There was a great question about transits in response to my post about “What Factors Impact Transit Shape” so I thought I’d answer in a blog post.
Jean Tate asked:
Question: In the image of Venus transiting the Sun, there are sunspots. How common are sunspots on the Sun-like stars in the Kepler field? How do sunspots change the transit light curve? How are sunspots modeled?
Starspots are dark blotches on the surface of the star and are regions of intense magnetic activity. Their temperature are lower than the rest of the photosphere which gives them their dark appearance. These blemishes are transitory and last anywhere from hours to months. They are an indication of the magnetic activity of the star, and the Sun goes through an 11-year cycle where the number of starspots (or sunspots as we call them on the Sun) changes. The more active the Sun, the more sunspots visible on its surface.
If you viewed the transit of Venus last July, there were several sunspots on the surface of the Sun which you can see in the image below.
On the Sun we can actually spatially resolve the sunspots, but on other stars we can’t. So for Kepler that is monitoring stars thousands of light years away, we detect starspots through the light curve. As the star rotates, starspots will come in and out view causing changes in the star’s brightness. The pattern in the star’s light curve will repeat once per rotation period of the star. At the equator, the Sun rotates every 24.47 days much longer than the short few-tens of hours that a planet transit lasts.
If the transiting planet doesn’t cross over a starspot we get a fairly rounded U-shaped symmetric bottom to the transit as you can see below for a set of simulated planet transits.
Because planet transits last a few to tens of hours and stars rotate over a period of days, you can think of the starspot as effectively stationary with the planet moving across it during the transit. The starspot is not as bright as the surrounding areas of the photosphere, so when the planet transits across that starspot the lightcurve gets a bit brighter than average and you don’t see a symmetric bottom to your transit. So you see a small positive bump in the transit lightcurve. In the observed transit shown below, the planet crosses a starspot during the second half of the transit.
Planets transiting starspots can be extremely useful. Those transits have been recently used to measure the alignment of the planet’s orbit to the rotation axis of the star. In our Solar System, the planets are about 8 degrees off from being aligned with the Sun’s rotation axis, but other planetary systems are severely misaligned.
If the planetary system is aligned with the star’s rotation axis, then the planet transit path is a chord that always crosses over the starspot when it is in view, so you will see many of the planet transits having the signature of a starspot crossing. Because the star is also rotating between transits, the starspot will be likely be in a different place on the star’s disk the next time the planet comes around so you will see the timing of the bump change from transit to transit. If the orbit is misaligned, then only when the starspot is in a position crossing the planet’s transit chord across the star’s surface will there be a positive bump in the transit lightcurve. So the next several transits the planet has extremely low chances of being timed such that the starspot is in the same position on the star’s disk for the event to repeat. So you should see no starspot signal in subsequent transits. You can see this effect below in the figure from Sanchis-Ojeda et al 2012.
Although Kepler was designed to find extrasolar planets, the Kepler light curves with their high temporal cadence and measurement precision is a rich data set for studying stellar astrophysics. Although the main goal of Planet Hunters is to search for new extrasolar planets, the Talk discussion tool was designed to enable volunteers to be able to identify other types of potentially interesting variable stars and oddball light curves that we weren’t necessarily looking for with the main classification interface.
With so many eyes looking through the light curves for 160,000 stars on the website, we’re bound to find an interesting star or two, and we have. Planet Hunters has helped discover a new RR Lyrae variable star. This is the second one spotted by Planet Hunters. Just like the first (which was spotted a year ago), this one was spotted by the keen eyes of our volunteers on Talk. It was reported to the Science Team, and Chris contacted the Kepler folks who study these sorts of thing, and it looks like it is indeed a new find. Congratulations to all involved. The RR Lyrae discovery is actually not the Kepler target star, but is nearby and contributing light into the photometric aperture, contaminating the actual Kepler star’s light curve with it’s changing brightness.
RR Lyrae’s are a special type of variable star. The radial pulsations cause the star to expand and contract producing observed changes in the star’s luminosity and subsequently the observed light curve. The American Association of Variable Star Observers (AAVSO) has a nice writeup describing the history and properties of RR Lyraes. Because the pulsations are supposedly simple radial expansion, these stars are often used as standard candles for measuring distances. But there is still a lot to learn about these stars. In particular, the underlying cause of Blazhko modulation, a periodic amplitude and/or phase variation of the pulsations with a period of typically 10-100x the typical pulsation period, that some of the RR Lyraes undergo is still an open question in stellar astrophysics.
