Transit method

In the case of a transit planet, continuous measurements reveal slight, periodic drops in brightness.

A planetary transit causes a decrease in brightness of the observed star.

The transit method is a photometric method for the detection of exoplanets . By 2019, around 80% of all known planets had been discovered using this method, which makes it the most successful method in the search for exoplanets. The planet is not observed directly, but only indirectly detected by observing the brightness of its star . Most of the discoveries made so far have been made with the help of the Kepler space telescope .

Summary

During a planetary transit , the planet obscures part of its parent star from a suitably situated observer, so that the brightness of the star is reduced during the transit. Continuous observation of the brightness of a star can detect these changes. The planet that causes a transit is called the transiting planet . Planets orbit their star on their orbit within a certain period, so that after one orbit another transit takes place. In order to rule out a random event, at least three transits with the same time interval must be observed. Although the probability of observing a transit with a randomly selected star is quite low with less than 1%, because the orbits can also be oriented in such a way that the planet never passes in front of its star when viewed from Earth, were with this star until August 2019 Method proven over 3000 exoplanets. Further information about the planet and the star can be obtained from observing the brightness gradients of the stars. For example, statements can be made about the period of revolution around its star, the planet's radius, the inclination of its orbit in relation to the line of sight and the direction of rotation of its star. In addition, from the knowledge of these values, statements can be made about the distance at which the planet orbits its star. In spectroscopic observation also data on the composition of his leave atmosphere and its albedo and temperature gain. At the beginning of 2017, the accuracy of the methods was not yet sufficient to obtain information about Earth-like planets, but measurements for Earth-like planets will be possible in the near future with the completion of high-intensity telescopes such as the James Webb Space Telescope .

Estimation of the probability of a transit

Consideration of the calculation of the observation probability using the solid angle for which a transit can be observed. The observer is assumed to be very distant.

Representation of the probability of transit depending on the star and planet orbit radius

Assuming that the orbits of the planets are randomly oriented in space, the probability that a distant observer can observe a transit can be calculated with the help of geometric considerations. A transit can only be observed when the planet passes in front of its star as seen by the observer and thus partially obscures the star. Under the assumption that the radius of the planet is negligible compared to the star radius , since mostly , the angular range of the orbit inclination in which a transit can take place can be estimated , assuming a circular path with the distance between the planet and the star is. However, since the plane of the path in the line of sight can also be oriented as desired in the case of an observable transit, the solid angle can be specified for the three-dimensional part of the orientation in which a transit takes place as . If it can be assumed that the orbits of the exoplanets are randomly oriented in space, the probability of being able to observe a transit can be given as a ratio to the total solid angle . This results in ${\displaystyle R_{p}}$ ${\displaystyle R_{*}}$ ${\displaystyle R_{p}\ll R_{*}}$ ${\displaystyle \vartheta \approx D_{*}/a_{p}=2R_{*}/a_{p}}$ ${\displaystyle a_{p}}$ ${\displaystyle \Omega _{\text{Transit}}=2\pi D_{*}/a_{p}}$ ${\displaystyle p}$ ${\displaystyle 4\pi }$

${\displaystyle p={\frac {\Omega _{\text{Transit}}}{\Omega _{\text{gesamt}}}}={\frac {R_{*}}{a_{p}}}}$.

The star radius can be obtained from the spectrum , its brightness and well-understood theoretical star models (see also Hertzsprung-Russell diagram ). For a given star radius, the probability is inversely proportional to the major semi-axis of the planet's orbit. With the help of Kepler's 3rd law and known star mass , which is also known from the star models, the probability can be rewritten ${\displaystyle R_{*}}$ ${\displaystyle M_{*}}$

