---
title: "Exoplanets"
book: "University Earth and Space Sciences — Year 3"
subject: earth
language: en
chapter: 28
exercises: 12
source: https://one-course.com/books/earth/5/en/chapter/28-exoplanets
license: CC-BY-NC-SA-4.0
credit: "One Earth and Space Sciences Book, One Course (one-course.com)"
---

# Chapter 28 — Exoplanets

In the autumn of 1994, two astronomers measuring the velocities of 142 stars found that one of them, 51 Pegasi, moved towards us and away from us with a period of about four days. The half-amplitude of the motion was $59\,\mathrm{m}\,\mathrm{s}^{-1}$, four times what their spectrograph could detect. The companion that pulled the star around had at least half the mass of Jupiter, on an orbit of $0.05\,\mathrm{au}$, eight times closer to its star than Mercury is to the Sun. No theory of planet formation had a place for such a body. The discovery, announced in October 1995, opened a new field of astronomy: three decades later, more than six thousand planets around other stars are confirmed, a few of them have been seen directly, and the air of some has been analysed. This chapter shows how planets too faint to be seen are found, what has been found, and how their atmospheres are read.

**You already know.**

The Doppler effect, the radial velocity, spectroscopic binaries, Kepler’s laws, the two-body problem and its centre of mass, Planck’s law and the scale height of an isothermal atmosphere are in the physics books. An exoplanet and the habitable zone of a star are in the school books. The flux and the luminosity of a source are in the Year 1 volume, in its chapter on telescopes and photometry; the equilibrium temperature, in its chapter on comparative planetology. Planet formation and the snow line are in the Year 2 volume. Jeans and [hydrodynamic escape](https://one-course.com/books/earth/5/en/chapter/27-planetary-atmospheres-and-habitability#def-b3-planetary-atmospheres-hydrodynamic) and the limits of the habitable zone are in this volume, in its chapter on planetary atmospheres.

## 28.1 The radial-velocity method

A planet does not orbit its star: both orbit their common centre of mass. The star, far heavier, moves on a small orbit of its own, and its light is Doppler-shifted back and forth with the period of the planet.

**Definition 28.1 (Radial-velocity method; minimum mass).**

The *radial-velocity method* detects a planet by the periodic variation of the radial velocity of its star, measured from the Doppler shift of the lines of the star’s spectrum. When the inclination $i$ of the orbit to the plane of the sky is unknown, it measures only the product $m_p\sin i$ of the planet’s mass by the sine of the inclination, the *minimum mass* of the planet.

**Theorem 28.2 (Radial-velocity semi-amplitude).**

A star of mass $M_\star$ and a planet of mass $m_p$ on a circular orbit of period $P$, inclined by $i$, give the star a radial velocity $v_r(t) = \gamma + K\cos\bigl(2\pi(t - t_0)/P\bigr)$, with $\gamma$ the velocity of the centre of mass and the semi-amplitude

$$
K = \left(\frac{2\pi G}{P}\right)^{1/3}
    \frac{m_p\sin i}{(M_\star + m_p)^{2/3}} .
$$

On an orbit of eccentricity $e$ the curve is no longer a sinusoid, $v_r = \gamma + K\bigl[\cos(\nu + \omega) + e\cos\omega\bigr]$, with $\nu$ the true anomaly and $\omega$ the argument of periastron, and $K$ is divided by $\sqrt{1 - e^2}$.

**Proof.** For the circular orbit: the separation $a$ of the two bodies obeys Kepler’s third law, $a^3 = G(M_\star + m_p)P^2/4\pi^2$. The star lies at the distance $a_\star = a\,m_p/(M_\star + m_p)$ from the centre of mass and covers its circle at the speed $v_\star = 2\pi a_\star/P$. Hence

$$
v_\star = \frac{2\pi}{P}\,\frac{m_p}{M_\star + m_p}
          \left(\frac{G(M_\star + m_p)P^2}{4\pi^2}\right)^{1/3}
        = \left(\frac{2\pi G}{P}\right)^{1/3}\frac{m_p}{(M_\star + m_p)^{2/3}} .
$$

The line of sight makes the angle $\pi/2 - i$ with the plane of the orbit, so only $v_\star\sin i$ of the star’s velocity lies along it, and that component turns with the star: $v_r - \gamma = v_\star\sin i\,
\cos\bigl(2\pi(t - t_0)/P\bigr)$. The eccentric form follows from the Keplerian ellipse and is admitted. ∎

Since the planet is much lighter than its star, $K$ fixes $m_p\sin i$ once the period is measured and the star’s mass is known from its spectrum. For 51 Pegasi b, $K = 59\,\mathrm{m}\,\mathrm{s}^{-1}$ and $P = 4.23\,\mathrm{d}$ around a star of one solar mass give $m_p\sin i = 0.47$ Jupiter masses. Jupiter moves the Sun at $12.5\,\mathrm{m}\,\mathrm{s}^{-1}$ over its $4331$-day orbit: a distant observer would see the Sun’s velocity swing by about $\pm 13\,\mathrm{m}\,\mathrm{s}^{-1}$ over twelve years. The Earth moves the Sun at $9\,\mathrm{cm}\,\mathrm{s}^{-1}$. The spectrograph that found 51 Pegasi b reached $13\,\mathrm{m}\,\mathrm{s}^{-1}$; its successors reach about $1\,\mathrm{m}\,\mathrm{s}^{-1}$ (a walking pace of $3.5\,\mathrm{km}/\mathrm{h}$), and the newest is built for about $10\,\mathrm{cm}\,\mathrm{s}^{-1}$, enough for an Earth-mass planet in the habitable zone of a low-mass star, where the planet’s orbit is short and its star is of low mass. The star itself sets a floor: the pulsations and spots of its surface, carried round by its rotation, shift its lines too, and a planet’s signal must be told apart from them.

![The radial velocity of a star with one planet, over two orbits, for an invented planet with K = 30\, m\, s-1 and a period of ten days. On a circular orbit (solid) the curve is a sinusoid of semi-amplitude K. On an eccentric orbit (dashed) the star moves fastest near periastron, and the curve becomes lopsided: its extremes are K(1 ± e ) and the peak-to-peak swing stays 2K.](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/fig-0720b0bc6a85.svg)

*The radial velocity of a star with one planet, over two orbits, for an invented planet with $K = 30\,\mathrm{m}\,\mathrm{s}^{-1}$ and a period of ten days. On a circular orbit (solid) the curve is a sinusoid of semi-amplitude $K$. On an eccentric orbit (dashed) the star moves fastest near periastron, and the curve becomes lopsided: its extremes are $K(1 \pm e\cos\omega)$ and the peak-to-peak swing stays $2K$.*

## 28.2 Transits

If the orbit is seen nearly edge-on, the planet passes in front of its star once per orbit and hides a small part of its disc.

**Definition 28.3 (Transit method; transit depth; impact parameter).**

The *transit method* detects a planet by the periodic dimming of its star as the planet crosses the stellar disc. The *transit depth* $\delta$ is the fractional drop of the star’s flux during the transit. The *impact parameter* $b = a\cos i/R_\star$ is the smallest distance between the centres of the planet and the star, as seen on the sky, in units of the stellar radius $R_\star$, for an orbit of radius $a$ and inclination $i$.