This class of variable stars is named for the prototype star, RR Lyrae, first identified to exhibit these oscillations and observed patterns of variation. The original RR Lyrae just so happens to be in the Kepler field as KIC 7198959. There are currently about 40 known RR Lyrae stars in the Kepler field, so this is indeed a rare find. This new find will subsequently be studied by the Kepler Cepheid & RR Lyrae Working Group, and hopefully eventually be included in publication featuring all RR Lyrae stars identified in the Kepler field.
Today we have a guest post from Willie Torres. Willie is an astronomer the Harvard-Smithsonian Center for Astrophysics (Cambridge, MA), and member of the Kepler team. His work for Kepler includes the statistical “validation” of transiting planet candidates that cannot be confirmed in the usual way, that is, by measuring their mass and showing that it is small enough to be a planet. He also works on determining fundamental parameters of stars in eclipsing or astrometric binaries, for testing models of stellar evolution.
Transiting circumbinary planets are interesting because they show us that planets can form in environments that are very different from our Solar System. Instead of being a single star, the central object is actually a pair of stars orbiting each other, and in these systems the planet can occasionally pass in front of one or even both stars, producing transit signals. For circumbinary planets such as those the Kepler Mission has announced (Kepler-16, Kepler-34, Kepler-35, Kepler-38, and most recently Kepler-47), the orbit of the two stars is such that they eclipse each other periodically, and these typically deep eclipses are what calls attention to them in the first place, in the light curves produced by Kepler. The neat thing about these transiting circumbinary systems is that they can also provide a wealth of information about the stars that is normally not available in regular transiting planet systems with a single host star. Two important stellar properties one can often measure are the masses and radii, from knowledge of the orbit of the stars around each other.
Masses and radii can of course also be determined in favorable eclipsing systems that don’t have planets, but when there is a transiting circumbinary planet, it’s even better. This is not hard to understand: as the planet passes in front of one or both stars, it is actually chasing a moving target because the two stars are revolving around each other. Each time the planet transits, the stars are in a different place in their orbit. This means that by measuring the precise times of these transits, we are actually mapping the orbit of the binary in a different way than would normally be done for a regular eclipsing binary. This provides extra information about the motion of the stars, and in particular it constrains the ratio of the masses between the two stars very well. It also helps to determine their sizes. Combining this with additional observations such as radial velocities measured from spectra of one or both stars, their masses and radii can be measured to high precision.
Astronomers care about the masses and radii of stars because these measurements allow them to test their models of how stars form and evolve. Theorists have come up with a fairly detailed prescription for how a star of a given mass and chemical composition changes its properties (radius, temperature, luminosity, etc.) as time goes by. But without real observations against which to check those predictions, we can’t be sure they’re right. This is important because astronomers often use those same models to infer properties of single stars that are much more difficult to measure directly. Or they may be interested in knowing the age of a star, which also relies on theoretical models. As it turns out, observations have shown that models for low mass stars (such as the cool M dwarfs) are not quite right: real stars tend to be a little bit larger and cooler than the models predict.
Circumbinary planets in which the eclipsing binary at the center contains an M dwarf are particularly interesting, because they allow us to test theory in this problematic low-mass regime. That happens to be the case for the recent exciting Planet Hunters discovery of KIC 4862625. The primary component in the eclipsing binary is an F star of about 1.3 solar masses, and the secondary is an mid M dwarf a little under 0.4 solar masses. They orbit each other every 20 days. The circumbinary planet goes around every 138 days. With other colleagues I’ve been working on determining the stellar properties of both stars as accurately as possible, and comparing them with several sets of stellar evolution models (since models are not all created equal). For getting the stellar properties we use not only high-quality spectra taken with the 10-meter Keck telescope in Hawaii, but also results from a very sophisticated modeling of the Kepler light curve that can reproduce all the binary eclipses as well as the transits of the circumbinary planet nearly perfectly. This tells us that we at least understand the dynamics of the system pretty well (i.e., how all the objects move).
But there are always complications. In this case, we took a high-resolution image of the system and discovered that there’s another star right next to eclipsing binary that (we realize now) is introducing contamination in the Kepler light curve. It’s only about 0.7 arcsec away from the eclipsing binary, and we believe it is physically associated. But wait, there’s more! The images show that this companion is actually a close binary itself! At the time of this writing we are still trying to figure out exactly how much extra light these new objects are contributing to the Kepler photometry, so that we can take that into account in order not to bias the measured properties of the eclipsing binary stars, or of the circumbinary planet.