${\displaystyle p={\frac {R_{*}}{T^{2/3}}}\cdot \left({\frac {4\pi ^{2}}{GM_{*}}}\right)^{1/3}}$,

where it was assumed that is, and describes the period of revolution of the planet. It is therefore to be expected that planets with a shorter orbital period can be observed with a higher probability than those with a long orbital period, which is actually shown in the data. ${\displaystyle M_{p}\ll M_{*}}$ ${\displaystyle T}$

Example calculation for our solar system

For planetary systems with a configuration similar to our solar system (star radius is the sun's radius , the planets are arranged in the same way), the following values ​​would result for the probability of observing a transit:

planet	major semi-axis / AE	probability
Mercury	00.387	1.203%
Venus	00.723	0.644%
earth	01.000	0.465%
Mars	01.520	0.306%
Jupiter	05.200	0.090%
Saturn	09.580	0.049%
Uranus	19.200	0.024%
Neptune	30.000	0.016%

For a planet in the habitable zone around a sun-like star, this results in an observation probability of around 0.5%.

Transit depth

Brightness measurements by the Kepler space telescope at the star Kepler-7 ; Zoom on a single transit. The depth of the transit is shown with , which in this example for the planet Kepler-7b is about 0.7%. ${\displaystyle \Delta F}$

If the transit planet covers its star, the brightness decreases. The relative decrease of the brightness during a transit with respect to the non-reduced brightness of the star is used as a transit depth (engl. Transit depth ), respectively. Assuming that the star emits light like a black body (which provides a good match in a first approximation), an expression for the transit depth can be given using the Stefan-Boltzmann law for the emitted power . The following applies to the area of ​​the covered area , to the total area of ​​the star disk . The Stefan-Boltzmann law can be used for these projection surfaces, although the surface of the star corresponds to that of a sphere, since the surface normal for the projection is also scaled, whereby the angle between the line of sight and the radial vector in the star is to the surface patch. The course of the brightness curve during a transit is independent of the distance to the observed planetary system, which results directly from the ratio of the radiated power during a transit and without taking into account the distance : ${\displaystyle \Delta F}$ ${\displaystyle P=\sigma AT^{4}}$ ${\displaystyle A_{p}=\pi R_{p}^{2}}$ ${\displaystyle A_{*}=\pi R_{*}^{2}}$ ${\displaystyle d{\vec {A}}}$ ${\displaystyle \cos(\phi )}$ ${\displaystyle \phi }$ ${\displaystyle P_{tr}}$ ${\displaystyle P_{0}}$

${\displaystyle {\frac {P_{tr}}{P_{0}}}={\frac {\sigma \left(A_{*}-A_{p}\right)T_{*}^{4}}{\sigma A_{*}T_{*}^{4}}}=1-{\frac {A_{p}}{A_{*}}}}$

The transit depth can therefore be specified as the ratio of the two radii

${\displaystyle \Delta F={\frac {A_{p}}{A_{*}}}={\frac {R_{p}^{2}}{R_{*}^{2}}}}$.

The relative decrease in brightness is therefore in a first approximation directly proportional to the hidden area and inversely proportional to the total area of the geometrically visible star disc. ${\displaystyle A_{p}}$ ${\displaystyle A_{*}}$

For our planetary system the following values ​​would result for the transit depth.

planet	Radius / km	Transit depth
Mercury	02,439	0.0012%
Venus	06,052	0.0076%
earth	06,378	0.0084%
Mars	03,386	0.0024%
Jupiter	69,170	1.01 00 %
Saturn	57,310	0.75 00 %
Uranus	25,270	0.135 0 %
Neptune	24,550	0.127 0 %

Disturbing influences

Edge darkening using the example of the sun, here during a Mercury transit. The edge areas appear a little darker than the central area of ​​the star. Well visible is a sunspot (left) that is even larger than Mercury.