**Proposition 28.4 (Transit depth).**

A dark planet of radius $R_p$ entirely in front of a star of uniform brightness gives the depth $\delta = (R_p/R_\star)^2$.

**Proof.** The planet hides the area $\pi R_p^2$ of a disc of area $\pi R_\star^2$ that is equally bright everywhere; the flux drops in proportion. ∎

Jupiter crossing the Sun would dim it by $1.06\,\%$, the Earth by $84$ parts per million: the first is within reach of a small telescope, the second needs a space telescope that measures thousands of stars for years. A stellar disc is not uniform, though.

**Definition 28.5 (Limb darkening).**

*Limb darkening* is the decrease of the surface brightness of a star from the centre of its disc to its edge, the limb. Near the limb the line of sight enters the star’s atmosphere obliquely and reaches unit optical depth in higher, cooler layers, which emit less. It is often written as a quadratic law in $\mu = \cos\theta =
\sqrt{1 - r^2}$, with $r$ the distance from the centre of the disc in stellar radii: $I(\mu)/I(1) = 1 - u_1(1 - \mu) - u_2(1 - \mu)^2$.

A planet near the centre of a limb-darkened disc hides brighter light than average and deepens the transit; near the limb it hides less. The bottom of the light curve is rounded, and its edges soften. The duration of a transit carries the geometry.

**Theorem 28.6 (Transit duration).**

For a circular orbit of period $P$, radius $a$ and inclination $i$, with $p = R_p/R_\star$ and [impact parameter](#def-b3-exoplanets-transit) $b$, the time from the first contact to the last is

$$
T_{14} = \frac{P}{\pi}\arcsin\!\left(\frac{R_\star}{a}\,
          \frac{\sqrt{(1 + p)^2 - b^2}}{\sin i}\right),
$$

and the time between the second and third contacts, when the planet is wholly inside the disc, $T_{23}$, is given by the same formula with $1 - p$ in place of $1 + p$.

**Proof.** Let $\varphi = 2\pi t/P$ be the orbital phase counted from mid-transit. Projected on the sky, with $x$ along the planet’s path and $y$ across it, the planet’s centre lies at $x = a\sin\varphi$ and $y = a\cos i\cos\varphi$ from the star’s centre. In stellar radii, the squared separation is

$$
z^2 = \left(\frac{a}{R_\star}\right)^2(\sin^2\varphi + \cos^2 i\cos^2\varphi)
    = \left(\frac{a}{R_\star}\right)^2\sin^2 i\,\sin^2\varphi + b^2 .
$$

The discs touch from outside when $z = 1 + p$, at $\sin\varphi = \sqrt{(1 + p)^2 - b^2}\,R_\star/(a\sin i)$, symmetric about mid-transit. The transit lasts the phase $2\varphi$, that is the time $2\varphi P/2\pi = P\varphi/\pi$. With $z = 1 - p$ the discs touch from inside. ∎

For a wide orbit the arcsine is its argument: $T_{14} \approx (P/\pi)
(R_\star/a)\sqrt{(1 + p)^2 - b^2}$. A central transit lasts longest; a grazing one, with $b$ near $1 + p$, is short and V-shaped. Measuring $T_{14}$ and $T_{23}$ on a light curve gives $b$ and $a/R_\star$, so the inclination: the transit removes the $\sin i$ that the radial velocity leaves unknown.

![Left: a planet with p = R_p/R_ = 0.1 crosses the disc of its star at the impact parameter b = 0.3, at the four contacts 1 to 4; the disc is darker towards the limb. Right: its light curve for a/R_ = 15 and a period of ten days, computed for a star with the quadratic limb-darkening coefficients u_1 = 0.40, u_2 = 0.25 (solid) and for a uniform star (dashed, depth exactly p2 = 1\,\%). Between contacts 2 and 3 the limb-darkened curve is rounded, and deeper than p2 near mid-transit.](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/fig-3e78496d1e52.svg)

![Left: a planet with p = R_p/R_ = 0.1 crosses the disc of its star at the impact parameter b = 0.3, at the four contacts 1 to 4; the disc is darker towards the limb. Right: its light curve for a/R_ = 15 and a period of ten days, computed for a star with the quadratic limb-darkening coefficients u_1 = 0.40, u_2 = 0.25 (solid) and for a uniform star (dashed, depth exactly p2 = 1\,\%). Between contacts 2 and 3 the limb-darkened curve is rounded, and deeper than p2 near mid-transit.](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/fig-9cf5eacc1288.svg)

*Left: a planet with $p = R_p/R_\star = 0.1$ crosses the disc of its star at the [impact parameter](#def-b3-exoplanets-transit) $b = 0.3$, at the four contacts 1 to 4; the disc is darker towards the limb. Right: its light curve for $a/R_\star = 15$ and a period of ten days, computed for a star with the quadratic limb-darkening coefficients $u_1 = 0.40$, $u_2 = 0.25$ (solid) and for a uniform star (dashed, depth exactly $p^2 = 1\,\%$). Between contacts 2 and 3 the limb-darkened curve is rounded, and deeper than $p^2$ near mid-transit.*

**Proposition 28.7 (Transit probability).**

A planet on a circular orbit of radius $a$, oriented at random, transits its star (with $b < 1$) with the probability $R_\star/a$.

**Proof.** For orbits oriented at random, $\cos i$ is uniformly distributed between 0 and 1. The planet transits when $b = (a/R_\star)\cos i < 1$, that is when $\cos i < R_\star/a$, which happens with the probability $R_\star/a$. ∎

A distant observer would see the Earth transit the Sun with a probability of $0.47\,\%$, Jupiter with $0.09\,\%$. A transit survey sees only a small fraction of the planets around the stars it watches, and mostly the close-in ones.

Each method alone leaves something out. The radial velocity gives $m_p\sin i$; the transit gives $R_p/R_\star$ and $i$. Together they give the true mass and the radius, hence the mean density $\bar\rho = 3m_p/4\pi R_p^3$, the first clue to what a planet is made of. The first transits, reported early in 2000, were those of a planet already known from its star’s velocity, HD 209458 b. Its two transits showed the [limb darkening](#def-b3-exoplanets-limb) of the star and gave a radius of 1.27 Jupiter radii at an inclination of $87.1{}^{\circ}$. With the archive’s mass and radius, its mean density is $0.34\,\mathrm{g}\,\mathrm{cm}^{-3}$, a third of water’s: a gas giant inflated by its star’s heat.

A transiting planet keeps time. If a second planet tugs on it, its transits come early or late in a pattern that repeats with the gravitational interaction.

**Definition 28.8 (Transit-timing variation).**

A *transit-timing variation* is the departure of the times of a planet’s transits from a strictly periodic sequence, caused by the gravitational pull of another body of the system; it can reveal, and weigh, planets that do not themselves transit.

A dip in a light curve is not always a planet. An eclipsing pair of stars that grazes, or a faint eclipsing pair behind the target whose light is diluted by the target’s, can mimic a transit. A candidate becomes a planet when its mass is measured, by radial velocity or by timing variations, or when every other explanation has been ruled out.

## 28.3 Imaging and microlensing

**Definition 28.9 (Direct imaging; coronagraph).**

*Direct imaging* detects a planet by its own light, reflected or emitted, separated on the sky from the light of its star. A *coronagraph* is an instrument that blocks the light of the star at the focus of the telescope, by a mask or by the interference of the light, so that faint sources close to the star can be seen.