It’s been an exciting week for exoplanets with the discovery of PH1 and the discovery of an Earth-mass planet around the closest star to Earth (Alpa Centauri B). In the coming days and weeks, you’ll hear more about the specifics of how we came out confirming and validating the discovery of PH1 and solidifying that it was a planet orbiting the two central stars in a four star system, but I wanted to give a brief summary of the data and results.
This effort has taken months and months from obtaining the telescope observations, to modeling the light curves, combining Kepler data with radial velocity observations, and applying stellar evolution models. Robert Gagliano and Kian Jek get enormous credit for the discovery and starting this process off and recognizing and spotting the planet transits.
Many collaborators have worked hard to confirm PH1 and study its properties, especially: Jerry Orosz, Josh Carter, Willie Torres, and Debra Fischer who have put tremendous effort (particularly in the past few weeks) to get us from so we found some transits to bona fide planet.
Everyone in this collaboration involved in the paper (including Kian and Robert who were coauthors on the discovery paper) are listed below:
To summarize PH1′s confirmation story, I thought I’d share my press talk slides:
Robert and Kian identified three transits in the light curve of an eclipsing binary . In the binary, there are two stars, one larger and hotter and one smaller and cooler. When the smaller cooler star crosses the face of the larger hotter star, you get some of the larger star’s blocked out and we call that a primary eclipse. When the smaller cooler star crosses behind the larger hotter star, we get a secondary eclipse where the light from the smaller cooler star is blocked out. So we see this big dip+ small dip pattern. Robert and Kian noticed three transits separated by ~137 days in Quarters 1-5 data superimposed on the light curve indicating a third small body in the system, and notified the science team of a possible circumbinary planet.
There’s a small chance that we’re seeing a false positive, where on the sky our Kepler eclipsing binary is aligned with a faint background eclipsing binary giving rise to the transit-like signal. If the transiting body is truly orbiting both stars, we have a way of checking. A body in a circumbinary orbit (orbiting around both stars in a stellar binary), is orbiting a moving target, so the positions and velocities of the two stars are different each transit. So this means that at each transit, there are slightly different gravitational tugs on the planet causing the timing and the duration of the transit to change. We see these changes and this gives us high confidence the planet is really orbiting the Kepler eclipsing binary and not some background source. If you look below at the 7 planet transits across the larger bigger star in the binary, you can convince yourself that the widths of the transits are changing. So bingo, planet!
We got radial velocity observations from the Keck telescopes on Mauna Kea. The radial velocity observations measure the wobble of the larger star in the binary that the planet orbits. With the precision of the observations and time duration we have on the observations, we cannot measure the wobble caused by the gravitational pull of the planet. What we are measuring is the wobble due to the gravitational tug of the smaller star as it orbits the larger star in the binary.
To our surprise we found two velocity contributions in the radial velocity observations. One is from the larger star in the eclipsing binary (solid points) with the model fit for the velocity observations shown in red. The 2nd component is stationary over the roughly 5 months we were taking radial velocity observations. This 2nd component is coming from a source that is providing some light in the spectra slit we place across the Kepler target when observing on Keck. It has the same value as the average or systemic velocity of the binary. So if you take the average value of the black points that’s the velocity of the eclipsing binary (and planet host stars) moving towards or away form us. This second component has the same velocity and random field stars have velocities in the galaxy ranging from ~20-60 km/s so to have a source that has to be very close to the eclipsing binary on the sky that we see it in the Keck observations and have the same average velocity as the eclipsing binary tells us that this source is bound to the eclipsing binary.
We used adaptive optics observations from the Keck II telescope to zoom in and look around our eclipsing binary for other stars that would be contamination the Kepler photometric aperture. We also used deep optical imaging to look for slightly further contamination stars as Bill Keel described in his blog post. As Bill discussed in his post, if there are stars providing extra light to the aperture that is summed up to make the Kepler light curve, it will give us the wrong planet radius. This is because the extra light will decrease the observed transit dips causing us to underestimate the size of the planet. We go these adaptive optics observations while we getting the radial velocity observations.
