Planets We Have Never Seen: What Wobbles and Shadows Really Tell Us — Epoche B2
Introduction Many people assume that modern telescopes have finally become powerful enough to photograph planets orbiting other stars. In reality, direct images exist for only a handful of young, giant planets far from their stars — the four planets of the star HR 8799, imaged by Marois and colleagues in 2008, are the textbook case. Almost every one of the more than five thousand confirmed exoplanets has never been seen. The obstacle is not, as one might expect, resolving power. A telescope's ability to separate two neighbouring points on the sky is set by the diffraction limit, roughly the observing wavelength divided by the mirror diameter: for an 8-metre telescope working in the near infrared at $1.6\ \mu\mathrm{m}$ this is about $2\times10^{-7}$ radians, or $0.04$ arcseconds, whereas a planet one astronomical unit from a star ten parsecs away subtends $0.1$ arcseconds. The geometry is comfortably within reach. The obstacle is contrast . A planet shines only by reflecting a small fraction of the starlight that falls on it, and the Earth reflects roughly one ten-billionth of the Sun's output towards a distant observer. Detecting that is like photographing a firefly beside a lighthouse. The few directly imaged planets escape this trap because they are young and still hot from their formation, so they glow in the infrared with their own heat rather than reflecting, and because they orbit far enough out for the starlight to be blocked by a mask. So since Mayor and Queloz announced a Jupiter-mass companion to the star 51 Pegasi in 1995, discovery has depended on two indirect techniques, both of which watch the star rather than the planet. Comparing them shows not only how they work, but also what each one cannot tell us. The Radial-Velocity Method: Measuring a Wobble A planet does not simply circle its star; both bodies orbit their common centre of mass, the balance point of the system, in the way that two skaters of unequal weight swinging around each other both turn about a point nearer the heavier one. The star therefore moves in a small circle of its own, and the component of that motion along our line of sight — the radial velocity — shifts the wavelength of its light through the Doppler effect, the same effect that lowers the pitch of a siren as it recedes. Light emitted at wavelength $\lambda$ by a source moving away at speed $v$ arrives at $\lambda(1 + v/c)$, where $c$ is the speed of light. The size of the wobble follows from two elementary facts. First, the balance condition: if the planet of mass $m_p$ sits at distance $a_p$ from the centre of mass and the star of mass $M_\star$ at distance $a_\star$, then $m_p a_p = M_\star a_\star$, so the star's orbit is smaller than the planet's by the mass ratio, $a_\star = a\,m_p/(M_\star + m_p)$ with $a$ the separation between the two bodies. Second, Kepler's third law in its Newtonian form, $a^3 = G(M_\star + m_p)P^2/4\pi^2$, which converts an orbital period $P$ into a separation. The star's orbital speed is $2\pi a_\star/P$, and only the fraction $\sin i$ of it points along our line of sight, where $i$ is the inclination — the tilt of the orbital plane, defined so that $i = 90^\circ$ means we view the orbit edge-on and $i = 0^\circ$ means face-on, with no motion towards or away from us at all. Combining the three gives the velocity semi-amplitude for a circular orbit: $$K = \left(\frac{2\pi G}{P}\right)^{1/3} \frac{m_p \sin i}{(M_\star + m_p)^{2/3}}.$$ The crucial limitation now sits visibly inside the formula: $m_p$ and $\sin i$ appear only as a product, so no amount of extra measurement of the same wobble can separate them. Because $i$ is unknown, the method gives a minimum mass — since $\sin i \le 1$, the true mass is at least the measured value, and could be much greater if the orbit happens to lie nearly face-on. Note also that $M_\star$ must be supplied from outside, from stellar models, so even the minimum mass is not free of assumption. For 51 Pegasi b, Mayor and Queloz measured a periodic velocity swing of amplitude $K \approx 59\ \mathrm{m\,s^{-1}}$ with a period of $4.23$ days. Putting $M_\star \approx 1\,M_\odot$ into the formula and solving for the mass term gives $$m_p \sin i = \frac{K\,M_\star^{2/3}}{(2\pi G/P)^{1/3}} \approx 8.9\times10^{26}\ \mathrm{kg} \approx 0.47\,M_{\mathrm{Jup}}.$$ Two features of that measurement deserve emphasis. The velocity is a walking pace: $59\ \mathrm{m\,s^{-1}}$ against the speed of light gives a fractional wavelength shift of $v/c \approx 2\times10^{-7}$, about $10^{-4}$ nanometres at visible wavelengths — far below the resolution of any single spectral line. It is recovered by cross-correlating thousands of absorption lines across the spectrum at once, so that the shift common to all of them emerges from noise that is not. And the four-day period was the true surprise of 1995: a Jupiter-mass body orbiting closer to its star than Mercury does to the Sun contradicted every expectation about where giant planets could exist, which is precisely why the result was initially met with scepticism. The Transit Method: Measuring a Shadow In contrast, the transit method watches for a small drop in a star's brightness when a planet crosses in front of it — an eclipse seen from far away, which blocks light in proportion to the area the planet covers. Since the star's disc has area $\pi R_\star^2$ and the planet's silhouette $\pi R_p^2$, the fractional dip in flux is a ratio of areas, hence of squared radii: $$\frac{\Delta F}{F} \approx \left(\frac{R_p}{R_\star}\right)^2.$$ For an Earth-sized planet crossing a Sun-like star, $R_p/R_\star = 6371/696{,}000 \approx 0.0092$, so the dip is about $8 \times 10^{-5}$ — 84 parts per million, a brightness change of less than a hundredth of one per cent, which is why space telescopes above the flickering atmosphere were needed to detect Earth-sized planets at all. Note that the transit yields only the ratio of radii; converting it to a physical size requires $R_\