Part IV: Frontiers | Chapter 1

Exoplanets

Worlds beyond our Solar System: detection methods, Kepler demographics, planetary atmospheres, and the search for habitable worlds

Overview

The discovery of exoplanets — planets orbiting other stars — has transformed our understanding of planetary systems. Since the first confirmed detection of 51 Pegasi b by Mayor and Queloz in 1995, over 5,700 exoplanets have been confirmed, revealing an astonishing diversity of planetary architectures. Hot Jupiters, super-Earths, sub-Neptunes, and circumbinary planets challenge the notion that our Solar System is typical.

In this chapter we derive the physics underlying the three main detection methods (transit, radial velocity, and direct imaging), analyze the statistical results from the Kepler mission, discuss planetary atmosphere characterization, and assess habitability criteria.

1. The Transit Method

When a planet passes in front of its host star as seen from Earth, it blocks a fraction of the starlight, producing a periodic dip in the light curve.

1.1 Transit Depth

The fractional decrease in stellar flux during a transit is simply the ratio of the projected areas:

$$\boxed{\delta = \left(\frac{R_p}{R_*}\right)^2}$$

For a Jupiter-sized planet transiting a Sun-like star: $\delta \approx (R_J/R_\odot)^2 \approx 0.01 = 1\%$. For an Earth-sized planet: $\delta \approx 8.4 \times 10^{-5} \approx 84$ ppm. The Kepler mission achieved photometric precision of $\sim 20$ ppm, enabling the detection of Earth-sized planets.

1.2 Transit Duration and Geometry

The total transit duration for a circular orbit is:

$$T_{\text{dur}} = \frac{P}{\pi}\arcsin\left(\frac{R_*}{a}\sqrt{(1+k)^2 - b^2}\right)$$

where $k = R_p/R_*$ is the radius ratio, $b = a\cos i/R_*$is the impact parameter, and $i$ is the orbital inclination. For a central transit ($b = 0$) and $k \ll 1$:

$$T_{\text{dur}} \approx \frac{R_* P}{\pi a} \approx 13\left(\frac{P}{1\,\text{yr}}\right)^{1/3}\left(\frac{M_*}{M_\odot}\right)^{-1/3}\left(\frac{R_*}{R_\odot}\right)\;\text{hours}$$

1.3 Geometric Transit Probability

The probability that a randomly oriented orbit produces a transit is:

$$\boxed{p_{\text{transit}} = \frac{R_*}{a} \approx 0.5\%\left(\frac{a}{1\,\text{AU}}\right)^{-1}\left(\frac{R_*}{R_\odot}\right)}$$

For hot Jupiters at $a = 0.05$ AU, $p_{\text{transit}} \approx 10\%$. For Earth analogues at 1 AU, $p_{\text{transit}} \approx 0.5\%$. This low probability means that transit surveys must monitor many thousands of stars.

2. The Radial Velocity Method

A planet orbiting a star causes the star to reflex-orbit the center of mass. The resulting Doppler shift of stellar spectral lines reveals the planet's presence.

2.1 The Radial Velocity Semi-Amplitude

For a circular orbit, the maximum radial velocity of the star is:

$$\boxed{K = \frac{M_p\sin i}{(M_* + M_p)^{2/3}}\left(\frac{2\pi G}{P}\right)^{1/3} \approx 28.4\left(\frac{M_p\sin i}{M_J}\right)\left(\frac{P}{1\,\text{yr}}\right)^{-1/3}\left(\frac{M_*}{M_\odot}\right)^{-2/3}\;\text{m/s}}$$

Jupiter induces a reflex velocity of $K \approx 12.5$ m/s on the Sun. Earth induces only $K \approx 0.09$ m/s — a formidable measurement challenge. Modern spectrographs (HARPS, ESPRESSO) achieve precisions of $\sim 0.3\text{--}1.0$ m/s.

2.2 The Minimum Mass Degeneracy

Radial velocities only measure $M_p\sin i$, where $i$ is the orbital inclination. Without independent knowledge of $i$, only a minimum mass is obtained. For randomly oriented orbits, the statistical correction is $\langle\sin i\rangle = \pi/4 \approx 0.79$. When a transit is also detected ($i \approx 90°$), the true mass is obtained.