The difficulty is double. An orbit of $1\,\mathrm{au}$ seen from $10\,\mathrm{pc}$ spans only $0.1{}^{\prime\prime}$, and the planet is many orders of magnitude fainter than its star. Young giant planets are the easiest targets: still hot from their formation, they glow in the infrared, where the contrast with their star is least. In 2008, images taken at two large telescopes showed three planets around the star HR 8799, at projected separations of 24, 38 and $68\,\mathrm{au}$, moving counter-clockwise from one year to the next; their low luminosity and the age of the system give them 5 to 13 Jupiter masses. The planet in the photograph below lies $0.82{}^{\prime\prime}$ from its star; the space telescope’s [coronagraphs](#def-b3-exoplanets-imaging) reach a contrast of $1 \times 10^{-5}$ at $1{}^{\prime\prime}$ at $4.4\,\text{µ}\mathrm{m}$.

![The first exoplanet imaged by the James Webb Space Telescope, HIP 65426 b, in four infrared filters (bottom; top, the star’s field in a ground-based sky survey); the light of the star, at the position of the star symbol, is blocked by the coronagraphs. The planet is a young gas giant of about 6 to 8 Jupiter masses, some 15 to 20 million years old, about a hundred times farther from its star than the Earth is from the Sun. Image NASA/ESA/CSA, A Carter (UCSC), the ERS 1386 team, and A. Pagan (STScI), CC BY 4.0.](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/img-68a55e0be4b3.jpg)

*The first exoplanet imaged by the James Webb Space Telescope, HIP 65426 b, in four infrared filters (bottom; top, the star’s field in a ground-based sky survey); the light of the star, at the position of the star symbol, is blocked by the [coronagraphs](#def-b3-exoplanets-imaging). The planet is a young gas giant of about 6 to 8 Jupiter masses, some 15 to 20 million years old, about a hundred times farther from its star than the Earth is from the Sun. Image NASA/ESA/CSA, A Carter (UCSC), the ERS 1386 team, and A. Pagan (STScI), CC BY 4.0.*

A fourth method uses gravity as a lens. When a star passes almost exactly in front of a more distant one, its gravity bends the light of the background star and brightens it for a few weeks; a planet of the lensing star adds a short blip.

**Definition 28.10 (Gravitational microlensing; Einstein radius).**

*Gravitational microlensing* is the temporary brightening of a background star when a foreground star, the lens, passes close to the line of sight; the images it forms are too close together to be resolved. The *Einstein radius* is the angular radius $\theta_\mathrm E$ of the ring into which a point lens of mass $M$ at the distance $D_L$ would turn a source at the distance $D_S$ exactly behind it:

$$
\theta_\mathrm E = \sqrt{\frac{4GM}{c^2}\,\frac{D_S - D_L}{D_L D_S}} .
$$

**Proposition 28.11 (Point-lens magnification).**

A source at the angular distance $u\,\theta_\mathrm E$ from a point lens is brightened by the factor

$$
A(u) = \frac{u^2 + 2}{u\sqrt{u^2 + 4}} .
$$

**Proof.** *Admitted at this level.* ∎

Both results come from the bending of light by a mass in general relativity. For a lens of half a solar mass halfway to a source $8\,\mathrm{kpc}$ away, $\theta_\mathrm E = 0.71\,\mathrm{mas}$, and the Einstein ring, at the distance of the lens, has a radius of $2.9\,\mathrm{au}$. At $u = 1$ the source is brightened 1.34 times, at $u = 0.1$ ten times. A planet near the Einstein ring of its star adds a short deviation to the light curve. Microlensing thus finds planets a few astronomical units from their stars, cold planets that transits and radial velocities miss, but each event happens once and is never repeated. In an event of 2003, published the next year, a deviation lasting about seven days was explained only by a companion of $0.0039$ times the mass of the lens: a planet of about 1.5 Jupiter masses some $3\,\mathrm{au}$ from its star.

![Left: the geometry of a gravitational lens, not to scale; rays from the source pass the lens at about the Einstein radius R_ E = _ E D_L and reach the observer, who sees the source displaced to the image. Right: the magnification of a point source by a point lens against time, in units of the time t_ E the lens takes to move by one Einstein radius, for three distances of closest approach u_0 (in Einstein radii).](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/fig-6b4bff9fb33e.svg)

![Left: the geometry of a gravitational lens, not to scale; rays from the source pass the lens at about the Einstein radius R_ E = _ E D_L and reach the observer, who sees the source displaced to the image. Right: the magnification of a point source by a point lens against time, in units of the time t_ E the lens takes to move by one Einstein radius, for three distances of closest approach u_0 (in Einstein radii).](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/fig-b75873f9f2f5.svg)

*Left: the geometry of a gravitational lens, not to scale; rays from the source pass the lens at about the [Einstein radius](#def-b3-exoplanets-microlensing) $R_\mathrm E = \theta_\mathrm E D_L$ and reach the observer, who sees the source displaced to the image. Right: the magnification of a point source by a point lens against time, in units of the time $t_\mathrm E$ the lens takes to move by one [Einstein radius](#def-b3-exoplanets-microlensing), for three distances of closest approach $u_0$ (in Einstein radii).*

Every method has its own bias, and the catalogue of known planets reflects them. Radial velocities favour massive planets close to their stars; transits favour large planets close to their stars, and need the orbit edge-on; microlensing finds planets of a few astronomical units; imaging finds young giants tens or hundreds of astronomical units out. The figure below shows the result: each method fills its own region, and the Solar System’s planets fall mostly in the gaps between them.

**Figures as of 2026.**

On 23 October 2026 the NASA Exoplanet Archive listed 6451 confirmed planets. By method of discovery: 4785 by transits, 1204 by radial velocity, 292 by microlensing, 98 by imaging, 29 by [transit-timing variations](#def-b3-exoplanets-ttv) and the rest by rarer methods, among them the first ones of all, found in 1992 around a pulsar by the timing of its pulses. The first discoveries by radial velocity date from 1995, by transit from 2002, by microlensing and imaging from 2004.

![Confirmed planets with a measured mass (or minimum mass) and semi-major axis, by method of discovery (NASA Exoplanet Archive, 23 October 2026: 1691 by transit, 1151 by radial velocity, 286 by microlensing, 94 by imaging), with the Solar System’s planets (black squares). Masses predicted from a mass–radius relation are left out. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/fig-695d866a5dc5.svg)

*Confirmed planets with a measured mass (or [minimum mass](#def-b3-exoplanets-rv)) and semi-major axis, by method of discovery (NASA Exoplanet Archive, 23 October 2026: 1691 by transit, 1151 by radial velocity, 286 by microlensing, 94 by imaging), with the Solar System’s planets (black squares). Masses predicted from a [mass–radius relation](#def-b3-exoplanets-mass-radius) are left out. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.*

## 28.4 What has been found

The first surprise was the [hot Jupiters](#def-b3-exoplanets-classes), giant planets on orbits of a few days. 51 Pegasi b cannot have formed so close to its star; it must have formed farther out, perhaps at a few astronomical units, beyond the snow line of its disc, and migrated inwards through its interaction with the disc, as theory had predicted.

