## 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].

## Einstein, Brahms, and Exoplanets

What do Albert Einstein, Johannes Brahms, and exoplanets have in common?  They are all great courses provided by The Great Courses.

Call me old fashioned, but I love a great lecture presented by an expert in the field.  What a wonderful way to get introduced to a new subject, or refamiliarize yourself with an old subject, or deepen your knowledge about a subject with which you are already familiar.

I recently finished watching the magnificent course “Albert Einstein: Physicist, Philosopher, Humanitarian” by Don Howard, Professor of Philosophy at the University of Notre Dame, former Director of Notre Dame’s Graduate Program in History and Philosophy of Science, and a Fellow of the University of Notre Dame’s Reilly Center for Science, Technology, and Values.

I have taken an interest in Einstein since I first encountered relativity in my early teens, and of course being a physics major in college I became much more familiar with Einstein’s remarkable scientific contributions.  But this course surprised and delighted me with many details about Einstein himself.  Howard obviously has a much deeper understanding of Einstein the person than most physicists do, and his enthusiasm for his subject comes through in every lecture.  I doubt you will find a more thorough treatment of Einstein anywhere short of reading a biography.  Recommended!

As luck would have it, while I was nearing the end of this course, Time came out with an updated reissue of its special edition, “Albert Einstein: The Enduring Legacy of a Modern Genius”.  Great photographs, great text.  Well worth every penny!

Robert Greenberg is music historian-in-residence with San Francisco Performances and has produced a lot of high-quality music courses for The Great Courses.  I am in the process of watching all of them (yes, really, they’re that good!).  Recently, I finished his course on Johannes Brahms, who is probably my all-time favorite composer.

The music of Brahms is well known by many, but how much do you know about Johannes Brahms the person, and the events of his life?  This course is the perfect introduction to those subjects, as well as his extraordinary compositions.

It is amazing to me that no one has yet made a feature-length film about the life of Johannes Brahms (1833-1897).  A historically accurate dramatic portrayal could easily become one of the most significant musical film biographies ever made.  Brahms was one of the greatest composers who ever lived, and he had an interesting life—there is much material to draw upon for the making of this movie.  Greenberg’s course is a great place to begin, and I would also recommend the definitive biography, “Brahms: His Life and Work” by Karl Geiringer.

You’ve just got to love The Great Courses.  This is what television could have been.  PBS is the only thing that even comes close.  I recently completed “The Search for Exoplanets: What Astronomers Know” presented by Joshua Winn, now Professor of Astrophysical Sciences at Princeton University.  Not since Carl Sagan or Neil deGrasse Tyson have I been this excited about an astronomy presenter.  Josh Winn presents his exoplanets course with enthusiasm, precision, and a delivery that really draws you in to the subject.  I hope we see much more of him in the future.

## 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).

## Stars Like Our Sun

The spectral type of our Sun is G2V, that is to say, a G2 main-sequence star.

Sun
Zodiacal Constellations
mv = -26.75, mb = -26.10, B-V = 0.65
Ecliptic
0.0000158 ly
Single star

Here are the brightest stars visible in the nighttime sky that have the same spectral type and therefore are, arguably, most like our Sun.  All have an apparent visual magnitude brighter than +6.00.

Rigil Kentaurus A, Alpha Centauri A (α Cen A)
Centaurus
mv = 0.01, mb = 0.72, B-V = 0.71
α2000 = 14h 39m 36s, δ2000 = -60° 50′ 02″
4.30 – 4.34 ly
Triple star system

Alula Australis B, Xi Ursae Majoris B (ξ UMa B)
Ursa Major
mv = 4.73, mb = 5.38, B-V = 0.65
α2000 = 11h 18m 11s, δ2000 = +31° 31′ 46″
28 – 30 ly
Quintuple star system

HR 4523 A
Centaurus
mv = 4.88, mb = 5.55, B-V = 0.67
α2000 = 11h 46m 31s, δ2000 = -40° 30′ 01″
30.0 – 30.1 ly
Binary star system; exoplanet

Eta Coronae Borealis A & B (η CrB A & B)
Corona Borealis
A: mv = 5.577, mb = 6.123, B-V = 0.546
B: mv = 5.95, mb = 6.48, B-V = 0.53
α2000 = 15h 23m 12s, δ2000 = +30° 17′ 18″
57 – 59 ly
Triple star system

HR 8323
Grus
mv = 5.58, mb = 6.18, B-V = 0.60
α2000 = 21h 48m 16s, δ2000 = -47° 18′ 13″
51.9 – 52.5 ly
Single star

Mu Velorum B (μ Vel B)
Vela
mv = 5.59, mb = 6.10, B-V = 0.51
α2000 = 10h 46m 46s, δ2000 = -49° 25′ 12″
116 – 119 ly
Binary star system

HR 7845 A
Capricornus
mv = 5.65, mb = 6.34, B-V = 0.69
α2000 = 20h 32m 24s, δ2000 = -09° 51′ 12″
79 – 80 ly
Binary star system

HR 3578
Hydra
mv = 5.86, mb = 6.39, B-V = 0.53
α2000 = 8h 58m 44s, δ2000 = -16° 07′ 58″
68 – 69 ly
Single star

HR 2007
Orion
mv = 5.97, mb = 6.61, B-V = 0.64
α2000 = 5h 48m 35s, δ2000 = -4° 05′ 41″
49.2 – 49.8 ly
Single star with exoplanet

The Eta Coronae Borealis system is noteworthy in that its two primary components are both G2V stars orbiting each other every 41.6 years.  The third component of this system is a distant infrared dwarf, spectral type L8V.

Two of these G2V stars host at least one exoplanet: HR 4523A in Centaurus and HR 2007 in Orion.

