Solar System dynamics and exoplanets

The motion of planets, their moons and exoplanets is governed by universal gravitation: $F = G\dfrac{Mm}{r^2}$. Kepler's laws follow from this law, and for a circular orbit around a body of mass $M$ the period is related to the semi-major axis: $T^2 = \frac{4\pi^2}{GM}\,a^3.$ Note that the period depends on $a$ and $M$ but not on the planet's own mass, so the observed $T$ and $a$ yield the mass of the central body. Real systems are not ideal: mutual perturbations, resonances, tides and migration cause small deviations from Keplerian motion, and it is precisely these deviations that let us measure planetary masses in the Solar System. Exoplanets are found mainly by indirect methods: the Doppler shift of stellar lines (radial velocities), the dimming of light during a transit, and also microlensing and direct imaging. The radial-velocity method gives the projected mass $m\sin i$, a transit gives the radius ratio $\delta = (R_p/R_*)^2$, and the two together give the planet's density and hence clues to its composition. The methods are sensitive to different planets: hot giants are easier to find than small cold ones, so the known exoplanet sample is biased.

A planet moves in a circular orbit around a star of mass $M = 1\,M_\odot$ with semi-major axis $a = 1.5$ AU. Find the orbital period $T$ in years and in days.

For a star of mass $M = 1\,M_\odot$, spectral lines shift periodically with amplitude $K = 12$ m/s and period $P = 4$ days. Estimate the planet's minimum mass $m\sin i$.

A planet transiting the disk of a star with radius $R_* = 1\,R_\odot$ dims the brightness by $\delta = 1\%$. Radial velocities at a nearly edge-on inclination ($i \approx 90^\circ$) give a mass $m \approx 0.9\,M_\text{Jup}$. Find the planet's radius and mean density, and decide whether it is rocky or gaseous.

A simulation instead of night-time observing. A real transit needs a telescope, clear weather and hours of time, so we work with a model or archival data.

1. Open an online simulator of transits and radial-velocity curves, or take archival data for some exoplanet from a public database and plot it in a spreadsheet.
2. Set a star of $1\,M_\odot$ and a planet on a circular orbit; record the resulting $P$ and $a$.
3. Using $M = \dfrac{4\pi^2 a^3}{G T^2}$, recover the stellar mass from the "observations" and compare it with the value you set.
4. From the transit depth find $R_p/R_*$, from the amplitude $K$ find $m\sin i$, compute the density and decide whether the planet is rocky or gaseous.
5. By varying the planet's radius and period, note at which parameters the signal falls below the noise level — this illustrates observational selection.

Safety. Never look at the Sun without a special filter; carry out any observation of bright objects only through a certified solar filter mounted in front of the aperture.

Show how one and the same orbit appears in two independent observational signals: the periodic shift of stellar lines and the periodic dimming of the star's brightness.

A schematic of two stacked graphs sharing a common time axis. The upper one is the radial-velocity curve $v_r(t) = K\sin(2\pi t/P)$ with the amplitude $K$ and period $P$ labelled. The lower one is the light curve with dips of depth $\delta = (R_p/R_*)^2$ at transit times; below one dip a small sketch shows the star and the planet crossing its disk. The dips coincide in time with the zero crossings of the sine curve.

A car's GPS receiver uses signals from satellites orbiting at an altitude of about $20\,200\ \text{km}$. Each satellite completes one orbit in about $11\ \text{h}\ 58\ \text{min}$, moving at roughly $3.9\ \text{km/s}$. The receiver measures signal travel times to within a few nanoseconds to fix the car's position to about $5\ \text{m}$.

The satellite's orbital radius and period are linked by Kepler's third law, $T^2 \propto a^3$, which follows from Newton's law of gravitation. The same laws allow us to calculate the orbits of exoplanets from their observed transit periods. The travel time of the signal also uses the speed of light, tying directly to distance measurement in astronomy.

At a coastal town, the difference between high and low tide is normally about $2.0\ \text{m}$. During a full moon, when the Sun and Moon pull in line, the tidal range can increase to about $3.0\ \text{m}$ (spring tide). The Moon is $384\,400\ \text{km}$ away; its tidal effect on Earth's oceans is about twice that of the Sun, even though the Sun is much more massive, because tidal force falls off as $1/r^3$.

This exercises Newton's law of gravitation, $F = G \frac{m_1 m_2}{r^2}$, and the concept of tidal forces, which arise from the difference in gravitational acceleration across a body. The same differential gravity causes a star to wobble due to an orbiting exoplanet, a signal detected by the radial-velocity method.

A 13-year-old uses a $100\ \text{mm}$ telescope and a smartphone camera to record the brightness of the star HD 209458 over $4$ hours. The star's brightness dips by about $1.5\%$ for $3$ hours when the planet HD 209458 b passes in front. The planet orbits every $3.5$ days.

The dip depth gives the ratio of planet radius to star radius. The transit duration and orbital period, combined with Kepler's third law, yield the semi-major axis and orbital inclination. This is the transit method, one of the main ways exoplanets are discovered and characterized.