3. Direct Imaging

Direct imaging involves spatially resolving the planet from its host star and detecting the planet's own thermal emission or reflected starlight.

3.1 The Contrast Challenge

The planet-to-star flux ratio depends on wavelength. In reflected light:

$$\frac{F_p}{F_*} = A_g\left(\frac{R_p}{a}\right)^2\phi(\alpha) \approx 10^{-10}\left(\frac{A_g}{0.3}\right)\left(\frac{R_p}{R_J}\right)^2\left(\frac{a}{5\,\text{AU}}\right)^{-2}$$

where $A_g$ is the geometric albedo and $\phi(\alpha)$ is the phase function. At thermal infrared wavelengths ($\sim 10\,\mu\text{m}$), the contrast improves to $\sim 10^{-7}$ for young giant planets.

3.2 Angular Resolution Requirements

The angular separation between planet and star must exceed the telescope resolution:

$$\theta = \frac{a}{d} = 0.1''\left(\frac{a}{5\,\text{AU}}\right)\left(\frac{d}{50\,\text{pc}}\right)^{-1}$$

Coronagraphs and adaptive optics systems on 8–10 m telescopes (GPI, SPHERE) achieve inner working angles of $\sim 0.1\text{--}0.2''$. The James Webb Space Telescope and future extremely large telescopes will push these limits further. Direct imaging currently detects only young ($< 100$ Myr), massive ($> 2\,M_J$) planets at wide separations ($> 10$ AU).

4. Kepler Demographics and Occurrence Rates

NASA's Kepler mission (2009–2018) monitored $\sim 200{,}000$ stars for four years, detecting over 2,600 confirmed planets and revealing the statistical properties of planetary systems.

4.1 The Planet Radius Distribution

The most striking result is the bimodal distribution of planet radii, with a gap (the "Fulton gap" or "radius valley") at $R_p \approx 1.5\text{--}2.0\,R_\oplus$. Planets cluster into two populations:

Super-Earths: $R_p \approx 1.0\text{--}1.5\,R_\oplus$, likely rocky with thin or no atmospheres.

Sub-Neptunes: $R_p \approx 2.0\text{--}3.5\,R_\oplus$, with significant hydrogen-helium envelopes.

The radius valley is explained by photoevaporation (XUV radiation from the host star strips atmospheres from close-in low-mass planets) or core-powered mass loss (residual heat from formation drives atmospheric escape).

4.2 Occurrence Rates

Correcting for the geometric transit probability and detection efficiency, Kepler results yield the following occurrence rates for FGK stars:

$$f_{\text{HJ}} \approx 0.5\text{--}1\% \;\;\text{(Hot Jupiters, P < 10 d)}$$

$$f_{\text{SE}} \approx 30\text{--}50\% \;\;\text{(Super-Earths, 1-4 R}_\oplus\text{, P < 100 d)}$$

$$\boxed{\eta_\oplus \approx 5\text{--}20\% \;\;\text{(Earth-sized planets in the habitable zone)}}$$

The parameter $\eta_\oplus$ is one of the most important numbers in astrobiology: it determines the abundance of potentially habitable worlds.

5. Atmospheric Characterization

Transit spectroscopy enables the study of exoplanet atmospheres by measuring the wavelength-dependent transit depth.

5.1 Transmission Spectroscopy

During transit, starlight passes through the planet's atmosphere, which absorbs at characteristic wavelengths. The effective transit depth varies with wavelength as:

$$\boxed{\delta(\lambda) = \left(\frac{R_p + N_H H}{R_*}\right)^2, \qquad H = \frac{k_B T}{\mu g}}$$

where $H$ is the atmospheric scale height, $\mu$ is the mean molecular weight, $g$ is the surface gravity, and $N_H \approx 5\text{--}10$ is the number of scale heights probed. For a hot Jupiter with $T = 1500$ K, $\mu = 2.3\,m_u$, and $g = 10$ m/s$^2$: $H \approx 500$ km, giving transit depth variations of $\sim 100$ ppm.