The course of the brightness curve is only as a first approximation a simple decrease in brightness by . Because of the darkening of the edges , the edge areas of a star disk appear darker than the center. In addition, the center of the star disk has a larger share in the blue spectrum than the areas near the edge. This comes from the optical density of the star's atmosphere , so that we see areas with a lower effective temperature at the edge than in the center. Since the radiated power is proportional to the fourth power of the temperature, small changes in temperature are already responsible for large changes in the radiated power, which is why the edge appears darker. If a planet now enters the star disk, it initially covers an area that is darker than the mean resulting from the total brightness. If the planet then continues to the center of the star disk, it covers an area that is brighter than the mean. The transit depth is therefore smaller at the beginning and towards the end of a transit than during the central phase, since at the beginning and towards the end areas with lower radiation power and in the central phase the area with the greatest radiation power is covered. Typical transit curves are therefore slightly convex during their valley phase and thus also confirm the occurrence of edge darkening in other stars. ${\displaystyle \Delta F}$

Since stars, just like the sun, have variable inhomogeneities in their magnetic field, star spots form (see sunspots ). These can reach an enormous size and appear darker, which is why they cause a decrease in brightness that can be in the order of magnitude of a transit. So that a star spot that is also moving across the star disk due to the rotation of the star is not mistakenly interpreted as a transit, the decrease in brightness must be detected several times, periodically and always with the same transit depth. Another indication of a star spot is the usually much slower rotation of the star, so that the decrease in brightness of a spot can last for several days, while a transit takes place in the range of hours.

Eclipsing binary stars that orbit one another in such a way that they do not cause a complete, but grazing mutual covering, cause brightness drops with a transit depth similar to that of a planet. So that observations of these darkenings are not rated as false positive findings, it is necessary to measure the course of the brightness curve precisely. Stars with a grazing eclipse create a V-shape, whereas planets create an extended valley phase. In addition, a shift in the intensity maximum of the emission spectrum during the eclipse can be measured in the case of double stars, the effective surface temperature of which is different, and a planet can thus be safely excluded.

Derivable parameters

With the help of this method some information can be obtained about both the transit planet and its parent star. It is important to determine the course of the brightness as precisely as possible. For most observations on planets in transit on the order of Hot Jupiters , observations from the earth's surface are sufficient. However, in order to be able to measure the course of brightness for planets similar to Earth, it is necessary to carry out the measurements from space in order to escape atmospheric disturbances. This is the main task of space telescopes like Kepler .

Orbital period and major semi-axis of the planet

Without explicit major disturbances , the orbit of a planet remains stable due to the conservation of angular momentum , just like in our solar system. If a transit is observed, another transit takes place after one cycle. The time interval between two transits of the same planet remains constant and corresponds exactly to the period of the planet's orbit around its star. With the help of Kepler's 3rd law and the knowledge of the star mass (which can be determined with the help of star models only from the spectrum and the luminosity of the star), the major semi-axis of the planet can be calculated.

In order to rule out that another object (such as a planemo ) accidentally passes by between the observer and the observed star and causes a decrease in brightness, at least three transits must be observed which took place at the same time interval. In order to detect a transit planet, its star must be observed for at least twice the time that the planet needs to orbit its parent star.

Orbit inclination

Illustration of the brightness curve of a planet transit with different centralities. The dashed line indicates the area covered by the planet.

If the brightness curve is measured precisely, the inclination of the orbit in relation to the line of sight can be determined using the analytical approximation of the brightness curve. The course of the curve depends on whether the planet moves centrally, offset, grazing or not at all over the star disc. Together with the mass of the star and the distance of the planet from it, the orbital inclination can be specified if you know how central the transit is. The distance to the center of the star disk is called the central parameter. The determining parameters can be approximated from the measured data points by using compensation calculation methods . The edges of the transit curve are particularly decisive for determining the inclination of the path.