**Definition 28.12 (Hot Jupiter; super-Earth; sub-Neptune).**

A *hot Jupiter* is a giant planet on a close orbit, usually taken as a [minimum mass](#def-b3-exoplanets-rv) above 0.1 Jupiter masses and a period shorter than ten days. A *super-Earth* is a planet larger than the Earth but smaller than the [radius valley](#def-b3-exoplanets-radius-valley), that is up to about 1.5 to 2 Earth radii, and therefore probably rocky. A *sub-Neptune* is a planet above the [radius valley](#def-b3-exoplanets-radius-valley) but smaller than Neptune, about 2 to 4 Earth radii, whose radius requires an envelope of light gases or volatiles.

**Definition 28.13 (Occurrence rate).**

The *occurrence rate* of a class of planets is the mean number of such planets per star, or the fraction of stars that host one, in a given range of size (or mass) and period, after the counts of a survey are corrected for the planets it could not detect: those whose orbits are not aligned for a transit, and those too small or too far to be seen.

[Hot Jupiters](#def-b3-exoplanets-classes), prominent in the catalogue because they are the easiest to find, are rare: a radial-velocity survey of nearby Sun-like stars found them around $(1.2 \pm 0.38)\,\%$ of the stars. Small planets are common. Searching the light curves of $42\,000$ stars watched by the Kepler space telescope, one study found 603 planets, ten of them Earth-sized (1 to 2 Earth radii) and receiving between one and four times the Earth’s sunlight; corrected for the transit probability and for the planets the search would miss, $(11 \pm 4)\,\%$ of Sun-like stars host such a planet.

The sizes of small planets are not spread evenly. Among close-in planets measured precisely, those of 1.5 to 2.0 Earth radii are at least twice rarer than those just smaller or just larger.

**Definition 28.14 (Radius valley).**

The *radius valley* is the deficit of planets with radii between about 1.5 and 2.0 Earth radii among planets with periods shorter than about a hundred days; it separates the [super-Earths](#def-b3-exoplanets-classes) from the [sub-Neptunes](#def-b3-exoplanets-classes).

The valley is read as the trace of escape. A rocky core born with a hydrogen and helium envelope of only a few percent of its mass has twice the radius of the bare core. Close to a young, active star, such an envelope is heated and lost by [hydrodynamic escape](https://one-course.com/books/earth/5/en/chapter/27-planetary-atmospheres-and-habitability#def-b3-planetary-atmospheres-hydrodynamic): by the star’s high-energy radiation during the first hundred million years (photoevaporation), or by the heat of the cooling core itself (core-powered mass loss). The models herd planets into bare cores of about 1.3 Earth radii and planets of twice that size; few are left in between. The two mechanisms depend differently on the star, and planets around stars of different types may tell them apart.

![(a) Transiting planets with radii known to better than 10\,\%, around stars smaller than two solar radii, with periods of 1 to 100 days (2370 planets, NASA Exoplanet Archive, 23 October 2026); shaded, the radius valley of 1.5 to 2.0 Earth radii. (b) The same planets counted in 20 bins of equal width in R. The counts are not corrected for the planets the surveys miss, which are more numerous among the small ones; the valley shows nonetheless.](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/fig-6f249489292c.svg)

*(a) Transiting planets with radii known to better than $10\,\%$, around stars smaller than two solar radii, with periods of 1 to 100 days (2370 planets, NASA Exoplanet Archive, 23 October 2026); shaded, the [radius valley](#def-b3-exoplanets-radius-valley) of 1.5 to 2.0 Earth radii. (b) The same planets counted in 20 bins of equal width in $\log R$. The counts are not corrected for the planets the surveys miss, which are more numerous among the small ones; the valley shows nonetheless.*

**Definition 28.15 (Mass–radius relation).**

The *mass–radius relation* of a composition is the radius of a planet of that composition as a function of its mass, computed from the equations of state of its materials in hydrostatic equilibrium; a measured planet is compared with the relations of candidate compositions.

A rocky planet is compressed by its own weight, and its radius grows more slowly than $M^{1/3}$. A fit to models of two-layer planets built on the Earth’s seismic model gives $R/R_\oplus = (1.07 - 0.21\,C)
(M/M_\oplus)^{1/3.7}$, for 1 to 8 Earth masses and a core mass fraction $C$ from 0 to 0.4; fitted to the Earth, Venus and a few small exoplanets, it gives $C = 0.26 \pm 0.07$. An iron planet is smaller at the same mass; a planet half water, larger; an envelope of hydrogen of only two percent of the mass makes a planet of five Earth masses $2.6$ Earth radii across, the size of a [sub-Neptune](#def-b3-exoplanets-classes). The degeneracy is real: a radius and a mass give one density, and several mixtures of iron, rock, water and gas can share it. The rocky planet Kepler-10 b, at 3.24 Earth masses and 1.47 Earth radii, has a mean density of $5.6\,\mathrm{g}\,\mathrm{cm}^{-3}$; the gas giant WASP-39 b, at 89 Earth masses and 14.3 Earth radii, $0.17\,\mathrm{g}\,\mathrm{cm}^{-3}$.

![Mass and radius of 1078 transiting planets whose mass is measured to better than 25\,\% and radius to better than 10\,\% (grey; NASA Exoplanet Archive, 23 October 2026), with model curves of L. Zeng and others (2016, 2019): pure iron; Earth-like rock, a third iron and two thirds silicate; half water on such a core, at 500\, K; an Earth-like core with a hydrogen envelope of 2\,\% of its mass, at 500\, K. Red squares: the Solar System’s planets (Mercury and Mars lie off the scale, to the left).](https://one-course.com/images/onecourse/chapters/earth-5/b3-exoplanets/fig-d45364ccf91c.svg)

*Mass and radius of 1078 transiting planets whose mass is measured to better than $25\,\%$ and radius to better than $10\,\%$ (grey; NASA Exoplanet Archive, 23 October 2026), with model curves of L. Zeng and others (2016, 2019): pure iron; Earth-like rock, a third iron and two thirds silicate; half water on such a core, at $500\,\mathrm{K}$; an Earth-like core with a hydrogen envelope of $2\,\%$ of its mass, at $500\,\mathrm{K}$. Red squares: the Solar System’s planets (Mercury and Mars lie off the scale, to the left).*

## 28.5 Atmospheres

A transiting planet can be weighed and measured; it can also be smelled. During a transit, a thin ring of the planet’s atmosphere stands against the star, and starlight filters through it. At wavelengths where a gas of the atmosphere absorbs, the ring is opaque higher up, and the planet looks larger.

**Definition 28.16 (Transmission spectroscopy; secondary eclipse).**

*Transmission spectroscopy* measures the [transit depth](#def-b3-exoplanets-transit) as a function of wavelength, to detect the gases of the atmosphere that the starlight crosses at the planet’s limb. A *secondary eclipse* is the passage of a planet behind its star, half an orbit after the transit on a circular orbit; the drop of the system’s flux then measures the light of the planet’s day side.

**Proposition 28.17 (Transmission signal).**

An atmosphere of scale height $H \ll R_p$, made opaque over $n$ more scale heights at the wavelength of an absorption band than at neighbouring wavelengths, deepens the transit there by

$$
\Delta\delta \approx n\,\frac{2HR_p}{R_\star^2} .
$$

**Proof.** The planet’s apparent radius grows from $R_p$ to $R_p + nH$, so the depth $(R_p + nH)^2/R_\star^2$ grows by $(2nHR_p + n^2H^2)/R_\star^2$, which is $2nHR_p/R_\star^2$ when $nH \ll R_p$. ∎

The scale height $H = kT/\bar m g$ is large for a hot planet of low gravity with light hydrogen air, small for a cool, dense planet with heavy air. For the Earth seen against the Sun, with $H = 8\,\mathrm{km}$, each scale height adds 0.2 parts per million; for a hot, puffy gas giant, several hundred.

