Astronomy/Exoplanets
Exoplanets, or extra-solar planets, are planets outside of Earth's solar system. Exoplanets is one of the topics for the 2024 Astronomy event, the others being Stellar Evolution and Star Formation.
Detection of Exoplanets
In the equations below, the subscript s denotes the star, and the subscript e denotes the exoplanet. [math]\displaystyle{ R }[/math] denotes the radius of the object, and [math]\displaystyle{ M }[/math] denotes the mass of the object.
Transit
The transit method uses the light blocked from the parent star by the host planet to determine various properties of the star and planet. When a planet comes across the plane of the star in the point of view of the Earth, the light given by the star will encounter a brief dip before then coming back up to the mean value; see this video for a demonstration of this. The size of this dip, called the transit depth, can be used to calculate the radius of the planet by comparing it with the radius of the star (which can be determined through other means) and the equation
where [math]\displaystyle{ \Delta F }[/math] is the change in flux or brightness of the star, and [math]\displaystyle{ F_0 }[/math] is the original flux.
These transits are often shown on light curves like the one in the image shown. These are plots of the brightness of the host star over time. Sometimes, the y axis is presented in a unit of flux like watts per square meter or photon counts. Other times, the y axis is presented in units of the mean flux of the star [math]\displaystyle{ F_0 }[/math] to make it easy to read off the transit depth [math]\displaystyle{ \frac{\Delta F}{F_0} }[/math] directly.
You might expect at first for the light curve to look flat at the bottom of transit: if a planet is already in front of the star, why should the brightness of the star change? The answer is that the brightness of the star is not the same across it: the star looks brighter to an observer near the middle, and darker near the edges. This effect is called limb darkening, and affects shorter wavelengths of light more than longer wavelengths (thus, some light curves look flatter at the bottom than others).
Since transits occur every time the planet passes in front of its star, you can use the time between two consecutive transits of the same exoplanet to measure its orbital period.
Not every planet is a transiting planet; for a planet to pass in front of its host star, its inclination (the tilt of the orbit in the sky) must be close to 90 degrees. One important factor is the likelihood of transit in terms of the properties of the star. The bigger the planet, the more likely the transit can happen which should make sense: the bigger the planet, the more area it covers in the sky and more importantly on the star itself, increasing the likelihood of a transit. The bigger the orbit, the less likely the transit can happen due to the inclination of the orbit. With a smaller orbital radius, there exists a greater range of inclinations that can result in a transit regardless of the size of the planet. However, with a larger orbital radius, there exists a lower range of inclination possibly resulting in a transit. Thus, planets with larger radii and closer orbits to their star are most likely to be able to be studied with the transit method.
As of September 2023, the transit method has discovered the largest number of exoplanets (about 3/4 of the total known exoplanets), particularly due to the Kepler/K2 and TESS (Transiting Exoplanet Survey Satellite) missions.
Transmission spectroscopy
Transmission spectroscopy (or "transit spectroscopy") is a method for studying exoplanet atmospheres using transits. It is one of several spectroscopic methods that can be used to detect the presence of molecules, clouds, and hazes in atmospheres. During transit, some of the light that reaches us is light from the star that has been transmitted to us through the atmosphere of the exoplanet. The amount of transmitted light depends on wavelength, since at certain wavelengths the components of the atmosphere will "block" more light via absorption or scattering. For instance, if there's a lot of water vapor in the atmosphere, then more light will be blocked at wavelengths corresponding to the absorption lines of water vapor.
To perform transmission spectroscopy, we measure the brightness over time of the star in many different small wavelength bands: in other words, create light curves for each band. Then, you can compare the transit depths in each band. Wavelengths at which things in the atmosphere block more light will cause the transit depths to be larger.
Secondary eclipses
Similar to a transit, a secondary eclipse occurs when a planet passes behind its host star rather than in front of it. This also results in a (smaller) drop in brightness of the system, as just before and after the secondary eclipse we see light from both the planet and star, while during secondary eclipse we only see light from the star. Thus, secondary eclipses tell us about the light coming from the planet, which may be reflected stellar light (dominates optical wavelengths) or emitted light from the planet (dominates infrared wavelengths). For example, in a process similar to transmission spectroscopy, we can construct an emission spectrum of the exoplanet by measuring the secondary eclipse depth across many wavelengths, which can give us information about the composition and temperature of the exoplanet's atmosphere.
Radial Velocity
Radial velocity, also known as Doppler spectroscopy, detects exoplanets by measuring Doppler shifts in the spectrum of the parent star. As of September 2023, about 19% of exoplanets were detected using radial velocity.
As discussed on the main Astronomy page, planets don't quite orbit the center of their host star. Instead, both the planet and host star orbit their shared center of mass (barycenter). Thus, as a planet moves, so does its host star. The host star moves the most for massive, close-orbiting planets, such as hot Jupiters, so the radial velocity method is most effective for these planets.
