Direct Imaging of Exoplanets Through Occultations

Planetary orbits are randomly oriented throughout our galaxy. The probability that an exoplanet’s orbit will be fortuitously aligned to allow that exoplanet to transit across the face of its parent star depends upon the radius of the star, the radius of the planet, and the distance of the planet from the star. In general, planets orbiting close-in are more likely to be seen transiting their star then planets orbiting further out.

The equation for the probability of observing a exoplanet transit event is

p_{tra} = \left (\frac{R_{\bigstar}+R_{p}}{a} \right )\left (\frac{1}{1-e^{2}} \right )

where ptra is the transit probability, R* is the radius of the star, Rp is the radius of the planet, a is the semi-major axis of the planetary orbit, and e is the eccentricity of the planetary orbit 

Utilizing the data in the NASA Exoplanet Archive for the 1,463 confirmed exoplanets where the above data is available (and assuming e = 0 when eccentricity is unavailable), we find that the median exoplanet transit probability is 0.0542. This means that, on average, 1 out of every 18 planetary systems will be favorably aligned to allow us to observe transits. However, keep in mind that our present sample of exoplanets is heavily biased towards large exoplanets orbiting close to their parent star. Considering a hypothetical sample of Earth-sized planets orbiting 1 AU from a Sun-sized star, the transit probability drops to 0.00469, which means that we would be able to detect only about 1 out of every 213 Earth-Sun analogs using the transit method.

How might we detect some of the other 99.5%? My admired colleague in England, Abdul Ahad, has written a paper about his intriguing idea: “Detecting Habitable Exoplanets During Asteroidal Occultations”. Abdul’s idea in a nutshell is to image the immediate environment around nearby stars while they are being occulted by asteroids or trans-Neptunian objects (TNOs) in order to detect planets orbiting around them. While there are many challenges (infrequency of observable events, narrow shadow path on the Earth’s surface, necessarily short exposure times, and extremely faint planetary magnitudes), I believe that his idea has merit and will one day soon be used to discover and characterize exoplanets orbiting nearby stars.

Ahad notes that the apparent visual magnitude of any given exoplanet will be directly proportional to the apparent visual magnitude of its parent star, since exoplanets shine by reflected light. Not only that, Earth-sized and Earth-like planets orbiting in the habitable zone of any star would shine by reflected light of the same intrinsic brightness, regardless of the brightness of the parent star. He also notes that the nearer the star is to us, the greater will be a given exoplanet’s angular distance from the occulted star. Thus, given both of these considerations (bright parent star + nearby parent star = increased likelihood of detection), nearby bright stars such as Alpha Centauri A & B, Sirius A, Procyon A, Altair, Vega, and Fomalhaut offer the best chance of exoplanet detection using this technique.

Since an exoplanet will be easiest to detect when it is at its greatest angular distance from its parent star, we will be seeing only about 50% of its total reflected light. An Earth analog orbiting Alpha Centauri A would thus shine at visual magnitude +23.7 at 0.94″ angular distance, and for Alpha Centauri B the values would be +24.9 and 0.55″.

Other considerations include the advantage of an extremely faint occulting solar system object (making it easier to detect faint exoplanets during the occultation event), and the signal boost offered by observing in the infrared, since exoplanets will be brightest at these wavelengths.

A distant (and therefore slow-moving) TNO would be ideal, but the angular size of the TNO needs to be larger than the angular size of the occulted star. However, slow-moving objects mean that occultation events will be rare.

The best chance of making this a usable technique for exoplanet discovery would be a space-based observatory that could be positioned at the center of the predicted shadow and would be able to move along with the shadow to increase exposure times (Ahad, personal communication). It would be an interesting challenge in orbital mechanics to design the optimal base orbit for such a spacecraft. The spacecraft orbit would be adjusted to match the position and velocity of the occultation shadow for each event using an ion drive or some other electric propulsion system.

One final thought on the imaging necessary to detect exoplanets using this technique. With a traditional CCD you would need to begin and end the exoplanet imaging exposure(s) only while the parent star is being occulted. This would not be easy to do, and would require two telescopes – one for the occultation event detection and one for the exoplanet imaging. A better approach would be to use a Geiger-mode avalanche photodiode (APD). Here’s a description of the device captured in 2016 on the MIT Lincoln Labs Advanced Imager Technology website:

A Geiger-mode avalanche photodiode (APD), on the other hand, can be used to build an all-digital pixel in which the arrival of each photon triggers a discrete electrical pulse. The photons are counted digitally within the pixel circuit, and the readout process is therefore noise-free. At low light levels, there is still noise in the image because photons arrive at random times so that the number of photon detection events during an exposure time has statistical variation. This noise is known as shot noise. One advantage of a pixel that can digitally count photons is that if shot noise is the only noise source, the image quality will be the best allowed by the laws of physics. Another advantage of an array of photon counting pixels is that, because of its noiseless readout, there is no penalty associated with reading the imager out frequently. If one reads out a thousand 1-ms exposures of a static scene and digitally adds them, one gets the same image quality as a single 1-s exposure. This would not be the case with a conventional imager that adds noise each time it is read out.

References
Ahad, A., “Detecting Habitable Exoplanets During Asteroidal Occultations”, International Journal of Scientific and Innovative Mathematical Research, Vol. 6(9), 25-30 (2018).
MIT Lincoln Labs, Advanced Imager Technology, https://www.ll.mit.edu/mission/electronics/ait/single-photon-sensitive-imagers/passive-photon-counting.html. Retrieved March 17, 2016.
NASA Exoplanet Archive https://exoplanetarchive.ipac.caltech.edu.
Winn, J.N., “Exoplanet Transits and Occultations,” in Exoplanets, ed. Seager, S., University of Arizona Press, Tucson (2011).

Exoplanets with Deep Transits

The list above shows the 35 stars presently known to dip in brightness by 0.02 magnitudes or more due to a transiting exoplanet.

The change in the star’s magnitude during transit is given by

\Delta m = 2.5\log_{10}\left ( 1+\delta \right )

where Δm is the drop in magnitude, and δ is the transit depth

The time between transits for these exoplanets ranges between 0.79 and 5.72 days, with a median period of 2.24 days.  You can generate your own ephemeris for any of these transiting exoplanets at:

https://exoplanetarchive.ipac.caltech.edu/cgi-bin/TransitView/nph-visibletbls?dataset=transits

The transit duration for these exoplanets ranges between 1.08 and 3.11 hours, with a median duration of 2.11 hours.

The exoplanets with the deepest transits, HATS-6 b at 0.035 magnitudes and Kepler-45 b at 0.034 magnitudes, cross stars that are 15.2 and 16.9 magnitude, respectively, so these events might be out of reach for most amateur photometrists.  The only other star hosting a transiting exoplanet with a Δm ≥ 0.03m is Tycho 5165-481-1 in Aquila (WASP-80 b) which at visual magnitude 11.9 is a better candidate for smaller instruments.  The brightest star on our list (by far) is HD 189733 in Vulpecula, magnitude 7.7, with a drop in brightness that is almost as good at 0.026 magnitudes.

References
Fakhouri, O. (2018). Exoplanet Orbit Database | Exoplanet Data Explorer. [online] Exoplanets.org. Available at: http://exoplanets.org/ [Accessed 11 Dec. 2018].