Planet radius and density

With the help of the transit depth and the star radius known from star models, the planet radius can be determined. The expression derived in the section Transit Depth can be used to find the planet radius: ${\displaystyle R_{p}}$

${\displaystyle R_{p}=R_{*}{\sqrt {\Delta F}}}$

For planets with a sufficiently large mass, the radial velocity method , which can otherwise only provide the minimum mass of an exoplanet, can now be used to determine the mass more precisely because of knowledge of the planet's orbital inclination. With the help of the radius of the planet and its mass, an indication of its average density is possible. This enables statements to be made about the nature of the planet, whether it is a gas planet or a rock planet.

Composition of the atmosphere of the planet

If the planet is in front of its parent star, it is possible to obtain spectroscopic information about the planet's atmosphere. During a transit it not only covers the light of the star, but its atmosphere is also shone through by the light of the star, similar to the Lomonosov effect . As with any gas through which a continuous spectrum shows, absorption lines appear . These can be identified in comparison with spectral measurements of the star outside a transit. The strength of the resulting absorption lines is in the order of magnitude of 0.001 to 0.01% of the Fraunhofer lines detectable in the star spectrum . The low value is due to the fact that the atmosphere shone through by the starlight makes up only about 0.001 to 0.01% of the projected area of ​​the star disk and therefore only a fraction of the light with information about the composition of the atmosphere is available. Most of the rest of the light is unchanged light from the star. The smaller the planet, the more measurements are needed to improve the signal-to-noise ratio and even to generate a usable data set. With the instruments available at the beginning of 2016, it was not yet possible to make statements about the atmosphere of Earth-like planets. With the completion of very bright telescopes such as the European Extremely Large Telescope or the James Webb Space Telescope and the application of new spectroscopic methods, this will be possible in the future.

Albedo and temperature of the planet

Course of brightness of the star HAT-P-7 during one orbit of its planet HAT-P-7b. The periodic change in the brightness curve, which is caused by the phase of the planet, is easy to see. The occultation of the planet by the star causes the second, smaller decrease in brightness.

The transit planet orbiting the star will not only cause a transit with a small eccentricity , i.e. an approximate circular orbit, but will also be covered by the star. Similar to an eclipsing star , an additional small decrease in brightness occurs when the planet is covered. Since the planet does not shine itself, but reflects the light from the star, there is also a modulation of the brightness curve due to the phase of the planet. If he turns his shadowy side towards the observer, which is the case directly before, during, or after a transit, only the brightness of the star reaches the observer. The further it walks around the star on its orbit, the more of its day side is visible to the observer and the brightness that can be measured increases. The larger the planet and the larger its albedo , the greater this effect. As soon as the planet moves behind its star, the brightness decreases by this contribution and an observer only receives the radiation flux of the star alone. This phase will cover (Engl. Eclipse ) because the observed object, is here covered the planet. After the cover, the brightness rises again as soon as the day side of the planet becomes visible again and decreases again in the course of time until the planet again transits in front of the star disc. For orbits with a large eccentricity, it is possible that the transit takes place in the periapsis and causes a transit, but a cover in the apoapsis cannot take place, since the large distance in the point distant from the star for a given orbit inclination is sufficient for a projection outside to lie on the star disc.

Light curves with modulation by the phase of the planet and an additional cover have already been proven for planets in the order of magnitude of Jupiter, see for example HAT-P-7b , and together with the also determinable radius reveal something about its reflectivity , the albedo . Together with the albedo and the distance to the star, which can also be calculated, statements can be made about its surface temperature . If it is also possible to measure the spectrum shortly before or shortly after an eclipse and compare this spectroscopic measurement with that during an eclipse, statements can even be made about the reflection spectrum of the planet. Such measurements for albedo, temperature and reflection spectrum have not yet been carried out for earth-like planets due to insufficient accuracy. This will be possible in the near future with new, powerful telescopes.