5.2 JWST Detections

The James Webb Space Telescope has revolutionized exoplanet atmosphere science. JWST has detected CO$_2$, H$_2$O, SO$_2$(a photochemical product), and other molecules in hot Jupiter and sub-Neptune atmospheres. For the TRAPPIST-1 system (seven rocky planets transiting an M dwarf), JWST is constraining whether the inner planets retain atmospheres.

Applications

The Habitable Zone

The habitable zone (HZ) is the range of orbital distances where liquid water can exist on a planet's surface. For a star of luminosity $L_*$, the HZ boundaries are approximately $a_{\text{HZ}} \propto \sqrt{L_*/L_\odot}$, ranging from $\sim 0.95\text{--}1.67$ AU for the Sun. The conservative HZ is bounded by the runaway greenhouse (inner edge) and maximum CO$_2$greenhouse (outer edge). TRAPPIST-1e and Proxima Centauri b are among the most promising rocky planets in their stars' habitable zones.

Planet Formation Theory

The observed diversity of exoplanetary systems constrains planet formation models. Core accretion (rocky cores grow by planetesimal accretion, then capture gas envelopes) explains most observed planets. Gravitational instability (direct fragmentation of the protoplanetary disk) may explain wide-orbit giant planets. Migration mechanisms (Type I and Type II) explain the presence of hot Jupiters and compact multi-planet systems.

The TRAPPIST-1 System

The TRAPPIST-1 system, discovered in 2017, contains seven Earth-sized planets orbiting an ultra-cool M dwarf star at just 12 pc distance. Three planets (e, f, g) lie in the habitable zone. The system is in a remarkable chain of near-resonant orbits (the periods form approximate integer ratios), indicating gentle migration through the protoplanetary disk. With transit depths of $\sim 0.5\text{--}0.8\%$(large because the host star has $R_* \approx 0.12\,R_\odot$), these planets are among the best targets for JWST atmospheric characterization of rocky worlds. Initial JWST observations have constrained the atmosphere of TRAPPIST-1b, finding it consistent with a bare rock with no atmosphere.

Gravitational Microlensing

Gravitational microlensing offers a complementary detection method sensitive to planets at $1\text{--}10$ AU, including free-floating (rogue) planets. When a foreground star passes in front of a background star, the gravitational lensing magnification produces a characteristic light curve. A planet orbiting the lens star creates an additional brief anomaly. Microlensing surveys (OGLE, MOA, KMTNet) have detected over 200 exoplanets and estimated that free-floating planets may be as common as bound planets in the Galaxy. The Roman Space Telescope will conduct a dedicated microlensing survey expected to detect thousands of cold planets and hundreds of free-floaters.

Historical Notes

The first confirmed exoplanets were found around the millisecond pulsar PSR B1257+12 by Aleksander Wolszczan and Dale Frail in 1992. The first planet around a main-sequence star, 51 Pegasi b, was discovered by Michel Mayor and Didier Queloz in 1995 using the radial velocity method — earning them the 2019 Nobel Prize in Physics (shared with James Peebles). The Kepler mission (launched 2009) detected thousands of candidates and established the field of exoplanet demographics. The TESS mission (launched 2018) surveys the nearest and brightest stars for transiting planets amenable to JWST follow-up. The discovery that rocky planets are common — perhaps outnumbering stars in the Galaxy — is one of the most profound findings in the history of astronomy.

The discovery of hot Jupiters (51 Peg b orbits at just 0.05 AU with a 4.2-day period) was a surprise: no theory had predicted gas giants so close to their parent stars. This led to the development of planetary migration theory, in which planets form at larger distances and migrate inward through interactions with the protoplanetary disk (Type I and Type II migration) or through dynamical scattering and tidal circularization. The rich diversity of planetary architectures discovered by Kepler — including compact systems of super-Earths with periods of days to weeks — continues to challenge and refine our understanding of planet formation and evolution.

The astrometric method, which measures the wobble of a star's position on the sky due to an orbiting planet, is now becoming competitive thanks to the extraordinary precision of the Gaia satellite. Gaia is expected to detect thousands of Jupiter-mass planets at orbital periods of 1–10 years around nearby stars, complementing the short-period sensitivity of the transit and RV methods. The combination of Gaia astrometry with radial velocity measurements will determine true 3D orbital architectures and provide model-independent masses for a large sample of planets. The upcoming census of exoplanets from Gaia, combined with TESS transit detections and JWST atmospheric characterization, will provide the most complete picture of planetary systems since the first exoplanet discovery three decades ago.