Due to the Doppler effect, spectral lines in the star's spectrum change their wavelength depending on the star's radial velocity (how fast the star is moving towards or away from us). When the star is moving towards us (negative radial velocity), the wavelengths of the spectral lines become shorter (blueshift), and when it's moving away from us (positive radial velocity), the wavelengths become longer (redshift). At any given moment in time, the radial velocity of the star can be calculated using the Doppler shift formula. By applying this formula to data over time of a spectral line in a star's spectrum (or multiple spectral lines), it's possible to plot a radial velocity curve. An example radial velocity curve is shown in the image below, and an animated radial velocity curve and spectrum can be found at this Astrobites article.
Several complications with using the radial velocity method are rotating spots on the star, granulation on the star's surface, oscillations on the star's surface, and the Rossiter–McLaughlin effect, which can all cause additional features in radial velocity curves (some of which can resemble exoplanet signals). In addition, the oscillation of the star can be complicated through pulsations on different planes on the star's surface.
Measuring mass with radial velocity
Radial velocity calculations can be quite complicated for elliptical orbits or when there are multiple exoplanets, but for the below math we'll assume a circular orbit and only one exoplanet.
The velocity and period of the planet's orbit and the mass of the planet can be calculated after radial velocity measurements.
- First, if the mass of the star [math]\displaystyle{ M_s }[/math] is not given, calculate it with the mass-luminosity relation (if it's a main sequence star).
- If the velocity of the star isn't given, calculate it by applying a modified form of the Doppler shift formula: [math]\displaystyle{ \frac{\Delta\lambda}{\lambda_{\mathrm{avg}}}=\frac{v}{c} }[/math], where [math]\displaystyle{ \lambda_{\mathrm{avg}} }[/math] is the average of the highest and lowest wavelength (of a given spectral line) and [math]\displaystyle{ \Delta\lambda }[/math] is the difference between the highest and the average wavelengths. Otherwise, if a radial velocity curve is given, you can determine the velocity of the star by looking at the amplitude of the radial velocity curve.
- Find the period [math]\displaystyle{ P }[/math] of the star's orbit from the radial velocity curve.
- Use the general form [math]\displaystyle{ \frac{a^3}{P^2}=\frac{GM_s}{4\pi^2} }[/math] of Kepler's Third Law to calculate [math]\displaystyle{ a }[/math], the length of the semi-major axis of the planet's orbit. Alternatively, one can use the specific form [math]\displaystyle{ \frac{a^3}{P^2}=M_s }[/math] where [math]\displaystyle{ r }[/math] is in AU, [math]\displaystyle{ P }[/math] is in years and [math]\displaystyle{ M_s }[/math] is in solar masses. (This form of Kepler's 3rd law is an approximation. We would normally need to use the total system mass--the mass of the star and planet together--but we assume the planet has a small enough mass to not include it here.)
- Find the velocity of the planet using the approximation [math]\displaystyle{ v_p=\frac{2\pi a}{P} }[/math].
- Finally, find the mass of the planet using [math]\displaystyle{ M_sv_s=M_ev_e }[/math].
This calculation gives us the minimum mass of the planet. The measured velocity of the star is less than its true velocity if the orbital plane is not perpendicular to the sky. Let [math]\displaystyle{ \alpha }[/math] denote the orbital inclination, then the true velocity of the star and true mass of the planet is given by [math]\displaystyle{ v_{s,\text{ true}}=\frac{v_s}{\sin\alpha}, M_{e, \text{true}}=\frac{M_e}{\sin\alpha}. }[/math] When an exoplanet transits the parent star, the inclination [math]\displaystyle{ \alpha }[/math] is very close to 90 degrees. Therefore, the calculation based on radial velocity is close to its true mass.
All of the above calculations can be summarized in the formula
which is an approximation of the Binary Mass Function for exoplanets in circular orbits. Using this equation, you can can solve for the mass of the exoplanet if the inclination is known (and the minimum mass [math]\displaystyle{ M_{e,\,\mathrm{true}} \sin \alpha }[/math] otherwise). Here, [math]\displaystyle{ K }[/math] is the amplitude of the radial velocity curve/the apparent orbital velocity from Doppler shift, and this formula can be modified for use with binary stars and eccentric orbits.
Direct Imaging
Direct imaging detects exoplanets by resolving the exoplanet from the star in an image. It is usually very difficult, because planets are much fainter than their parent stars. Coronagraphs are used to block the light from the star so the planets can be resolved. Most direct imaged exoplanets are relatively close to the Earth, widely separated from their parent star, and are especially large and hot. Images are made in the infrared, where the radiation from the planet is the strongest. (Although planets also reflect light from their host stars in the optical wavelengths, which we could in theory detect, these are also the wavelengths at which the star is brightest, meaning it's currently too difficult to directly image planets using reflected light.) Since planets are usually hottest when they are young, direct imaging is useful for detecting young planets and studying planet formation.
Types of Exoplanets
Gas Giant
Planets composed mainly of hydrogen and helium. They may possibly have rocky or icy cores. They have masses greater than 10 Earth masses.