**Proposition 28.18 (Secondary-eclipse depth).**

If the day side of a planet radiates as a black body at the temperature $T_d$ and the star as one at $T_\star$, the [secondary eclipse](#def-b3-exoplanets-transmission) at the wavelength $\lambda$ has the depth

$$
\delta_\mathrm{ecl} \approx \left(\frac{R_p}{R_\star}\right)^2
  \frac{B_\lambda(T_d)}{B_\lambda(T_\star)},
$$

with $B_\lambda$ Planck’s law, the planet’s reflected light neglected.

**Proof.** Just before the eclipse the observer receives the star’s light, in proportion to $\pi R_\star^2 B_\lambda(T_\star)$, plus the day side’s, in proportion to $\pi R_p^2 B_\lambda(T_d)$; during the eclipse only the star’s. The fractional drop is the ratio of the planet’s term to the star’s, the planet’s being small. ∎

The eclipse is deepest in the infrared, where the cool planet is brightest relative to its star: it gives the day side’s temperature, and a hot day side next to a cool night side reveals a thin or absent atmosphere that does not carry heat around.

**Figures as of 2023.**

In 2022 the transmission spectrum of the gas giant WASP-39 b, measured between 3.0 and $5.5\,\text{µ}\mathrm{m}$ by the James Webb Space Telescope, showed a prominent absorption band of carbon dioxide at $4.3\,\text{µ}\mathrm{m}$, at 26 standard deviations, where earlier photometry had only hinted at the gas: its first clear identification in an exoplanet’s atmosphere. In 2023 five [secondary eclipses](#def-b3-exoplanets-transmission) of the Earth-sized TRAPPIST-1 b, which receives four times the Earth’s sunlight from a cool, small star, were measured at $15\,\text{µ}\mathrm{m}$. They fit heat re-radiated from the day side alone, with no detectable carbon dioxide: the planet has little or no atmosphere to carry heat to its night side.

The rocky planets of the habitable zones of small stars are within reach of these methods, at the limit. Whether they kept their air against the escape processes of the previous chapter is now an observational question.

## 28.6 Exercises

**Exercise 28.1 ★.**

(a) Compute the radial-velocity semi-amplitude of the Sun caused by Jupiter (mass $1898 \times 10^{24}\,\mathrm{kg}$, period $4331$ days, eccentricity 0.049) and by the Earth (mass $5.97 \times 10^{24}\,\mathrm{kg}$, period $365.2$ days, eccentricity 0.017). The Sun’s mass is $1.9884 \times 10^{30}\,\mathrm{kg}$. (b) Convert the precisions $3.5\,\mathrm{km}/\mathrm{h}$ and $0.35\,\mathrm{km}/\mathrm{h}$ into metres per second, and compare.

**Solution of Exercise 28.1.**

(a) $K = (2\pi G/P)^{1/3}m_p/\bigl((M_\odot + m_p)^{2/3}\sqrt{1 - e^2}\bigr)$. Jupiter: $P = 3.742 \times 10^{8}\,\mathrm{s}$, $(2\pi G/P)^{1/3} = 1.039 \times 10^{-6}$ in SI units, $K = 1.039 \times 10^{-6}\times1.898 \times 10^{27}/(1.990 \times 10^{30})^{2/3}/0.9988
= 12.5\,\mathrm{m}\,\mathrm{s}^{-1}$. The Earth: $P = 3.155 \times 10^{7}\,\mathrm{s}$, $K =
0.089\,\mathrm{m}\,\mathrm{s}^{-1}$, about $9\,\mathrm{cm}\,\mathrm{s}^{-1}$. (b) $3.5\,\mathrm{km}/\mathrm{h}$ $=
0.97\,\mathrm{m}\,\mathrm{s}^{-1}$ and $0.35\,\mathrm{km}/\mathrm{h}$ $= 0.097\,\mathrm{m}\,\mathrm{s}^{-1}$. Jupiter’s signal is thirteen times the first precision; the Earth’s is below even the second, and the star’s own noise comes on top.

**Exercise 28.2 ★.**

(a) Compute the [transit depths](#def-b3-exoplanets-transit) of Jupiter (radius $71\,492\,\mathrm{km}$) and of the Earth (radius $6378\,\mathrm{km}$) across the Sun (radius $695\,700\,\mathrm{km}$). (b) Compute the probability that a randomly placed observer sees each of them transit, with orbital radii of $778.5$ and $149.6\,\mathrm{M}\mathrm{km}$.

**Solution of Exercise 28.2.**

(a) $(71492/695700)^2 = 0.0106$, a dip of $1.06\,\%$; $(6378/695700)^2 = 8.4 \times 10^{-5}$, 84 parts per million. (b) $R_\star/a$: the Earth $695700/1.496 \times 10^{8} = 4.7 \times 10^{-3}$, about one chance in 215; Jupiter $695700/7.785 \times 10^{8} = 8.9 \times 10^{-4}$, one chance in 1100.

**Exercise 28.3 ★.**

51 Pegasi b gives its star a radial-velocity semi-amplitude of $59\,\mathrm{m}\,\mathrm{s}^{-1}$, with a period of $4.23\,\mathrm{d}$. Taking the star’s mass equal to the Sun’s, compute the planet’s [minimum mass](#def-b3-exoplanets-rv), in Jupiter masses and in Earth masses, and the radius of its orbit.

**Solution of Exercise 28.3.**

$m_p\sin i \approx K M_\odot^{2/3}(P/2\pi G)^{1/3}$ with $P =
3.655 \times 10^{5}\,\mathrm{s}$: $(P/2\pi G)^{1/3} = 9.55 \times 10^{4}$ in SI units, so $m_p
\sin i = 59\times(1.9884 \times 10^{30})^{2/3}\times9.55 \times 10^{4} =
8.9 \times 10^{26}\,\mathrm{kg}$, that is 0.47 Jupiter masses or 149 Earth masses (the correction for $m_p$ in $M_\star + m_p$ is below one part in a thousand). $a = \bigl(GM_\odot P^2/4\pi^2\bigr)^{1/3} = 7.66 \times 10^{9}\,\mathrm{m} =
0.051\,\mathrm{au}$.

**Exercise 28.4 ★.**

A star of half a solar mass at $4\,\mathrm{kpc}$ lenses a star at $8\,\mathrm{kpc}$ ($1\,\mathrm{pc} = 3.0857 \times 10^{16}\,\mathrm{m}$). (a) Compute its [Einstein radius](#def-b3-exoplanets-microlensing) in milliarcseconds and the radius of the Einstein ring, in astronomical units, at the distance of the lens. (b) Compute the magnification at $u = 1$, $0.5$ and $0.1$. (c) How long does the lens take to cross one [Einstein radius](#def-b3-exoplanets-microlensing) if it moves across the line of sight at $200\,\mathrm{km}\,\mathrm{s}^{-1}$?