In the adaptive optics observations there is a source 0.7” away (or about 1000 AU) from the eclipsing binary, and we knew about its existence when we were analyzing the Keck radial velocity observations. There is was the ah-ha moment, where we went oh, this source in the adaptive optics observations must be that second velocity source we see in the radial velocity observations, that we think is gravitationally bound to the eclipsing binary orbiting well outside the planet’s orbit. To our surprise the adaptive optics observations revealed that this source is elongated in one direction (which you can see in the slide below). What we think this means is that we’re just barely resolving the source as a visual binary (2 stars!). So that we have a pair of stars, getting 2 for the price of 1 – (getting us to four stars in the system!) orbiting the eclipsing binary. Our best guess from the Keck observations is that the 2 stars in the distant binary are separated by no more than 40-50 AU.
Combining all of these observations, we went after obtaining the properties of the planet and the stars in the PH1 system. Here below are the properties that come out of the combined photometric-dynamical model that uses the radial velocity observations and Kepler light curve for both PH1 and the stars,. The age we estimate for the entire system from spectral modeling. We estimate the mass of the planet by the fact that it not massive enough to pull on its parents stars sufficiently for us to see slight changes in the timings of when the stars eclipse each other (so when the smaller cooler star crosses the face of the larger hotter star and when the smaller cooler star crosses behind the larger hotter star).
We confirmed with modeling the eclipsing binary properties with just the Kepler light curve, and we’re confident in our estimation for the planet and host star properties. The planet is a gas giant, a bit bigger than Neptune and slightly smaller than Saturn. PH1 orbits its parent stars at a distance between Mercury and Venus in our own Solar System. The planetary system is stable. The planet happily orbits the eclipsing binary every ~138 days not really noticing there’s a second pair of stars out at 1000 AU .
PH1 is the 7th circumbinary planet, and the 6th circumbinary system. Below are the orbits of all the circumbianry Kepler planets and PH1 (not depicting the second distant pair of stars).So why do we care? Well, circumbinary planets are the extremes of planet formation, and we need to understand how they form if we are really understand how we form planets in our own solar system. Planet formation models need to be able to explain both environments, and each of the systems detected gives us another puzzle piece to this picture.
Today we have a guest post from Bill Keel. Bill is a member of the science team for Galaxy Zoo, and is more accustomed to dealing with stars by the billion than one at a time. He is a University of Alabama astronomer, weekend trombonist, and occasional photographer, being gradually trained by two cats with names out of Tolkien. Both his Twitter stream and his posts on the Galaxy Zoo forum can be found under the name NGC3314, and his other professional exploits may be found at http://astronomy.ua.edu/keel.
Kepler is sometimes most effective when properly backed up by other instruments, since its design was tightly optimized for precision in measuring bright stars at the expense of other things (such as angular resolution). Here’s a case showing how interpretation of Kepler results on planetary transits can be assisted by fairly routine ground-based measurements.In late June, I got an email request from Meg Schwamb:
“We’ve found a planet with ~130 days orbit going around a eclipsing binary. The eclipsing binary has a 20 day orbit so the planet is circumbinary and there’s a third star in the binary+planet system orbiting out at ~1000 AU with a period of 10E4-10E5 years. We’ve been following up the system with Keck observations.“ [We didn't yet know at the time that this third star would itself turn out to be a binary star].
The region around this star from our perspective is very busy (like the whole Kepler field), and the Kepler measurement includes light from additional faint stars. One, in particular, appears about 3 arcseconds away from the star of interest, well within the 6-acsecond radius of a Kepler measurement. Knowing its brightness would help narrow down the planet’s properties, making sure we have the right starting points in brightness for the Kepler target star my itself.
My institution is a partner in the SARA consortium, which operates telescopes in Arizona and Chile remotely. As a result, I have fairly regular nights scheduled, and indeed there were a couple of nights I could use at our northern telescope, a 0.9m instrument on Kitt Peak, Arizona, in July (just before shutdown for the monsoon season). After a couple of tries when the weather didn’t quite cooperate, including one night that was clear but the air to turbulent for this project, I got an hour’s worth of images on the evening of July 17. The image quality (seeing, in astronomical jargon) was 1.5-1.8 arcseconds, meaning that these values give the diameter across which a stars image drops to half its peak intensity due to atmospheric turbulence. That makes separating stars 3 arcseconds apart tractable. The timing worked out well during the night – the field was within 15 degrees of the zenith, minimizing atmospheric and tracking problems.