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  Scheme for the planetary system around the star HAT-P-7

More options

Direction of rotation of the star

During a transit, the planet moves into the star disk from one side and initially covers part of the peripheral area. It moves on over the central area of ​​the star disk and will emerge again from the star disk on the side opposite the entry point. It is thanks to this fact that we can obtain information about the rotation of the star. The absorption lines of a star are broadened. From the width of the spectral lines in the light of a star, conclusions can be drawn about the tangential velocity and thus the rotation of the star with the help of the Doppler effect . If the planet covers the edge area, the gas of which is moving towards the observer, the portion of the light shifted into the blue decreases. The mean of the line seems to move into the red. While the planet is centrally located in front of the star, this shift disappears. On the opposite side, it covers areas that are shifted into red because they are moving away from the observer. The mean shifts into the blue (see also Rossiter-McLaughlin effect ). This shift of the line mean during a transit enables statements about the minimum rotation speed of the star. It can also be decided whether the star rotates in the same direction as its planet orbits it, or whether the star rotates in the opposite direction. If a shift into the red is detected upon entry and a shift into the blue upon exiting, the star rotates in the same direction as its companion and vice versa.

Investigation of the star's magnetic field

Star spots that are covered by the planet during a transit can be seen in the brightness curve.

Star spots are caused by inhomogeneities in the star's magnetic field when the field lines emerge from the star's surface. These appear darker and, like a transit, reduce the brightness of the star. If the orbit inclination of a transit planet is known, its path across the star disc is known. If there is a star spot on this line, the planet will also cover the spot during its passage. Since the spot has a lower brightness than the surface of the star surrounding it, the total decrease in brightness is smaller when the spot is covered than when the planet and the spot are visible. The brightness curve shows an increase in the presence of a star spot on the path of the planet. If it is a transit planet with a short orbit period in the range of a few days, this increase can be observed several times, since spots on the star take from several days to weeks for one orbit. A rotation of the star and a displacement of the spot on the surface of the star can also be detected if the increase visible in the brightness curve moves on over the course of several transits. If a star has transit planets with a very short orbital period, data about its star spots can be obtained by evaluating as many transits as possible. Conclusions about the magnetic activity of the star can be drawn from the frequency of occurrence and the size of the star spots observed.

Transit timing variation (TTV)

Using transit timing variation it is possible to infer the existence of other planets in this system by observing the transit of one or more planets in a system. The orbits of the transit planets are influenced by orbits. Even if the planet causing the orbital disruption cannot be detected by a transit, these influences can be detected by changing the orbital time and thus the time interval between the transits. Using model calculations, these variations can be traced back to another celestial body in this planetary system. The longer a planetary system is observed, the more precisely one can deduce the additional or additional planets. The possible parameters for the causing planet can also be restricted by longer observations, so that with sufficiently long and precise observation the orbit, phase and mass of the planet can be roughly determined, although it never causes a transit. The method allows even non-transit planets with a mass similar to that of Earth to be detectable.

Trojans

With the transit method, Trojans can be detected in other planetary systems. These clusters of asteroids orbit the star in the same orbit as a planet and are located in Lagrange points L4 and L5. If many orbits of a planet are continuously observed, these brightness curves can be superimposed and statistically averaged. The expected large number of asteroids in the Lagrange points results in a measurable decrease in brightness that cannot be distinguished from noise in a single measurement process, but can be detected when many measurements are superimposed. Due to the transit of the planet accompanied by the Trojans, the times at which the Trojans can be expected in front of the star can be calculated. At these times, it can be observed specifically. Likewise, Trojans from a massive planet can, in the sum of all individual asteroids in this area, reach the total mass of a small planet like Mercury or Mars. In this way, if another planet exists, whose transit can be observed, small orbital disturbances and variations in the orbital time can be detected. These variations in the orbital time allow conclusions to be drawn about Trojans on other planets in the observed planetary system.

Exomonde

If an exoplanet has one or more moons , these moons can also cause a decrease in brightness, which always takes place at the time of the planet's transit. This passage usually has a shallower transit depth than that of the planet, but can be detected for large moons. Because the moon orbits its planet, if several transits are observed, this small additional decrease in brightness will begin some time earlier or later or will fail completely, depending on where the moon is currently in its orbit. If enough such events are observed, its size and period around its planet can be determined from the additional transit depth and from the observation of the respective position of the moon relative to the planet.

