar planetary system |

Formation in a disk around a planet accounts for the shared direction and plane of regular satellites. Capture is a possible origin for irregular satellites; their orbital properties alone do not identify the history of each moon.

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

Kepler-11 illustrates a compact multiplanet system. The lecture’s statement that most exoplanetary systems are compact describes the detected sample. Close-in planets transit more often and produce larger stellar radial-velocity signals, so this sample needs a selection correction before it can describe all planetary systems.

## Transit

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

A transit reduces the observed stellar brightness when a planet crosses the stellar disk. A time series of brightness measurements is a _light curve_. Repeated transits give an orbital period; their depth and shape constrain the planet-to-star radius ratio and orbital geometry. Converting a radius ratio into a planet radius requires the stellar radius.

Define inclination $i$ between the orbital angular-momentum axis and our line of sight. An edge-on orbit has $i=90^\circ$; a face-on orbit has $i=0^\circ$. A transit requires an orbit sufficiently close to edge-on for the planet’s projected path to cross the star.

### transit depth.

For an opaque planet crossing a uniformly bright star, with the planet fully inside the stellar disk,

$$
Z=\frac{F_{\mathrm{out}}-F_{\mathrm{in}}}{F_{\mathrm{out}}}
  =\frac{\pi R_p^2}{\pi R_*^2}
  =\left(\frac{R_p}{R_*}\right)^2.
$$

Here $Z$ is the fractional transit depth, $R_p$ the planet radius and $R_*$ the stellar radius. Doubling $R_p$ gives four times the depth for the same star. A grazing transit or extra unresolved light changes this interpretation.

### limb-darkening

The stellar disk usually appears fainter toward its edge. In a photosphere whose temperature falls outward, a line of sight near the limb samples higher, cooler layers. The brightness profile depends on the stellar temperature structure and observing wavelength.

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

For planets around the same star, a deeper transit generally indicates a larger radius, subject to the transit chord and limb-darkening. Different chords sample different parts of the same brightness profile.

Orbital distance does not enter the simple area-ratio depth formula. It affects the probability and duration of a transit. Duration also depends on stellar size, inclination and eccentricity, so duration alone does not determine the period.

### known transiting expolanets

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

For randomly oriented circular orbits, the geometric transit probability is

$$
P_{\mathrm{tr}}=\frac{R_*+R_p}{a}
\approx\frac{R_*}{a}
\approx 0.5\%\left(\frac{R_*}{R_\odot}\right)
\left(\frac{a}{1\,\mathrm{au}}\right)^{-1},
$$

where $a$ is the orbital separation and $R_\odot$ is the Sun’s radius. The approximation assumes $R_p\ll R_*$. Doubling the separation halves the geometric probability. A survey also needs enough observing time to catch transits, which further favours short periods.

## Transit Timing Variations

Mutual gravitational perturbations can make successive transits arrive earlier or later than a constant-period prediction. For transit number $n$, the timing residual is

$$
\delta t_n=t_n-(t_0+nP).
$$

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

B has the larger timing variations in this figure. Check the vertical scales: A varies by roughly a minute, B by roughly ten minutes. A larger perturbing mass can produce larger timing shifts in its companion, which motivates the lecture’s mass question. Inferring that A must be more massive from these amplitudes alone goes too far: the periods, proximity to orbital resonances and eccentricities also affect the signal. Mass estimates require a dynamical fit. See [Deck and Agol’s TTV analysis](https://ntrs.nasa.gov/citations/20230001197).

## Radial velocity

Time-resolved spectroscopy measures shifts in stellar absorption lines. The planet and star orbit their common centre of mass, so the star alternately approaches and recedes from us. The line wavelengths supply the line-of-sight component of that motion.

With positive $v_{\mathrm{rad}}$ defined away from the observer, the special-relativistic Doppler relation for motion along the line of sight is

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

At stellar reflex speeds, $|v_{\mathrm{rad}}|\ll c$, giving $\Delta\lambda/\lambda_{\mathrm{ref}}\approx v_{\mathrm{rad}}/c$. A fitted RV curve constrains the orbital period, eccentricity and a mass–inclination combination. It does not require a large star; bright targets with stable, sharp spectral lines are easier to measure precisely.

