Simulate the transit of extrasolar planets

by @ulaulaman about #exoplanets #planetary_transit #kepler_mission #nasa #astronomy

The search for extrasolar planets (or exoplanets) had its first success in 1991 with the discovery of some planets orbiting around the pulsar PSR1257+12^{(1, 2, 3)}, measuring the variations of the radio pulses coming from the star. The second important milestone in exoplanet research takes place in 1995, with the discovery around the star 51 Pegasi (a star like our Sun) of a Jupiter-like planet, found at a distance closer than Mercury's orbit in our Solar System⁽⁴⁾.

(51 Pegasi via BBC)

These initial discoveries, and many other up to 2009^{(5, 6)} were made using the method of radial velocity or Doppler oscillation, in practice it is assumed that the radial velocity of a star is affected by the presence of a planet orbiting the star itself. In this way, the frequence from the star will be blue when the planet moves in its orbit toward the Earth, tending to red when the planet moves away⁽⁷⁾. With the radial velocity, however, is rather difficult to determine the exact orbit of a planet (or at least something that comes close), and then effectively allows us to determine the period of rotation around the star and the orbital eccentricity (i.e. the deviation by a circle) of the orbit of the planet itself. And I must remember that the method is effective especially for massive planets.
Luckly, the transit method is a more effective method to discover and study exoplanets: it is based on the examination of the light emitted by the studied star and it is considered by Dimitar Sasselov⁽⁸⁾ the method of observation more fruitful: when this light decreases, this means that the star is passing in front of an object. In this way it is possible to determine the radius of a planet and its orbital period. Using essentially the same tools used for the detection of the planet, it is also possible to study the atmosphere of the planet itself, determining its composition, temperature and the presence and formation of clouds.

(comparison between radial velocity and transit⁽⁷⁾)

The Kepler mission is based just on this latter method: launched on March 6, 2009⁽⁵⁾, has found 3845 candidate planets to April 2014, with the first findings published in Science in 2010⁽⁵⁾ and the first Earth-like planet announced a couple of days ago:

Looking the following plots and the plunet-hunting infographic, we can see that astronomers are looking for a kind of light holes, which allow them to determine a bit of data from the planet candidate as its mass, the orbital period and other typically stuff:

(The first transit of Kepler⁽⁵⁾)

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(Transits of the planets of Kepler-11⁽⁹⁾)

The research about Kepler-11⁽⁹⁾ is interesting because, in addition to combining the data for the studied system, it also collects information on a particular planet, Kepler-11g, and also offers deductions, based on the data, about the composition and formation of the planetary system, thus providing, as well as a number of interesting scientific data, a good example of the potential of the mission in general.
The Kepler mission, therefore, was in the last years, a source of interesting news and it can be didactically interesting to use Kepler to start bringing astronomy into the classroom. The advantages of this approach with regard to the teaching of physics are multiple and we can emphasize some of these rather than others based on the order and degree of the school. For example, we can introduce students to the study of the direct data of astronomical experiments, almost all public and freely available, some even in simple formats to read even with the usual text editor. In this way, for example, they get used to the development of real data and to their statistical processing^{(11, 12)}, but it is also possible to create a sort of miniature Kepler mission⁽¹⁰⁾:

In this setup, a hollow translucent plastic sphere (7.5 cm) with a light bulb inside acts as the star and opaque balls with diameters of 4 and 2.75 cm represent two exoplanets of different sizes but of the same mass (but different density). The sphere and one of balls are linked by a horizontally placed solid plastic tube which is in turn attached firmly and perpendicularly to a stiff vertical axis placed at the supposed 'centre of mass' of the two-object system. The 'centre of mass' can be moved along the tube depending on the relative mass of the star to that of the planet. The vertical axis is rotated by a motor leading to the rotation of both objects around the centre of mass. Periodically the ball (planet) crosses the light path from the plastic sphere (star) to the detector made up of eight photodiodes (each a silicon planar photodiode in a standard TO-18 hermetically sealed metal case with a glass lens) arranged horizontally and wired to the datalogger. The detector's voltage rises and drops according to the intensity of light impinging on it. The changing intensity is due mainly to the transiting of the ball (planet) moving past the plastic sphere (star).

(Experimental set-up)

During the installation of the experimental set-up, with the passage of the planet in front of the star and the light collection through the tools, it is useful to start a first discussion on the observations, such as noting the detected star movement detected, or to stimulate a discussion about possible improvements to the experimental apparatus; or possible effects of the actual size of the planet on the star's brightness. Another point that can be the object of the study is to compare the effect of the radius of the planet on the time of the transit.
The proposed experiment is certainly very interesting and it can be a great way to introduce students to the moments of construction, design, implementation and data taking of the experiment, even fascinating as it may be to Kepler, beyond the imperfections detected by same group of researchers in the project:

(...) the short distance between the star and planet does not reflect reality. The planet is unrealistically large compared to the star and our star does not rotate or turn around its axis.