Protoplanetary Disks: The Birthplaces of Planets

ALMA has revolutionized our understanding of planet formation by resolving the structure of protoplanetary disks around young stars. The iconic image of HL Tauri revealed a series of concentric rings and gaps in the dust distribution, likely carved by forming planets. The gap width and depth constrain the planet mass through the relation $\Delta \propto (M_p/M_*)^{1/2}(H/r)^{-1}$, where $H/r$is the disk aspect ratio. Hundreds of disks have now been imaged at high resolution, showing ubiquitous substructure that suggests planet formation begins very early ($< 1$ Myr) in the disk lifetime. This challenges the classical core accretion model, which predicts formation timescales of several million years for giant planets.

Mass-Radius Diagram and Planetary Interiors

When both the mass (from RV) and radius (from transit) of a planet are measured, the bulk density $\bar{\rho} = 3M/(4\pi R^3)$ constrains the interior composition.

Composition Constraints

The mass-radius relation for different compositions follows approximately:

$$R \propto M^{0.27}\;\;\text{(rocky/iron)}, \qquad R \propto M^{-0.04}\;\;\text{(gas giant, degenerate core)}$$

Pure iron planets lie below the Earth composition line, while water worlds lie above it. Planets with significant H/He envelopes (sub-Neptunes) have much larger radii for their mass. The mass-radius diagram reveals that planets with $R_p > 1.6\,R_\oplus$generally require volatile envelopes, consistent with the radius valley interpretation. For rocky planets, the iron-to-silicate ratio can be constrained, with Mercury-like compositions ($\sim 70\%$ iron core) producing denser, more compact planets than Earth-like mixtures ($\sim 33\%$ iron core).

Atmospheric Escape and Evolution

Close-in planets experience intense XUV irradiation from their host star, driving hydrodynamic escape of their atmospheres. The energy-limited escape rate is:

$$\dot{M} = \frac{\eta\,\pi R_p^3\,F_{\text{XUV}}}{G M_p K_{\text{tide}}}$$

where $\eta$ is the heating efficiency ($\sim 0.1\text{--}0.3$),$F_{\text{XUV}}$ is the stellar XUV flux, and $K_{\text{tide}}$is a tidal correction factor. This mechanism can strip the entire H/He envelope of a low-mass planet ($M_p \lesssim 5\,M_\oplus$) within the first Gyr, transforming a sub-Neptune into a bare rocky super-Earth and producing the observed radius valley.

The Mass-Radius Diagram and Planet Types

The mass-radius diagram reveals distinct planet populations. Planets below the $R \propto M^{0.27}$ Earth composition line are iron-enriched (Mercury-like). Planets above this line contain increasing fractions of water ice or volatile envelopes. The transition from rocky super-Earths to volatile-rich sub-Neptunes occurs at approximately $M_p \approx 5\text{--}10\,M_\oplus$. Above $\sim 0.3\,M_J$, giant planets follow a nearly flat mass-radius relation ($R \approx R_J$) due to electron degeneracy in the interior. The mass-radius relation, combined with interior structure models, allows inference of core mass, envelope fraction, and bulk water content for individual planets.

Emission Spectroscopy and Thermal Phase Curves

During secondary eclipse (when the planet passes behind its star), the planet's thermal emission can be isolated by measuring the drop in total system flux. The eclipse depth at wavelength $\lambda$ gives the planet's brightness temperature $T_B(\lambda)$. Phase curve observations, measuring the flux variation over a full orbit, map the day-night temperature contrast and identify hotspot offsets from the substellar point caused by atmospheric circulation. Spitzer and now JWST have measured phase curves for dozens of hot Jupiters, revealing that most have eastward-shifted hotspots (consistent with equatorial superrotation) and large day-night contrasts ($\Delta T \sim 500\text{--}1500$ K).

Computational Exploration

The following simulation generates transit light curves, computes radial velocity curves, models the Kepler planet radius distribution, and illustrates the habitable zone as a function of stellar type.