Trying to get precision measurements of bright and faint stars simultaneously takes some care – good data on the faint star isn’t much help if the bright star is hopelessly saturated in the data. So instead of one long exposure, I took 60 1-minute observations, using a red filter to roughly match the midpoint of the very broad spectral band used by Kepler. For further analysis, that gave both the grand average of all 60, and I also used averages of subsets of 10 to help estimate certain sources of error in the processing.
Even though the fainter interfering star was clearly separated from the bright one in these images, there was enough spillover to need correction.I tried several procedures or this – the most successful took as a reference point a similarly bright star with no companion in that direction, subtracting variously scaled versions of its image to eliminate as much of the bright star’s light as possible (the subtracted images looked a little odd in the middle – much later I realized that might come from the very close companion star seen in other data).
To make sure we understood how our brightness measurements relate to the Kepler data, I checked published magnitudes for Kepler stars in this neighborhood. This gave me some bad moments until I realized that the published values were often based on short exposures with a telescope no bigger than I was using – bright stars are OK, faint stars become quickly much less accurate. Phew. Now I know this, so that if it comes up again, I’m ready.
The result? That fainter star has magnitude R=18.73, making it only 1.02% as bright as the Kepler target with planet. Other contaminating stars are still fainter, down to 0.03% of the target star’s red-light intensity.
Today we have a guest post by fellow Planet Hunter Daryll (nighthawk_black) updating us on the search for dwarf novae and cataclysmic variables. Daryll’s here to talk about a dwarf nova candidate found in PH Talk.
Hi Planet Hunters,
Following the guest post from GO Director Martin Still, a review of light some light curves discussed on PH Talk turned up an interesting target somewhat similar to the serendipitous Dwarf Nova known as NIK 1. First noted by myself and several volunteers as a possible cataclysmic, we believe this to be another SU UMA type variant with over 50 quasi-periodic brightness changes observed and a defined superoutburst, in the public Q6 data.
The activity is not visible in all quarters. An examination of the accompanying target pixel files (these are files created by the Kepler processing pipeline that show the brightness over time for each of the pixels that are added up to make a Kepler light curve and those surrounding that don’t go into making the light curve – they can help you see if the features in the Kepler light curve come from the target star or something nearby that is contaminating the target’s star aperture) reveal that the true source of the dwarf nova candidate lies in the background and likely originates from an adjacent source tagged as KID-11412049, leaving how much activity we see in the original light curve dependent on the differing aperture pixel masks used for each Quarterly roll. Unfortunately it does not appear to be an eclipsing arrangement nor has it displayed any transiting circumbinary companions.
We asked the science team to take a look at this star and they think it looks like a good dwarf nova candidate. The PH science team has applied for Directors Discretionary Time seeking additional observations in the coming Quarter (we’re all waiting to hear back if the Planet Hunters proposal has been approved) to learn more about this system including its outburst supercycle, accretion disc stability and component compositions. Early analysis indicates high mass transfer with a notably short orbital period of 76 minutes; a GALEX survey shows this location also appears to be associated with a UV source.
Screening out background binaries from transit candidates is something the community has gotten pretty sharp at and I believe more of Martin’s missing Dwarf Nova will turn up. If confirmed, this will be the 5th Superoutbursting DN in the Kepler FOV and the 17th total, so well done and keep up the eagle-eyed hunting!
We have a guest post from Martin Still. Martin is Deputy Science Team Lead and Guest Observer Office Director for Kepler. He’s writing today to tell you about an interesting class of objects you might encounter when classifying Kepler light curves
I think one of the hardest types of light curves to classify in terms of variability as well as identifying transits in are the pulsating ones or pulsators as the Science Team has dubbed them. There are many types of pulsating stars but we’re referring to those where the star’s brightness is rapidly oscillating up and down over the 30 day period with many cycles in the span of 5 days
One example is SPH10074728:
You can see there’s a a discernible up-down (nearly vertical) changes in the star’s brightness. When classifying I would have said this was variable and selected the pulsating button or
I find these stars the hardest to identify transits in. For the example above, I would say there are no transits, and all those dips are due to the natural variability of the star. At least one of our current planet candidates is from a pulsator like this, so there can be transit signals in these light curves., They’re just a little harder to spot than the quiet curves. One thing to keep in mind when classifying that may help on these types of curves is that the dip from a transit typically lasts a few hours to ~10 hours, so if you see dips lasting days, those aren’t due to orbiting bodies. But go with your gut, if you think there’s a transit definitely mark it, if it’s real others will mark it too.