**Solution of Exercise 28.4.**

(a) $4GM/c^2 = 4\times6.674 \times 10^{-11}\times9.94 \times 10^{29}/(2.998 \times 10^{8})^2 =
2953\,\mathrm{m}$; $(D_S - D_L)/(D_LD_S) = 1/D_S\times(D_S - D_L)/D_L =
1/(8\,\mathrm{kpc}) = 1/2.469 \times 10^{20}\,\mathrm{m}$. $\theta_\mathrm E =
\sqrt{2953/2.469 \times 10^{20}} = 3.46 \times 10^{-9}\,\mathrm{rad} = 0.71\,\mathrm{mas}$. At the lens, $R_\mathrm E = \theta_\mathrm E D_L = 3.46 \times 10^{-9}\times
1.234 \times 10^{20}\,\mathrm{m} = 4.27 \times 10^{11}\,\mathrm{m} = 2.85\,\mathrm{au}$. (b) $A(1) =
3/\sqrt5 = 1.34$; $A(0.5) = 2.25/(0.5\times2.062) = 2.18$; $A(0.1) =
2.01/(0.1\times2.002) = 10.0$. (c) $t_\mathrm E = 4.27 \times 10^{11}/2 \times 10^{5}
= 2.13 \times 10^{6}\,\mathrm{s}$, about 25 days: microlensing events last weeks.

**Exercise 28.5 ★★.**

A planet of radius ratio $p = 0.12$ orbits a star of one solar mass and one solar radius in $3.0\,\mathrm{d}$. (a) Compute $a/R_\star$. (b) Compute $T_{14}$ and $T_{23}$ for $b = 0$ and for $b = 0.7$, and the inclination in the second case. (c) How does the shape of the light curve tell a central transit from a grazing one?

**Solution of Exercise 28.5.**

(a) $a = (GM_\odot P^2/4\pi^2)^{1/3} = 6.09 \times 10^{9}\,\mathrm{m}$ ($0.0407\,\mathrm{au}$), so $a/R_\star = 8.75$. (b) With $P/\pi = 22.9\,\mathrm{h}$, for $b = 0$:

$$
T_{14} = \frac P\pi\arcsin\frac{1.12}{8.75} = 2.94\,\mathrm{h}, \qquad
T_{23} = \frac P\pi\arcsin\frac{0.88}{8.75} = 2.31\,\mathrm{h}.
$$

For $b = 0.7$: $\cos i = 0.7/8.75$, $i = 85.4{}^{\circ}$, $(a/R_\star)\sin i =
8.726$, and

$$
T_{14} = \frac P\pi\arcsin\frac{\sqrt{1.2544 - 0.49}}{8.726} = 2.30\,\mathrm{h},
\qquad
T_{23} = \frac P\pi\arcsin\frac{\sqrt{0.7744 - 0.49}}{8.726} = 1.40\,\mathrm{h}.
$$

(c) Ingress and egress take $(T_{14} - T_{23})/T_{14} = 21\,\%$ of the transit when it is central, $39\,\%$ at $b = 0.7$; a grazing transit, $b \to 1$, has no flat bottom at all and looks like a V.

**Exercise 28.6 ★★.**

From the masses and radii of the Exoplanet Archive, compute the mean densities of TRAPPIST-1 b (1.374 Earth masses, 1.116 Earth radii), Kepler-10 b (3.24, 1.47), WASP-39 b (89.31, 14.34) and HD 209458 b (232.0, 15.58), with $M_\oplus = 5.97 \times 10^{24}\,\mathrm{kg}$ and $R_\oplus =
6378\,\mathrm{km}$. Which are rocky, which gaseous?

**Solution of Exercise 28.6.**

$\bar\rho = M/\tfrac43\pi R^3$, with $\tfrac43\pi R_\oplus^3 =
1.087 \times 10^{21}\,\mathrm{m}^{3}$: TRAPPIST-1 b $1.374\times5.97 \times 10^{24}/(1.116^3\times
1.087 \times 10^{21}) = 5.43\,\mathrm{g}\,\mathrm{cm}^{-3}$; Kepler-10 b $5.60\,\mathrm{g}\,\mathrm{cm}^{-3}$; WASP-39 b $0.167\,\mathrm{g}\,\mathrm{cm}^{-3}$; HD 209458 b $0.337\,\mathrm{g}\,\mathrm{cm}^{-3}$. The first two are as dense as the Earth ($5.49\,\mathrm{g}\,\mathrm{cm}^{-3}$ with the same constants) and rocky; the last two are gas giants, lighter than water, WASP-39 b eight times lighter than Jupiter.

**Exercise 28.7 ★★.**

The radial velocity of a star with one planet on an eccentric orbit ranges from $+50.4\,\mathrm{m}\,\mathrm{s}^{-1}$ to $-9.6\,\mathrm{m}\,\mathrm{s}^{-1}$, and the velocity of the system’s centre of mass is $12.4\,\mathrm{m}\,\mathrm{s}^{-1}$. Using the eccentric form of the theorem, compute $K$ and $e\cos\omega$. Why can the mean of the two extremes differ from $\gamma$?

**Solution of Exercise 28.7.**

The extremes are $\gamma + K(1 + e\cos\omega)$ and $\gamma - K(1 -
e\cos\omega)$. Their difference is $2K = 60.0\,\mathrm{m}\,\mathrm{s}^{-1}$, so $K =
30.0\,\mathrm{m}\,\mathrm{s}^{-1}$; their mean is $\gamma + Ke\cos\omega =
20.4\,\mathrm{m}\,\mathrm{s}^{-1}$, so $Ke\cos\omega = 8.0\,\mathrm{m}\,\mathrm{s}^{-1}$ and $e\cos\omega = 0.27$. On an eccentric orbit the star lingers near apastron and rushes through periastron, so the curve is lopsided and the midpoint of its extremes is not its time average $\gamma$.

**Exercise 28.8 ★★.**

WASP-39 b has 89.31 Earth masses and 14.34 Earth radii; its star 0.939 solar radii. Take a temperature of $1100\,\mathrm{K}$ and a mean molecular mass of $2.3\,\mathrm{u}$ for its atmosphere. (a) Compute its gravity, its scale height and its [transit depth](#def-b3-exoplanets-transit). (b) Compute the transmission signal of one scale height, and of an absorption band two scale heights high, in parts per million.

**Solution of Exercise 28.8.**

(a) $g = GM/R^2 = 6.674 \times 10^{-11}\times5.332 \times 10^{26}/(9.144 \times 10^{7})^2 =
4.26\,\mathrm{m}\,\mathrm{s}^{-2}$; $H = kT/\bar m g = 1.381 \times 10^{-23}\times1100/(2.3
\times1.661 \times 10^{-27}\times4.26) = 934\,\mathrm{km}$. With $R_\star =
6.533 \times 10^{8}\,\mathrm{m}$, the depth is $(9.144 \times 10^{7}/6.533 \times 10^{8})^2 =
0.0196$. (b) $2HR_p/R_\star^2 = 2\times9.34 \times 10^{5}\times
9.144 \times 10^{7}/(6.533 \times 10^{8})^2 = 4.0 \times 10^{-4}$, 400 parts per million; two scale heights, 800: a gas giant of low gravity is the easiest target for [transmission spectroscopy](#def-b3-exoplanets-transmission).