The stellar RV semi-amplitude is

$$
K=\left(\frac{2\pi G}{P}\right)^{1/3}
\frac{M_p\sin i}{(M_*+M_p)^{2/3}\sqrt{1-e^2}}.
$$

For $M_p\ll M_*$ and a circular orbit ($e=0$), this becomes the lecture’s expression:

$$
K\approx M_p\sin i\left(\frac{2\pi G}{P M_*^2}\right)^{1/3}.
$$

Here $M_p$ is the planet mass, $M_*$ the stellar mass and $P$ the period. RV gives approximately $M_p\sin i$, a **minimum mass**, because $\sin i\le 1$.[^rv-mass] A face-on orbit has no line-of-sight reflex signal; an edge-on orbit has the largest signal for a fixed mass and orbit.

_Derivation for circular orbits_

Let $a_*$ and $a_p$ be the stellar and planetary orbital radii about their common centre of mass, and $a=a_*+a_p$ their separation. Then

$$
a_*M_*=a_pM_p,\qquad
P^2=\frac{4\pi^2a^3}{G(M_*+M_p)},\qquad
K=\frac{2\pi a_*}{P}\sin i.
$$

Substitution gives the circular RV expression above. Larger planet masses and shorter periods increase the signal; a smaller host-star mass also increases it. Stellar activity can produce competing spectral shifts.

A transit constrains $i$, allowing RV to give the planet mass instead of only its minimum. Combined with the radius, this yields bulk density:

$$
\rho_p=\frac{3M_p}{4\pi R_p^3}.
$$

[^rv-mass]: Slide 59 says “upper limit”, contradicting “minimum mass” on slide 53 and the inclination dependence. [NASA’s catalogue documentation](https://heasarc.gsfc.nasa.gov/W3Browse/catalog/exoplanets.html) confirms the minimum-mass convention.

## Astrometry

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

Astrometry measures a star’s position on the sky. The model must separate proper motion (the star’s motion through space), annual parallax (the observer’s changing position) and orbital motion about the system’s centre of mass.

### Aside - Parallax

A parsec is the distance at which $1\,\mathrm{au}$ subtends one arcsecond. It is about $3.26$ light-years. Annual parallax uses the Earth–Sun baseline; the parallax angle $p$ is the angular semimajor axis of the annual parallactic ellipse. For the full displacement between opposite extremes, use $2p$. The distance $d$ follows from the reciprocal relation below. See [ESA’s distance definitions](https://www.esa.int/Science_Exploration/Space_Science/Gaia/Cosmic_distances).

$$
1\,\mathrm{arcsec}=\frac{1}{3600}{}^\circ,
\qquad
\frac{d}{\mathrm{pc}}=\left(\frac{p}{\mathrm{arcsec}}\right)^{-1}.
$$

For a small angular separation $\theta$ and a system at distance $d$, the projected physical separation is

$$
x\approx d\theta
=1\,\mathrm{au}\left(\frac{d}{1\,\mathrm{pc}}\right)
\left(\frac{\theta}{1\,\mathrm{arcsec}}\right).
$$

The expression $d\theta$ requires $\theta$ in radians; the right-hand expression handles the parsec–arcsecond conversion. This is a projected separation, which need not equal an orbital semimajor axis.

The star’s astrometric orbital scale is $\alpha=a_*/d$. For $M_p\ll M_*$,

$$
\alpha\approx\frac{M_p}{d}
\left(\frac{GP^2}{4\pi^2M_*^2}\right)^{1/3}.
$$

With consistent physical units, $\alpha$ is in radians. The measured orbit is projected onto the sky; fitting its shape constrains inclination. The signal grows with planet mass and orbital size and shrinks with distance. Long periods give larger orbits, though observing enough of one requires a longer survey.

### Gravitational Microlensing

A foreground star bends the light of a more distant background star. As their alignment changes, the background source brightens and fades. A planet orbiting the foreground lens can add a shorter deviation to that light curve.

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

The lecture notes that sensitivity is greatest near the lens’s Einstein ring, often corresponding to projected separations of one to a few astronomical units. The inferred separation is projected on the sky. A lensing event generally does not repeat, so follow-up cannot simply wait for the next orbit to reproduce it.

## Direct imaging

The sixth method in slides 95–102 detects light from the planet separately from the star. A coronagraph suppresses starlight; residual glare and limited angular resolution still hide faint, close companions. The lecture’s HR 8799 example contains four directly imaged planets.

Young, massive planets at wide separations are favourable targets: they emit heat left from formation and can be resolved farther from the star’s glare. An image measures projected separation and brightness. Recovering mass from brightness requires an age estimate and a cooling model. [NASA’s detection overview](https://science.nasa.gov/exoplanets/how-we-find-and-characterize/) puts this method alongside transits, RV, astrometry and microlensing.