Transit Light Curves, Radial Velocities, and Planet Demographics

Python

script.py250 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a0a')
for ax in axes.flat:
    ax.set_facecolor('#0f0f0f')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

rng = np.random.default_rng(42)

# --- Plot 1: Transit light curves ---
ax1 = axes[0, 0]

def transit_lightcurve(t, P, T0, R_p_R_star, a_R_star, inc_deg, u1=0.3, u2=0.1):
    """Simplified Mandel-Agol transit light curve with limb darkening"""
    inc = inc_deg * np.pi / 180
    phase = 2 * np.pi * (t - T0) / P
    # Sky-projected separation
    x = a_R_star * np.sin(phase)
    y = a_R_star * np.cos(phase) * np.cos(inc)
    z_sep = np.sqrt(x**2 + y**2)

k = R_p_R_star
    flux = np.ones_like(t)

for i in range(len(t)):
        d = z_sep[i]
        if d >= 1 + k:
            flux[i] = 1.0
        elif d <= 1 - k:
            # Full transit
            flux[i] = 1 - k**2
        elif d < 1 + k and d > abs(1 - k):
            # Partial transit (simplified)
            area = k**2 * np.arccos((d**2 + k**2 - 1)/(2*d*k)) +                    np.arccos((d**2 + 1 - k**2)/(2*d)) -                    0.5*np.sqrt(max(0, (-d+k+1)*(d+k-1)*(d-k+1)*(d+k+1)))
            flux[i] = 1 - area / np.pi
        # Behind star
        if a_R_star * np.cos(phase[i]) > 0 and d < 1 + k:
            pass  # Only block when in front

# Simple limb darkening correction
    # Approximate: increase depth by ~10% for typical LD
    depth = 1 - flux
    flux = 1 - depth * (1 + 0.1)

return flux

t_hours = np.linspace(-5, 5, 1000)

# Hot Jupiter: R_p/R* = 0.1, a/R* = 8
planets = [
    ('Hot Jupiter (1.0 R_J)', 3.0*24, 0, 0.10, 8.0, 90, '#ff4444'),
    ('Neptune-sized (0.35 R_J)', 10.0*24, 0, 0.035, 20.0, 89, '#ffa500'),
    ('Earth-sized (0.01 R_J)', 365.25*24, 0, 0.009, 215.0, 89.8, '#44bbff'),
]

for label, P, T0, k, a_R, inc, col in planets:
    flux = transit_lightcurve(t_hours, P, T0, k, a_R, inc)
    # Scale transit duration to fit in window
    t_scaled = t_hours * (a_R / 8.0)  # normalize durations

offset = {'#ff4444': 0, '#ffa500': -0.015, '#44bbff': -0.03}[col]
    ax1.plot(t_scaled, flux + offset, color=col, linewidth=2, label=label)

ax1.set_xlabel('Time from mid-transit (hours)', fontsize=12)
ax1.set_ylabel('Relative Flux', fontsize=12)
ax1.set_title('Transit Light Curves', fontsize=14, fontweight='bold')
ax1.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax1.set_xlim(-5, 5)
ax1.set_ylim(0.955, 1.005)
ax1.grid(True, alpha=0.15)

# Annotate depths
ax1.annotate('1% dip', xy=(0, 0.99), xytext=(2, 0.99), color='#ff4444', fontsize=9,
            arrowprops=dict(arrowstyle='->', color='#ff4444'))

# --- Plot 2: Radial velocity curve ---
ax2 = axes[0, 1]

t_days = np.linspace(0, 30, 500)

# Multiple planets
rv_total = np.zeros_like(t_days)

planet_rv = [
    ('Planet b (hot Jup)', 3.5, 80.0, 0.0, '#ff4444'),
    ('Planet c (warm Nep)', 12.0, 8.0, 0.15, '#ffa500'),
]

for label, P, K, ecc, col in planet_rv:
    phase = 2 * np.pi * t_days / P
    # Keplerian: for low ecc, v_r ~ K * (cos(f + omega) + e*cos(omega))
    rv = K * np.cos(phase)  # circular approximation
    rv_total += rv
    ax2.plot(t_days, rv, '--', color=col, linewidth=1.5, alpha=0.5, label=f'{label}: K={K} m/s')