**Exercise 28.9 ★★.**

TRAPPIST-1 b has a radius of 1.116 Earth radii; its star, 0.1192 solar radii and $2566\,\mathrm{K}$. Compute the depth of its [secondary eclipse](#def-b3-exoplanets-transmission) at $15\,\text{µ}\mathrm{m}$ for a day side at $400\,\mathrm{K}$ and at $500\,\mathrm{K}$ ($hc/k = 0.014\,388\,\mathrm{m}\,\mathrm{K}$). Why is a measurement of the depth a thermometer?

**Solution of Exercise 28.9.**

$p = 1.116\times6378/(0.1192\times695700) = 0.0858$, $p^2 = 7.37 \times 10^{-3}$. $B_\lambda \propto 1/(e^{hc/\lambda kT} - 1)$ with $hc/\lambda k =
959\,\mathrm{K}$ at $15\,\text{µ}\mathrm{m}$: the star $1/(e^{0.374} - 1) = 2.20$; a day side at $400\,\mathrm{K}$, $1/(e^{2.40} - 1) = 0.0998$, ratio $0.0453$, depth $3.3 \times 10^{-4}$ (334 parts per million); at $500\,\mathrm{K}$, $1/(e^{1.92} -
1) = 0.172$, ratio $0.078$, depth $5.7 \times 10^{-4}$. A hundred kelvins change the depth by $70\,\%$: the measured depth fixes the day-side temperature closely, and a hot day side means that little heat is carried to the night side.

**Exercise 28.10 ★★★.**

An invented red dwarf has 0.2 solar masses, 0.2 solar radii and a luminosity of $0.005$ solar luminosities. (a) At what distance does a planet receive the Earth’s stellar flux, and what is its period? (b) Compute the semi-amplitude caused by a planet of one Earth mass there, and compare it with the Earth’s effect on the Sun. (c) Compute the [transit depth](#def-b3-exoplanets-transit) of an Earth-sized planet. Why are such stars favoured targets in the search for rocky planets in habitable zones?

**Solution of Exercise 28.10.**

(a) $S \propto L/d^2$, so $d = \sqrt{0.005}\,\mathrm{au} = 0.0707\,\mathrm{au}$; $P = 2\pi\sqrt{a^3/GM} = 1.33 \times 10^{6}\,\mathrm{s} = 15.4\,\mathrm{d}$. (b) $K = (2\pi G/P)^{1/3}M_\oplus/(0.2M_\odot)^{2/3} = 0.75\,\mathrm{m}\,\mathrm{s}^{-1}$, 8.4 times the Earth’s effect on the Sun: the light star moves more, and the short period helps too. (c) $(6378/139140)^2 = 2.1 \times 10^{-3}$, 25 times the Earth’s transit across the Sun. Both signals are larger, the transits come every two weeks, and the transit probability $R_\star/a =
0.013$ is three times the Earth’s: small stars give rocky planets in their habitable zones a chance to be found and studied.

**Exercise 28.11 ★★★.**

For the quadratic limb-darkening law, show that the mean intensity of the disc is $I(1)(1 - u_1/3 - u_2/6)$. Deduce the depth, in units of $p^2$, of a small planet at $r = 0$, $0.5$ and $0.9$ for $u_1 = 0.40$, $u_2 = 0.25$. What error would a uniform-disc model make on the radius of a planet observed at mid-transit with $b = 0$?

**Solution of Exercise 28.11.**

With $r\,dr = -\mu\,d\mu$, the mean intensity is $\frac1\pi\int_0^1
I\,2\pi r\,dr = 2\int_0^1 I(\mu)\mu\,d\mu = 2\bigl[\tfrac12 - u_1(\tfrac12 -
\tfrac13) - u_2\int_0^1(1 - \mu)^2\mu\,d\mu\bigr]$. The last integral is $\tfrac12 - \tfrac23 + \tfrac14 = \tfrac1{12}$, so the mean is $1 - u_1/3 -
u_2/6$, here $0.825$. A small planet hides $\pi p^2 I(\mu)$ out of $\pi\times
0.825$: depth $p^2 I(\mu)/0.825$. At $r = 0$, $I = 1$: $1.212\,p^2$; at $r = 0.5$ ($\mu = 0.866$), $I = 0.942$: $1.142\,p^2$; at $r = 0.9$ ($\mu =
0.436$), $I = 0.695$: $0.842\,p^2$. A uniform-disc model would read $1.212\,p^2$ as $p^2$ and overestimate the radius by $\sqrt{1.212} = 1.10$, $10\,\%$.

**Exercise 28.12 ★★★.**

A survey watched $42\,000$ Sun-like stars and searched them for Earth-sized planets receiving one to four times the Earth’s sunlight, that is on orbits of 0.5 to $1\,\mathrm{au}$. (a) Compute the transit probability at 0.5, 0.7 and $1\,\mathrm{au}$. (b) If $11\,\%$ of the stars host one such planet, at about $0.7\,\mathrm{au}$, how many transit? (c) The survey found ten. What does the difference measure, and why must the [occurrence rate](#def-b3-exoplanets-occurrence) be corrected for it?

**Solution of Exercise 28.12.**

(a) $R_\odot/a$: $9.3 \times 10^{-3}$ at $0.5\,\mathrm{au}$, $6.6 \times 10^{-3}$ at $0.7\,\mathrm{au}$, $4.7 \times 10^{-3}$ at $1\,\mathrm{au}$. (b) $42000\times0.11\times
6.6 \times 10^{-3} = 31$ planets transit. (c) Only about a third of them were found: the others gave transits too shallow (84 parts per million for an Earth) or too few (one or two a year) to rise above the noise. That fraction is the detection efficiency; without correcting for it, and for the transit probability, the counts would underestimate how common such planets are by a factor of about 460.

## 28.7 Problem: A Planet Around Selene’s Lamp

**Problem 28.1.**

Weekend problem — weighing, sizing and sniffing an invented planet

Selene’s Lamp is an invented K dwarf of mass 0.80 solar masses, radius 0.76 solar radii and effective temperature $5000\,\mathrm{K}$. A radial-velocity survey finds that its velocity varies sinusoidally with a period of $12.0\,\mathrm{d}$ and a semi-amplitude of $2.59\,\mathrm{m}\,\mathrm{s}^{-1}$. A space telescope then sees transits of the same period, of depth $838$ parts per million and total duration $T_{14} = 3.23\,\mathrm{h}$. Take $M_\odot = 1.9884 \times 10^{30}\,\mathrm{kg}$, $R_\odot = 695\,700\,\mathrm{km}$, $L_\odot = 3.828 \times 10^{26}\,\mathrm{W}$, $M_\oplus = 5.97 \times 10^{24}\,\mathrm{kg}$, $R_\oplus = 6378\,\mathrm{km}$.

**Part I — The radial velocity.**

1. Compute the radius of the orbit, in metres and in astronomical units, neglecting the planet’s mass against the star’s.
2. Compute the planet’s orbital speed.
3. Compute the [minimum mass](#def-b3-exoplanets-rv) of the planet, in Earth masses.
4. Compute the radius of the star’s own orbit about the centre of mass, and check that it gives the observed semi-amplitude.
5. By how many times does the signal exceed a precision of $1\,\mathrm{m}\,\mathrm{s}^{-1}$ , and of $0.1\,\mathrm{m}\,\mathrm{s}^{-1}$ ?
6. If the orbit had $e = 0.2$ with the same $K$ , by how much would the [minimum mass](#def-b3-exoplanets-rv) change?