If the proposal of the Thai group allows us to bring in classroom a small portion of the universe, Samuel J. George suggests that we can consider as the next step to download and process the actual experimental data⁽¹¹⁾ by taking example from the Extrasolar Planets Encyclopedia. Extracting data from a specific star for a specific planet is possible to propose a work to derive the properties of that particular planet. For example, with regard to the size of the planet, it first determines the depth of the transit using the formula: \[\Delta F = \frac{B_{\text{max}} - B_{\text{min}}}{B_{\text{max}}}\] Already here there are the first difficulties: first of all we must to determine the beginning and end of the transit, and then to determine the light intensity outside the transit ($B_{\text{max}}$) and inside ($B_{\text{min}}$). This ratio is directly proportional to the ratio between the areas: \[\left(\frac{R_p}{R_s}\right)^2\] and then it is possible to determine the radius of the planet $R_p$ from the radius of the star $R_s$.
Regarding the orbit, first of all we must to assume that this is circular with radius $a$ greater than the radius of the star $R$. At this point we can simply choose if to give to the students the equation to determine the radius $a$: \[\tau = \frac{RP}{\pi a}\] where $\tau$ is the time of transition, $P$ the orbital period⁽¹³⁾.
Finally, it is possible to estimate the mass of the star by applying Kepler's third law: \[P = \frac{2 \pi}{\sqrt{G M_s}} a^{\frac{3}{2}}\]

(Data about the transit of HD209458)

Another way to examine the data from the stars is, as mentioned, the radial velocity based on the Doppler effect⁽¹²⁾. In this case, however, to process and interpret experimental data is necessary to introduce the trigonometric functions, and we must also considere a small eccentricity. The equations used by Lo Presto and McKay⁽¹²⁾, however, seem to reproduce very well the experimental data chosen and they use a development in Fourier series.

In this case, therefore, if the experience can be more useful for a first or second year of university, it can still have its uses even for a fourth or fifth year of european high school, for example, because many of the advanced concepts used during the experience are (or should be) part of the school curriculum in those classes.
The experience with the transit method, however, would, in my view, also be brought to the lower classes, having the advantage of allowing the students to begin to become familiar with real data.
More generally, experiences of this kind are well suited e.g. for multi-disciplinary projects that also involve studies of the biosphere, the atmosphere and planetary chemistry.

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(2) Bisnovatyi-Kogan, G. S. Planetary System around the Pulsar PSR:1257+12 Astronomy and Astrophysics, Vol.275, no. 1, p.161 (1993)
(3) Wolszczan, A. (1994). Confirmation of Earth-Mass Planets Orbiting the Millisecond Pulsar PSR B1257 + 12 Science, 264 (5158), 538-542 DOI: 10.1126/science.264.5158.538 (pdf)

(4) Mayor, M., & Queloz, D. (1995). A Jupiter-mass companion to a solar-type star Nature, 378 (6555), 355-359 DOI: 10.1038/378355a0 (pdf | html)

(5) Borucki, W., Koch, D., Basri, G., Batalha, N., Brown, T., Caldwell, D., Caldwell, J., Christensen-Dalsgaard, J., Cochran, W., DeVore, E., Dunham, E., Dupree, A., Gautier, T., Geary, J., Gilliland, R., Gould, A., Howell, S., Jenkins, J., Kondo, Y., Latham, D., Marcy, G., Meibom, S., Kjeldsen, H., Lissauer, J., Monet, D., Morrison, D., Sasselov, D., Tarter, J., Boss, A., Brownlee, D., Owen, T., Buzasi, D., Charbonneau, D., Doyle, L., Fortney, J., Ford, E., Holman, M., Seager, S., Steffen, J., Welsh, W., Rowe, J., Anderson, H., Buchhave, L., Ciardi, D., Walkowicz, L., Sherry, W., Horch, E., Isaacson, H., Everett, M., Fischer, D., Torres, G., Johnson, J., Endl, M., MacQueen, P., Bryson, S., Dotson, J., Haas, M., Kolodziejczak, J., Van Cleve, J., Chandrasekaran, H., Twicken, J., Quintana, E., Clarke, B., Allen, C., Li, J., Wu, H., Tenenbaum, P., Verner, E., Bruhweiler, F., Barnes, J., & Prsa, A. (2010). Kepler Planet-Detection Mission: Introduction and First Results Science, 327 (5968), 977-980 DOI: 10.1126/science.1185402
(6) Astronomy: Beyond the stars by Eugenie Samuel Reich
(7) Astronomy: Exoplanets on the cheap by Lee Billings
(8) Astronomy: Extrasolar planets by Dimitar Sasselov
(9) Lissauer, J., Fabrycky, D., Ford, E., Borucki, W., Fressin, F., Marcy, G., Orosz, J., Rowe, J., Torres, G., Welsh, W., Batalha, N., Bryson, S., Buchhave, L., Caldwell, D., Carter, J., Charbonneau, D., Christiansen, J., Cochran, W., Desert, J., Dunham, E., Fanelli, M., Fortney, J., Gautier III, T., Geary, J., Gilliland, R., Haas, M., Hall, J., Holman, M., Koch, D., Latham, D., Lopez, E., McCauliff, S., Miller, N., Morehead, R., Quintana, E., Ragozzine, D., Sasselov, D., Short, D., & Steffen, J. (2011). A closely packed system of low-mass, low-density planets transiting Kepler-11 Nature, 470 (7332), 53-58 DOI: 10.1038/nature09760
(10) Choopan, W., Ketpichainarong, W., Laosinchai, P., & Panijpan, B. (2011). A demonstration setup to simulate detection of planets outside the solar system Physics Education, 46 (5), 554-558 DOI: 10.1088/0031-9120/46/5/007
(11) George, S. (2011). Extrasolar planets in the classroom Physics Education, 46 (4), 403-406 DOI: 10.1088/0031-9120/46/4/004
(12) LoPresto, M., & McKay, R. (2005). An introductory physics exercise using real extrasolar planet data Physics Education, 40 (1), 46-50 DOI: 10.1088/0031-9120/40/1/003 (pdf)
(13) The idea is to combine the equation for the period of the orbit \[P = \frac{2 \pi a}{v_c}\] where $v_c$ is the circular velocity of the planet, with \[\tau = \frac{2R}{v_c}\] the expected transit time.