Hot Jupiters
Gas giants that orbit very close to their host star. One theory for their formation is that hot Jupiters formed farther away and migrated inward. Migration is a change in orbit due to interactions with a disk of gas or planetesimals. Hot Jupiters are found within .05-.5 AU of the host star. They are extremely hot, with temperatures as high as 2400 K. They were initially the most common type of exoplanet found because they are the easiest to detect with the transit and radial velocity methods (because they are huge and close to the host star), but as detection methods have improved smaller exoplanets and exoplanets farther from their host stars have been discovered as well.
Ice Giant
Composed primarily of volatile substances heavier than helium, such as oxygen, carbon, nitrogen, and sulfur. Ice giants have significantly less helium and hydrogen than gas giants and they are also smaller. Uranus and Neptune are ice giants. According to some planetary models, these two giant planets may have layers of superionic ice under relatively shallow hydrogen and helium atmospheres, which would explain their unusual magnetic fields.
Terrestrial Planet
Composed primarily of silicate minerals or metals.
Super-Earth
Defined exclusively by mass with upper and lower limits. Super Earths are ‘potentially’ rocky planets with up to 10 times the mass of Earth. The term ‘Super Earth’ simply refers to the mass of the planet and not to any planetary conditions, so some of these may actually be gas dwarfs. The Kepler Mission defined a Super-Earth as a planet bigger than Earth-like planets (.8-1.25 Earth radii), but smaller than mini-Neptunes (2-4 Earth radii).
Mini-Neptune
Also known as a gas dwarf or transitional planet. Mini-Neptunes are planets with a mass up to 10 Earth masses. They are less massive than Uranus and Neptune (shocker) and have thick hydrogen/helium atmospheres.
Pulsar Planet
A planet that orbits a pulsar, a rapidly rotating neutron star. Pulsar planets are discovered through anomalies in pulsar timing measurements. Pulsars rotate at a regular speed, so any bodies orbiting the pulsar will cause regular changes in its pulsation. The changes can be detected with precise timing measurements.
Goldilocks Planet
Planet that falls within a star's habitable zone, which basically means it has the potential to support liquid water on its surface.
Rogue Planet
Also known as interstellar planet, nomad planet, free-floating planet, orphan planet, wandering planet or starless planet. A planet without a host star that orbits the galaxy directly.
Puffy Planet
A planet with a large radius but very low density. Puffy planets expand because they are being warmed from the inside out. This warming may be from the star's heat reaches the planet's core, or from stellar winds carrying ions and heat that reach deeper into the planet. The ions are attracted to the planet's magnetic field. Friction is generated by winds blowing past ions being held by the magnetic field, creating heat that will warm the planet from the inside and causing it to expand.
Chthonian Planet
The rocky core left behind when a hot Jupiter orbits too close to their star. The star's heat and extreme gravity can rip away the planet's water or atmosphere.
Water Worlds
An exoplanet completely covered in water. Simulations suggest that these planets formed from ice-rich debris further from their host star. As they migrated inward, the water melted and covered the planet in a giant ocean.
Temperature of Exoplanets
In calculation of temperature of exoplanets, the star is often assumed to be a blackbody. The exoplanet is assumed to reflect some of the radiation, have no heating from its core, and have emissivity close to 1.
Let the temperature of the exoplanet and the star be [math]\displaystyle{ T_e }[/math] and [math]\displaystyle{ T_s }[/math], and the radius be [math]\displaystyle{ R_e }[/math] and [math]\displaystyle{ R_s }[/math]. They are separated by a distance of [math]\displaystyle{ D }[/math]. Then, by Stefan-Boltzmann Law, the radiation from the star and the exoplanet are
where [math]\displaystyle{ \sigma }[/math] is the Stefan-Boltzmann constant.
Only a fraction of the star's radiation reaches the exoplanet, and only a fraction of those radiation is absorbed. The ratio of radiation that reaches the exoplanet is [math]\displaystyle{ \frac{\pi R_e^2}{4\pi D^2} }[/math] by considering the sphere centered at the sun that crosses the exoplanet, and [math]\displaystyle{ 1-A }[/math] of those is absorbed, where [math]\displaystyle{ A }[/math] is the Albedo of the planet. Therefore, [math]\displaystyle{ L_e=\frac{\pi R_e^2(1-A)}{4\pi D^2}L_s. }[/math]
Expanding [math]\displaystyle{ L_e }[/math] and [math]\displaystyle{ L_s }[/math] and simplifying, we find
For example, if both the sun and the Earth are assumed to be blackbodies ([math]\displaystyle{ A=0 }[/math]), the temperature of the Earth would be
This equation can also be adjusted to account for the presence of an atmosphere with greenhouse gases, which gives a better prediction of temperature for some types of exoplanets.
External Links
- NASA Exoplanet Exploration Website (Exoplanet information for the general public)
- NASA Exoplanet Archive (Database and exoplanet news)
- The Extrasolar Planets Encyclopaedia (Another database)
- NASA Exoplanet Watch (Collect and work with exoplanet data!)