In addition, it is theoretically possible to estimate the mass of the moon: Since both bodies revolve around a common center of gravity , the mass ratio of the two bodies can be determined from the slightly different entry times of the planet into the star disk. If the mass of the transit planet is known, the mass of the moon follows. With the help of the transit depth of the moon, its density can also be determined.

See also: extrasolar moon

See also

• Passage

Trivia

• The planet HD 209458 b , discovered in 1999, was the first exoplanet to be detected using this method.
• Kepler-88 b was the first exoplanet in which irregularities in the transit ( transit timing variations ) gave indications of the further exoplanet Kepler-88c .
• Many transit planets discovered by 2018 are located in the constellations Swan and Lyra . The Kepler space telescope observed a section of the sky there and until then had detected most of the exoplanets.
• HD 189733 is a relatively bright star (7,676 mag) whose exoplanet HD 189733 b can also be detected by amateur astronomers.

literature

• Mathias Scholz: Planetology of extrasolar planets. Springer Spectrum, Springer-Verlag Berlin Heidelberg 2014, ISBN 978-3-642-41748-1 , p. 112ff.
• Valerio Bozza, Luigi Mancini, Alessandro Sozzetti: Methods of Detecting Exoplanets: 1st Advanced School on Exoplanetary Science . Springer, 2016, ISBN 978-3-319-27456-0 .

Web links

• Interactive web applet for simulating various transit curves (Adobe Flash)
• Exoplanet-style transit light curve of Venus on YouTube . Time-lapse image of the Venus transit 2012 with a brightness curve
• Are there extrasolar planets? from the alpha-Centauri television series(approx. 15 minutes). First broadcast on Jan 17, 1999 (brief explanation of the method towards the end of the episode)
• Methods of Exoplanet Discovery - The Transit Method. Article and applet for simulating exoplanet transits on beltoforion.de.