Today’s post is brought to you by guest blogger Charles Baldner, who will be writing a few blog posts this summer on topics related to stellar structure, asteroseismology, and stelalr activity. Charles is a graduate student in the Astronomy Department at Yale University. In his research, he uses helioseismology to study links between the interior of the Sun and solar activity.
Kepler is, first and foremost, an instrument designed to discover and investigate planets around other stars. It will probably not surprise you, however, if I tell you that Kepler data also provides an astounding amount of information about the stars themselves. What the planet hunter sees as noise – that annoying scatter in the data that hides or confuses the telltale signs of a planet – is music to another scientist. I mean that almost literally: like drums, flutes, bagpipes, or guitar strings, stars ‘ring’ at a variety of specific pitches, encoding information about all sorts of stellar properties. Using these `sounds’ to study stars is the science called asteroseismology.
A star is, more or less, a giant sphere of hot gas. Just like in the Earth’s atmosphere or oceans, waves can propagate through a star’s interior. These waves can reflect at the surface, causing it to move up and down, or to brighten or dim. If you can measure the velocity of the surface of a star very precisely, or measure the changes in brightness at the surface, you can detect these waves. If you take enough measurements, you can perhaps see the star ringing just like a musical instrument. In many stars, in fact, the waves you are seeing are sound waves, bouncing back and forth in the stellar interior just as they do inside an organ pipe.
We have used this kind of study to probe the inside of the Sun for more than thirty years. This is called helioseismology, and we have used it to determine the structure of the Sun very precisely. We can measure to great accuracy, for example, exactly where the interior of the Sun changes from `radiative’ to `convective’ (to learn more about the structure of the Sun, you can of course start at Wikipedia: http://en.wikipedia.org/wiki/Sun). We can also see the effects of rotation — different layers and different latitudes of the Sun rotate at different speeds, and we can measure this with helioseismology. Today, I use the tools of helioseismology to probe the regions just beneath sunspots.
In stars, as you can perhaps imagine, measuring these oscillations is much more challenging than it is in the Sun. After all, for most asteroseismic pulsations, we’re talking about minute changes
in velocity or brightness. But that, of course, is precisely what planet search instruments are built to measure, and the Kepler mission is providing us with an immense trove of data with which to use
asteroseismology to study large numbers of stars. In a future post, I’ll go over a few of the sorts of things we hope to glean from Kepler’s asteroseismic measurements.
Image Credit: NASA/ESA/SOHO http://sohowww.nascom.nasa.gov/data/realtime/eit_304/512/
I was talking to last week’s seminar speaker, and we were talking about Planet Hunters and some of the things that might be lurking in the Kepler data. One cool thought is there might be inverse transits so instead of dimming events, instead the star actually appears brighter.
There are lots of eclipsing binaries that you’ve probably seen as you’ve been classified, but another interesting type of eclipsing binary might be a transiting white dwarf orbiting a main sequence star. White dwarfs are about the same size or a little bit bigger than the Earth about half as massive as the Sun. Depending on where the white dwarf orbits, there could be magnification causing a brightening as the white dwarf crosses in front it’s companion star. This magnification is caused by gravitational microlensing, where a massive object bends light of a background source resulting in images of the source that are magnified and distorted. Transiting exoplanets are not massive enough to bend and distort the light of their companion stars significantly. For eclipsing binaries it looks white dwarfs are in the sweet spot, if they are orbiting extremely close to their partner main sequence star. Papers in 2003 by Sahu and Gilliland (2003) and Farmer and Agol predicted that Kepler might be able to detect such events. In these cases during the transiting event, the ligthcurve gets brighter rather than fainter. These events last as long as the transit does so only a few hours (if the white dwarf is orbiting at 1 AU the event is ~10 hours in duration).
Here’s some examples from a paper by Sahu and Gilliland (2003) .
A transiting 0.6 solar mass white dwarf orbiting at 1 AU
0.6 solar mass white dwarf at different orbital radii from a solar-type star
There are some estimates of how many might be there ranging from a few to a about a hundred or so events in the Kepler monitored stars, but we really don’t know. No one has detected them, and there could be 1 or none but with so many eyeballs staring at the data, we might uncover them if they’re there. Anyone seen anything like this in the light curves you’ve classified? It would be very exciting if we found one, it would be the first such discovery – if you see an inverse transit like the examples above, please share on Talk and let us know about your discovery!