# Add noise
rv_obs_t = np.linspace(0.5, 29.5, 40)
rv_obs = np.interp(rv_obs_t, t_days, rv_total)
rv_obs += rng.normal(0, 2.0, len(rv_obs))

ax2.plot(t_days, rv_total, color='#ffa500', linewidth=2.5, label='Total RV')
ax2.errorbar(rv_obs_t, rv_obs, yerr=2.0, fmt='o', color='#ffdd00', markersize=4,
             capsize=2, ecolor='#ffdd00', alpha=0.7, label='Observations')

ax2.axhline(y=0, color='#888', linestyle=':', linewidth=1)
ax2.set_xlabel('Time (days)', fontsize=12)
ax2.set_ylabel('Radial Velocity (m/s)', fontsize=12)
ax2.set_title('Radial Velocity Curve', fontsize=14, fontweight='bold')
ax2.legend(fontsize=8, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax2.grid(True, alpha=0.15)

# --- Plot 3: Kepler planet radius distribution ---
ax3 = axes[1, 0]

# Generate synthetic radius distribution matching Kepler results
# Bimodal: super-Earths peak at 1.3 Re, sub-Neptunes peak at 2.5 Re
N_se = 800
N_sn = 600
N_nep = 200
N_jup = 50

R_se = rng.normal(1.3, 0.2, N_se)
R_sn = rng.normal(2.5, 0.4, N_sn)
R_nep = rng.normal(4.0, 0.8, N_nep)
R_jup = rng.normal(11.0, 2.0, N_jup)

R_all = np.concatenate([R_se, R_sn, R_nep, R_jup])
R_all = R_all[R_all > 0.5]

bins = np.linspace(0.5, 20, 60)
ax3.hist(R_all, bins=bins, color='#ffa500', alpha=0.7, edgecolor='#333',
         density=True, label='Simulated Kepler distribution')

# Mark radius valley
ax3.axvline(x=1.7, color='#ff4444', linestyle='--', linewidth=2, alpha=0.7, label='Radius valley')
ax3.axvspan(1.5, 2.0, alpha=0.1, color='#ff4444')

# Mark categories
ax3.text(1.2, 0.35, 'Super-\nEarths', color='#44bbff', fontsize=10, ha='center')
ax3.text(2.8, 0.2, 'Sub-\nNeptunes', color='#ffa500', fontsize=10, ha='center')
ax3.text(4.5, 0.05, 'Neptunes', color='#ffdd00', fontsize=10, ha='center')
ax3.text(11, 0.02, 'Giants', color='#ff4444', fontsize=10, ha='center')

ax3.set_xlabel('Planet Radius (R_Earth)', fontsize=12)
ax3.set_ylabel('Probability Density', fontsize=12)
ax3.set_title('Planet Radius Distribution (Kepler)', fontsize=14, fontweight='bold')
ax3.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax3.set_xlim(0.5, 18)
ax3.set_ylim(0, 0.5)
ax3.grid(True, alpha=0.15)

# --- Plot 4: Habitable zone vs stellar type ---
ax4 = axes[1, 1]

# Stellar parameters
spectral_types = ['M5', 'M2', 'M0', 'K5', 'K0', 'G2', 'F5', 'F0', 'A5']
T_eff = [3000, 3500, 3800, 4400, 5200, 5778, 6500, 7200, 8200]
L_star = [0.003, 0.03, 0.08, 0.2, 0.6, 1.0, 2.5, 6.0, 20.0]  # L_sun
R_star = [0.2, 0.4, 0.6, 0.7, 0.85, 1.0, 1.3, 1.5, 1.8]

y_pos = np.arange(len(spectral_types))

for i, (sp, L, T, R) in enumerate(zip(spectral_types, L_star, T_eff, R_star)):
    # Conservative HZ
    a_inner = 0.95 * np.sqrt(L)
    a_outer = 1.67 * np.sqrt(L)
    # Optimistic HZ
    a_inner_opt = 0.75 * np.sqrt(L)
    a_outer_opt = 1.77 * np.sqrt(L)