Individual evidence

1. ↑ ^{a b} Exoplanet and Candidate Statistics. In: NASA Exoplanet Archive . Retrieved August 4, 2019 . 
2. ↑ Will all the stars Kepler observes have transiting planets? In: Kepler FAQ . ( nasa.gov [accessed July 8, 2016]).
3. ↑ ^{a b c} Jeff Hecht: The truth about exoplanets . In: Nature . tape 503 , February 18, 2016, p. 272-274 , doi : 10.1038 / 530272a .
4. ↑ ^{a b} Frequently Asked Questions from the Public about the Kepler Mission . ( nasa.gov [accessed July 8, 2016]).
5. ↑ ^{a b c d e f g h} Mathias Scholz: Planetology of extrasolar planets . Springer Spectrum, Springer-Verlag Berlin Heidelberg 2014, ISBN 978-3-642-41748-1 , p. 112-173 .
6. ↑ exoplanets.org Histogram of the orbital period of all transit planets (filter: "Transit = 1") can be plotted.
7. ↑ ^{a b} About Transits . In: About Kepler . ( nasa.gov [accessed July 8, 2016]).
8. ↑ Don't the stars vary more than the change caused by a transit? In: Kepler FAQ . ( nasa.gov [accessed July 8, 2016]).
9. ↑ Valerio Bozza, Luigi Mancini, Alessandro Sozzetti: Methods of Detecting Exoplanets: 1st Advanced School on Exoplanetary Science . Springer, 2016, ISBN 978-3-319-27456-0 , pp. 117 .
10. ↑ Do you need several transits to find a planet? In: Kepler FAQ . ( nasa.gov [accessed July 8, 2016]).
11. ↑ ^{a b} Radial Velocity: The First Method that Worked . ( planetary.org [accessed July 8, 2016]).
12. ↑ Atmospheres of exoplanets . ( exoplanets.ch homepage of the Observatoire de Genève at the University of Geneva [accessed on August 4, 2019]).
13. ↑ Brief description of the methodology under the menu item of the transit method. ( Exoplanets.nasa.gov [accessed on July 8, 2016])
14. ↑ Bruce L. Gary: HAT-P-7: AXA Light Curves & Finder Charts & All-Sky Photometry Results ( brucegary.net [accessed July 8, 2016])
15. ↑ WJ Borucki et al .: Kepler's Optical Phase Curve of the Exoplanet HAT-P-7b . In: Science . Vol. 325, August 7, 2009, pp. 709 f ., doi : 10.1126 / science.1178312 ( sciencemag.org ).
16. ^ ^{A b} Jason W. Barnes: Transit Lightcurves of Extrasolar Planets Orbiting Rapidly-Rotating Stars . In: The Astrophysical Journal . tape 705 , no. 1 , 2009, p. 683-692 , doi : 10.1088 / 0004-637X / 705/1/683 , arxiv : 0909.1752 .
17. ↑ Adriana Valio: Starspot detection from planetary transits-observed by CoRoT . In: RevMexAA (Serie de Conferencias) . 2009 ( researchgate.net ).
18. ↑ Jason A. Dittmann, Laird M. Close, Elizabeth M. Green, Mike Fenwick: A Tentative Detection of a Starspot During Consecutive Transits of an Extrasolar Planet from the Ground: No Evidence of a Double Transiting Planet System Around TrES-1 . In: Astrophys. J. tape 701 , 2009, p. 756-763 , doi : 10.1088 / 0004-637X / 701/1/756 , arxiv : 0906.4320 .
19. ↑ James RA Davenport, Leslie Hebb, Suzanne L. Hawley: Using Transiting Planets to Model Starspot Evolution . 2014, arxiv : 1408.5201 .
20. ↑ Jordi Miralda-Escudé: Orbital Perturbations of Transiting Planets: A Possible Method to Measure Stellar Quadrupoles and to Detect Earth-Mass Planets . In: The Astrophysical Journal . tape 564 , no. 2 , 2002, p. 1019-1023 , doi : 10.1086 / 324279 .
21. ^ Matthew J. Holman, Norman W. Murray: The Use of Transit Timing to Detect Extrasolar Planets with Masses as Small as Earth . In: Science . 2005, arxiv : astro-ph / 0412028 .
22. ↑ Markus Janson: A Systematic Search for Trojan Planets in the Kepler data . doi : 10.1088 / 0004-637X / 774/2/156 , arxiv : 1307.7161 .
23. ↑ Michael Hippke, Daniel Angerhausen: A statistical search for a population of Exo-Trojans in the Kepler dataset . In: The Astrophysical Journal Letters . tape 811 , no. 1 , 2015, doi : 10.1088 / 0004-637X / 811/1/1 , arxiv : 1508.00427 .
24. ↑ Eric B. Ford, Matthew J. Holman: Using Transit Timing Observations to Search for Trojans of Transiting Extrasolar Planets . In: The Astrophysical Journal Letters . tape 664 , no. 1 , 2007, p. L51-L54 , doi : 10.1086 / 520579 , arxiv : 0705.0356 .
25. ↑ A. Simon, K. Szatmáry, and Gy. M. Szabó: Determination of the size, mass, and density of "exomoons" from photometric transit timing variations . In: Astronomy & Astrophysics . tape 470 , no. 2 , 2007, p. 727-731 , doi : 10.1051 / 0004-6361: 20066560 .
26. ↑ Kepler's Field Of View In Targeted Star Field nasa.gov
27. ↑ David Schneider: DIY Exoplanet Detector - You don't need a high-powered telescope to spot the signature of an alien world. In: IEEE Spectrum. November 28, 2014, accessed February 18, 2018 .