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).

Bonner Durchmusterung und Gaia

As our civilization and technology continue to evolve, it seems we take far too much for granted.  We neglect to consider how incredibly hard people used to work years ago to achieve results we today would pass off as almost trivial.  But history has many lessons to teach us, if only we would listen.

As an example, Prussian astronomer Friedrich Wilhelm August Argelander (1799-1875) at the age of 60 began publishing the most comprehensive star catalogue and atlas ever compiled, as of that date.  From 1852 to 1859, Argelander and his assistants carefully and accurately recorded the position and brightness of over 324,000 stars using a 3-inch (!) telescope in Bonn, Germany.  Employing the Earth’s rotation, star positions were measured as each star drifted across the eyepiece reticle in the stationary meridian telescope by carefully recording when each star crossed the line, and where along the line the crossing point was.

Stars Transiting in a Meridian Telescope

One person observed through the telescope and called off the star’s brightness as each star crossed the line, noting the exact position along the reticle on a pad with a cardboard template so that the numbers could be written down without looking away from the telescope.  A second person, the recorder, noted the exact time of reticle crossing and the brightness called out by the observer.  In this way, two people were able to record the position and brightness of every star.

Each star was observed at least twice so that any errors could be detected and corrected.  In some areas of the Milky Way, as many as 30 stars would cross the reticle each minute.  What stamina and dedication it must have taken Argelander and his staff to make over 700,000 observations in just seven years!  Argelander’s catalogue is called the Bonner Durchmusterung and is still used by astronomers even today.  It was the last major star catalogue to be produced without the aid of photography.

Like Argelander’s small meridian telescope, the European Space Agency’s Gaia astrometric space observatory is currently measuring tens of thousands of stars each minute (down to mv = 20) as they transit across a large CCD array—the modern day equivalent of an eyepiece reticle.  But instead of utilizing the Earth’s rotation period relative to the background stars of 23h56m04s, Gaia’s twin telescopes separated by exactly 106.5° sweep across the stars as Gaia rotates once every six hours.  A slight precession in Gaia’s orientation ensures that the field of view is shifted so that there is only a little overlap during the next six-hour rotation.

When Gaia completes its ongoing mission, it will have measured the positions, distances, and 3D space motions of around a billion stars, not just twice but 70 times!

Though electronic computers do most of the work these days, someone still has to program them.  Some 450 scientists and software experts are immersed in the challenging task of converting raw data into scientifically useful information.

I’d like to conclude this entry with a quotation from Albert Einstein (1879-1955), who was born and died exactly 80 years after Argelander.

Many times a day I realize how much of my outer and inner life is built upon the labors of my fellowmen, both living and dead, and how earnestly I must exert myself in order to give as much as I have received.

I love that quote.  Words to live by.

Separating Observer from Observed

One of the most difficult things to do in observational science is to separate the observer from the observed.  For example, in CCD astronomy, we apply bias, dark, and flat-field corrections as well as utilize median combines of shifted images to yield an image that is, ideally, free of any CCD chip defects including differences in pixel sensitivity and zero-point.

We as observers are constrained by other limitations.  For example, when we look at a particular galaxy, we observe it from a single vantage point in space and time, a vantage point we cannot change due to our great distance from the object and our existence within an exceedingly short interval of time.

Yet another limitation is a phenomenon that astronomers often call “observational selection”.  Put simply, we are most likely to see what is easiest to see.  For example, many of the exoplanets we have discovered thus far are “hot Jupiters”.  Is this because massive planets that orbit very close to a star are common?  Not necessarily.  The radial velocity technique we use to detect many exoplanets is biased towards finding massive planets with short-period orbits because such planets cause the biggest radial velocity fluctuations in their parent star over the shortest period of time.  Planets like the Earth with its relatively small mass and long orbital period (1 year) are much more difficult to detect using the radial velocity technique.  The same holds true for the transit method.  Planets orbiting close to a star will transit more often—and are more likely to transit—than comparable planets further out.  Larger planets will exhibit a larger Δm than smaller planets, regardless of their location.  It may be that Earthlike planets are much more prevalent than hot Jupiters, but we can’t really conclude that looking at the data collected so far (though Kepler has helped recently to make a stronger case for abundant terrestrial planets).

Here’s another important observational selection effect to consider in astronomy: the farther away a celestial object is the brighter that object must be for us to even see it.  In other words, many far-away objects cannot be observed because they are too dim.  This means that when we look at a given volume of space, intrinsically bright objects are over-represented.  The average luminosity of objects seems to increase with increasing distance.  This is called the Malmquist bias, named after the Swedish astronomer Gunnar Malmquist (1893-1982).