HR 4523A has a planet midway in mass between Uranus and Neptune orbiting every 122 days between 0.30 and 0.62 AU from the star (similar to orbital distance of the planet Mercury in our own solar system).  The other stellar component of this system. HR 4523B, is a distant M4V star currently orbiting about 211 AU from HR 4523A.

HR 2007, a single star like the Sun, has a planet about 78% more massive than Neptune, orbiting every 407 days, more or less.  If this planet were in our own solar system, it would range between the orbits of Venus and Mars, roughly.

## Metallicity

No, it’s not the name of a rock band. Astronomers (unlike everybody else) consider all elements besides hydrogen and helium to be metals. For example, our Sun has a metallicity of at least 2% by mass (Vagnozzi 2016). That means as much as 98% of the mass of the Sun is hydrogen (~73%) and helium (~25%), with 2% being everything else.

Traditionally, elemental abundances in the Sun have been measured using spectroscopy of the Sun’s photosphere.  In principle, stronger spectral lines (usually absorption) of an element indicate a greater abundance of that element, but deriving the correct proportions from the cacophony of spectral lines is challenging.

A more direct approach to measuring the Sun’s elemental abundances is analyzing the composition of the solar wind, though the material blown away from the surface of the Sun that we measure near Earth’s orbit may be somewhat different from the actual photospheric composition.  The solar wind appears to best reflect the composition of the Sun’s photosphere in the solar polar regions near solar minimum.  The Ulysses spacecraft made solar wind measurements above both the Sun’s north and south polar regions during the 1994-1995 solar minimum.  Analysis of these Ulysses data indicate the most abundant elements are (after hydrogen and helium, in order of abundance): oxygen, carbon, nitrogen, magnesium, silicon, neon, iron, and sulfur—though one analysis of the data shows that neon is the third most abundant element (after carbon).

The elephant in the room is, of course, are the photospheric abundances we measure using spectroscopy or the collection of solar wind particles indicative of the Sun’s composition as a whole?  As it turns out, we do have ways to probe the interior of the Sun.  Both helioseismology and the flux of neutrinos emanating from the Sun are sensitive to metal abundances within the Sun.  Helioseismology is the study of the propagation of acoustic pressure waves (p-waves) within the Sun.  Neutrino flux is devilishly hard to measure since neutrinos so seldom interact with the matter in our instruments.  Our studies of the interior of the Sun (except for sophisticated computer models) are still in their infancy.

You might imagine that if measuring the metallicity of the Sun in our own front yard is this difficult, then measuring it for other stars presents an even more formidable challenge.

In practice, metallicity is usually expressed as the abundance of iron relative to hydrogen.  Even though iron is only the seventh most abundant metal (in the Sun, at least), it has 26 electrons, leading to the formation of many spectral lines corresponding to the various ionization states within a wide range of temperature and pressure regimes.  Of the metals having a higher abundance than iron, silicon has the largest number of electrons, only 14, and it does not form nearly as many spectral lines in the visible part of the spectrum as does iron.  Thus defined, the metallicity of the Sun [Fe/H] = 0.00 by definition.  It is a logarithmic scale: [Fe/H] = -1.0 indicates an abundance of iron relative to hydrogen just 1/10 that of the Sun.  [Fe/H] = +1.0 indicates an abundance of iron relative to hydrogen 10 times that of the Sun.

The relationship between stellar metallicity and the existence and nature of exoplanets is an active topic of research.  It is complicated by the fact that we can never say for certain that a star does not have planets, since our observational techniques are strongly biased towards detecting planets with an orbital plane near our line of sight to the star.

References
Vagnozzi, S. 2016, 51st Recontres de Moriond, Cosmology, At La Thuile

Many times each week, all manner of asteroids and trans-Neptunian objects pass in front of stars, casting shadows a few miles wide all over the Earth.  There are several potential events each week at any particular location.  I use the word “potential” because there is still significant uncertainty in the paths for many of these events.  The orbits of most small solar system objects are not yet precisely known, and, to a lesser extent, there is some uncertainty in the position of the occulted (obscured) star.

On Sunday evening, November 20, I got lucky.  Not only did I record a 1.02 second occultation event, but I was lucky to see it at all as I was significantly south of the predicted path.

The star affected was Tycho 5182-758-1 (also known as BD -3° 5037) in Aquarius and the object that moved in front of it was the asteroid 430 Hybris, a space rock about 20 miles across that orbits once around the Sun every 4.8 years.  Many asteroids have interesting names, and Hybris is no exception.  In Greek mythology, Hybris is a spirit of insolence, violence, and outrageous behavior.  It is also an alternative form of the word hubris.  All quite appropriate given the outcome of the U.S. presidential election less than two weeks earlier.

Here is the video I recorded of the event:

Occultation of the star Tycho 5182-758-1 in Aquarius by the asteroid 430 Hybris

And here is the light curve I derived from the video which clearly shows the event:

Steve Messner (near Northfield, Minnesota) and I were the only ones to observe this event.  It was a miss for Steve, and he was much closer to the predicted path!

Why do we do it? Even a single positive observation can greatly improve our knowledge of the orbit of the asteroid or trans-Neptunian object.  More than one positive observation gives us valuable information about its size and shape.  We can discover asteroid/TNO satellites and even rings!  But that’s not all.  These occultation events can also give us valuable information about the star.  Its size, position, and the separation and position angle of new or known companion stars.  Someday, we may even be able to use these events to discover exoplanets!

If you love observational astronomy and would like to contribute scientifically valuable observations by observing occultation events, contact me and I will help you get started.  The more observers we have, the more valuable our scientific contribution will be.