**Part II — The transit.**

7. Compute the radius ratio $p = R_p/R_\star$ .
8. Compute the radius of the planet, in kilometres and in Earth radii.
9. Compute $a/R_\star$ .
10. Compute the duration of a central transit ( $b = 0$ ).
11. From the observed $T_{14}$ , compute the [impact parameter](#def-b3-exoplanets-transit) .
12. Compute the inclination of the orbit.
13. What was the probability that this planet transits, seen from a random direction?

**Part III — What is it made of?**

14. Compute the true mass of the planet.
15. Compute its mean density.
16. On the Earth-like rocky model curve, a planet of this mass would have a radius of 1.74 Earth radii. Compare, and say what the planet must contain besides rock.
17. Where does the planet lie with respect to the [radius valley](#def-b3-exoplanets-radius-valley) ? Is it a [super-Earth](#def-b3-exoplanets-classes) or a [sub-Neptune](#def-b3-exoplanets-classes) ?
18. Compute its surface gravity.

**Part IV — Its atmosphere.**

19. Compute the luminosity of the star, in watts and in solar luminosities.
20. Compute the stellar flux the planet receives, relative to the Earth’s. Is it in the habitable zone?
21. With a Bond albedo of 0.3, compute its equilibrium temperature.
22. Compute the scale height of an atmosphere of hydrogen and helium (mean molecular mass $2.3\,\mathrm{u}$ ) at that temperature.
23. Compute the transmission signal of one scale height, and of a water band two scale heights high, in parts per million.
24. Compute the same for an atmosphere of steam ( $18\,\mathrm{u}$ ). Which of the two could a space telescope with a precision of about ten parts per million tell apart from a flat spectrum?
25. Give the planet’s mean density, in grams per cubic centimetre, to two significant figures.

**Solution of Problem 28.1.**

**1.** $M_\star = 0.80\times1.9884 \times 10^{30} = 1.591 \times 10^{30}\,\mathrm{kg}$, $P =
1.0368 \times 10^{6}\,\mathrm{s}$: $a = (GM_\star P^2/4\pi^2)^{1/3} = 1.4245 \times 10^{10}\,\mathrm{m}
= 0.0952\,\mathrm{au}$.

**2.** $v = 2\pi a/P = 86.3\,\mathrm{km}\,\mathrm{s}^{-1}$.

**3.** $m_p\sin i = KM_\star^{2/3}(P/2\pi G)^{1/3} = 2.59\times1.363 \times 10^{20}
\times1.352 \times 10^{5} = 4.77 \times 10^{25}\,\mathrm{kg}$, $7.99$ Earth masses.

**4.** $a_\star = a\,m_p/M_\star = 1.4245 \times 10^{10}\times4.77 \times 10^{25}/
1.591 \times 10^{30} = 427\,\mathrm{km}$; $2\pi a_\star/P = 2.59\,\mathrm{m}\,\mathrm{s}^{-1}$, the observed $K$.

**5.** $2.59$ times a precision of $1\,\mathrm{m}\,\mathrm{s}^{-1}$, 26 times one of $0.1\,\mathrm{m}\,\mathrm{s}^{-1}$: the planet is within reach of the best instruments, barely of the previous generation.

**6.** $m_p\sin i \propto K\sqrt{1 - e^2}$: $\times0.980$, 7.83 Earth masses, $2\,\%$ less.

**7.** $p = \sqrt{8.38 \times 10^{-4}} = 0.02895$.

**8.** $R_\star = 0.76\times695\,700\,\mathrm{km} = 528\,700\,\mathrm{km}$; $R_p =
15\,310\,\mathrm{km} = 2.40$ Earth radii.

**9.** $a/R_\star = 1.4245 \times 10^{10}/5.287 \times 10^{8} = 26.94$.

**10.** $T_{14} = (288\,\mathrm{h}/\pi)\arcsin(1.02895/26.94) = 3.50\,\mathrm{h}$.

**11.** $\sqrt{(1 + p)^2 - b^2} = (a/R_\star)\sin(\pi T_{14}/P) = 26.94\times
\sin(\pi\times3.23/288) = 0.949$ (with $\sin i \approx 1$); $b^2 =
1.0587 - 0.9008 = 0.158$, $b = 0.40$.

**12.** $\cos i = b\,R_\star/a = 0.40/26.94 = 0.0148$, $i = 89.15{}^{\circ}$.

**13.** $R_\star/a = 1/26.94 = 0.037$: one chance in 27.

**14.** $\sin i = 0.99989$: the true mass is 7.99 Earth masses, the [minimum mass](#def-b3-exoplanets-rv) within $1 \times 10^{-4}$.

**15.** $\bar\rho = 4.77 \times 10^{25}/\bigl(\tfrac43\pi(1.531 \times 10^{7})^3\bigr) =
4.77 \times 10^{25}/1.502 \times 10^{22} = 3180\,\mathrm{kg}\,\mathrm{m}^{-3}$, $3.18\,\mathrm{g}\,\mathrm{cm}^{-3}$.

**16.** The planet is $2.40/1.74 = 1.38$ times larger than a rocky planet of its mass, 2.6 times its volume. Rock alone cannot make it: it needs an envelope of light gas, a few percent of its mass in hydrogen and helium, or a large fraction of water.

**17.** At 2.40 Earth radii it lies above the valley, in the 2–3 Earth-radii peak: a [sub-Neptune](#def-b3-exoplanets-classes).

**18.** $g = Gm_p/R_p^2 = 6.674 \times 10^{-11}\times4.77 \times 10^{25}/(1.531 \times 10^{7})^2 =
13.6\,\mathrm{m}\,\mathrm{s}^{-2}$.

**19.** $L = 4\pi R_\star^2\sigma T^4 = 4\pi(5.287 \times 10^{8})^2\times5.670 \times 10^{-8}
\times5000^4 = 1.245 \times 10^{26}\,\mathrm{W}$, 0.325 solar luminosities.

**20.** $S/S_\oplus = 0.325/0.0952^2 = 35.9$: far inside the inner edge of the habitable zone.

**21.** $L/4\pi a^2 = 4.88 \times 10^{4}\,\mathrm{W}\,\mathrm{m}^{-2}$; $T_\mathrm{eq} =
\bigl(4.88 \times 10^{4}\times0.7/(4\sigma)\bigr)^{1/4} = 623\,\mathrm{K}$.

**22.** $H = kT/\bar m g = 1.381 \times 10^{-23}\times623/(2.3\times1.661 \times 10^{-27}
\times13.6) = 166\,\mathrm{km}$.

**23.** $2HR_p/R_\star^2 = 2\times1.66 \times 10^{5}\times1.531 \times 10^{7}/
(5.287 \times 10^{8})^2 = 1.82 \times 10^{-5}$, 18 parts per million; a band two scale heights high, 36 parts per million.

**24.** $H$ is $18/2.3 = 7.8$ times smaller, $21\,\mathrm{km}$: 2.3 parts per million per scale height, 4.6 for the band. At ten parts per million, the hydrogen-rich atmosphere shows its water band at more than three times the noise; a steam atmosphere looks flat, like a cloudy one or a planet without air.

**25.** **$3.2\,\mathrm{g}/\mathrm{cm}^{3}$**.