# Optimistic (lighter)
    ax4.barh(i, a_outer_opt - a_inner_opt, left=a_inner_opt, height=0.5,
             color='#44bbff', alpha=0.2)
    # Conservative (darker)
    ax4.barh(i, a_outer - a_inner, left=a_inner, height=0.5,
             color='#ffa500', alpha=0.5)

# Mark known HZ planets
hz_planets = [
    ('Proxima b', 0.048, 'M5', '#ff4444'),
    ('TRAPPIST-1e', 0.029, 'M5', '#ffdd00'),
    ('Kepler-442b', 0.409, 'K5', '#44bbff'),
    ('Kepler-186f', 0.356, 'M0', '#ff69b4'),
]

for name, a, sp_type, col in hz_planets:
    y_idx = spectral_types.index(sp_type) if sp_type in spectral_types else 0
    ax4.plot(a, y_idx, '*', color=col, markersize=15, zorder=10)
    ax4.text(a + 0.05, y_idx + 0.3, name, color=col, fontsize=7)

ax4.set_yticks(y_pos)
ax4.set_yticklabels(spectral_types)
ax4.set_xlabel('Semi-major axis (AU)', fontsize=12)
ax4.set_ylabel('Spectral Type', fontsize=12)
ax4.set_title('Habitable Zone vs Stellar Type', fontsize=14, fontweight='bold')
ax4.set_xscale('log')
ax4.set_xlim(0.01, 10)
ax4.grid(True, alpha=0.15, axis='x')

# Custom legend
from matplotlib.patches import Patch
legend_elements = [Patch(facecolor='#ffa500', alpha=0.5, label='Conservative HZ'),
                   Patch(facecolor='#44bbff', alpha=0.2, label='Optimistic HZ')]
ax4.legend(handles=legend_elements, fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')

# Print summary
print("=" * 65)
print("EXOPLANETS: KEY RESULTS")
print("=" * 65)

print("\nTransit depths:")
for name, k in [("Jupiter-Sun", 0.10), ("Neptune-Sun", 0.035), ("Earth-Sun", 0.009)]:
    delta = k**2
    print(f"  {name:15s}: delta = {delta:.6f} = {delta*1e6:.0f} ppm")

print("\nRadial velocity semi-amplitudes:")
for name, Mp_MJ, P_yr in [("Jupiter", 1.0, 11.86), ("Saturn", 0.30, 29.46),
                           ("Neptune", 0.054, 164.8), ("Earth", 0.00315, 1.0)]:
    K = 28.4 * Mp_MJ * P_yr**(-1.0/3.0)
    print(f"  {name:10s}: K = {K:8.2f} m/s")

print("\nGeometric transit probability:")
for a_AU in [0.05, 0.2, 1.0, 5.0]:
    p_tr = 0.005 / a_AU
    print(f"  a = {a_AU} AU: p_transit = {p_tr*100:.2f}%")

print("\nOccurrence rates (Kepler):")
print("  Hot Jupiters (P<10d): 0.5-1%")
print("  Super-Earths (1-4 Re, P<100d): 30-50%")
print("  eta_Earth (HZ rocky): 5-20%")

print("\nHabitable zone boundaries (conservative):")
for sp, L in zip(spectral_types[:6], L_star[:6]):
    a_in = 0.95 * np.sqrt(L)
    a_out = 1.67 * np.sqrt(L)
    print(f"  {sp}: {a_in:.3f} - {a_out:.3f} AU")

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
print("\nPlot saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Chapter Summary

Exoplanets are detected primarily through transits (measuring $\delta = (R_p/R_*)^2$) and radial velocities (measuring $K \propto M_p\sin i\,P^{-1/3}$). Direct imaging complements these methods for young giant planets at wide separations.

Kepler revealed that small planets are far more common than gas giants, with a bimodal radius distribution separated by the radius valley at $\sim 1.7\,R_\oplus$. Rocky planets in the habitable zone occur around $5\text{--}20\%$ of Sun-like stars.

Transmission spectroscopy with JWST is now characterizing exoplanet atmospheres, detecting molecules like H$_2$O, CO$_2$, and SO$_2$. The search for biosignatures in the atmospheres of rocky habitable-zone planets is the next frontier.