| Astronomical Observations & Research |
Magnitude of an exoplanet orbiting within the habitable zone.[An astronomical paper by A. Ahad] Copyright © 2004 Abdul Ahad. All rights reserved. 
Article posted: 1 November 2004 Abstract Extrasolar planets ("exoplanets") orbiting around their parent stars in the life-supporting habitable zones may some day be directly photographed by using a technique whereby an occulting disk placed inside the telescope would mask out the glare of the star itself, thus enabling the faint planetary body to be imaged. This may prove especially feasible for nearby stars, where the angular separation between the planet and its parent star is greatest. But at what visual magnitude would such an exoplanet ("New Earth") be expected to shine when it has been successfully isolated from its parent star in the manner described? This paper sets out a numerical model for evaluating such possible magnitudes for New Earth, based on it having assumed Earth or Jupiter-like photometric properties, a range of potential sizes and % phase illuminations as the planet revolves about its parent star. Basis of Model The brightness ratio, R, between two objects of magnitudes m1 and m2 is given by: R = 10^0.4*(m2-m1)......................................(1) If the Earth were placed at a standard distance of 1 AU from the observer in its heliocentric orbit and it exhibited a phase of 100% (full disk), the planet would shine with an apparent visual magnitude of -3.86. This quantitity is denoted V(1,0) and is an IAU approved photometric parameter for each planet listed in NASA factsheets [1]. Now, the Sun shines at a visual magnitude of -26.8 as viewed from a standard distance of 1 AU. Hence, by equation (1) above we note that the Earth is overpowered by the Sun by a total brightness factor of some 1,499,684,836 to 1. Through a similar calculation, we find that Jupiter's brightness ratio to the Sun (if it were placed at a standard distance of 1 AU from both the observer AND the Sun and it exhibited a phase of 100%, V(1,0)=-9.40) would be about 9,120,108 to 1 Taking logarithms on both sides and rearranging equation (1) above, we have:- m2 = 5/2 * Log10 R + m1 ......................................(2) In equation (2) above, let m1 be the visual magnitude of the brighter source (i.e the Sun or the parent star) and let m2 be the visual magnitude of the fainter source (i.e. the planet). Imagine if the Sun were viewed from a remote location in space at the distance of nearby stars when it would be totally 'starlike' in appearance and shine at a magnitude of m1. Since we know the brightness ratios, equation (2) above can be used to accurately determine the expected visual magnitude, m2, of either an Earth-like planet or a Jupiter-like planet located in the habitable zone around our Sun, as seen from such a location. Application of Model to Determine Exoplanet Magnitudes If the Earth were hypothetically placed in the habitable zone around another star, then by the default definition of a "habitable zone", it would experience a total light flux exactly equivalent to that which it experiences in our own solar system. Hence, the brightness ratio between that star and the Earth will always be constant, irrespective of the candidate star's own intrinsic or apparent brightness. Likewise, for Jupiter, we would expect an identical brightness ratio to that experienced in our solar system. Hence, this model can be used to predict the apparent visual magnitude of an exoplanet orbiting *any* star within its habitable zone as seen from Earth. Conversely, if the apparent magnitude of such a planet were to be estimated by direct observation, we can deduce an approximation for its size/mass based on the assumed Earth/Jupiter photometric comparisons.
For example if the Earth were placed in the habitable zone around the Sun-like star Alpha Centauri 'A', which shines at a visual apparent magnitude of -0.01, then the magnitude of Earth in the neighbourhood of that star as seen from Earth would be:- m2 = 5/2 * Log10 (1,499,684,836) + (-0.01) = 22.9 That figure of course assumes the disk of the planet is presented at 100% phase. Due to its changing orbital orientation relative to the observer, if the planet were viewed at 50% phase then we can apply equation (1) above to reduce its brightness by an equivalent factor to give an expected fainter magnitude of 23.7 The tables below give a range of projected magnitudes for hypothetical planets of varying sizes orbiting at distances of 1.3 AU and 0.7 AU [2] around the two nearby stars Alpha Centauri 'A' and 'B', respectively (within their individual habitable zones):-
Projected Magnitudes for an Exoplanet Orbiting in the Habitable Zone around Alpha Centauri 'B'
Viewed from Earth, a planet circling around Alpha Centauri 'A' within its habitable zone would have a maximum elongation of 0.986 arc second; one circling around its companion star Alpha Centauri 'B' would have a maximum elongation of 0.561 arc second. Thus both are within easy resolution range of even amateur-sized instruments. The magnitudes listed in the table above for a potentially larger sized object are also comfortably within the grasp of moderate sized telescopes. What makes the detection of exoplanets so difficult is the problem of containing the comparatively vast amount of scattered light coming from the star itself, which drowns out the faint planetary body with an overwhelming amount of 'noise' in the CCD imaging instruments. NASA's Terrestrial Planet Finder [3] mission promises to search for New Earths from an orbital location, where with no atmospheric blurring, the vacuum of space will facilitate vastly improved capabilities for containing the degree of light scatter emanating from the star itself and enable much fainter magnitude detection limits for directly imaging the planets. Some Limitations Worth Noting Since the above model and its results are being projected purely in *visual* magnitude terms based on the light flux from our own Sun (of G-type spectral classification), depending on the spectral classification of the particular parent star concerned, a 'bolometric correction'[4] may be needed as pointed out here . This will be especially significant in the case of hot "O" or cool "M" type stars that are far removed from the mid-range classification of our own Sun on the H-R diagram. The particular wavelengths at which the planet detection instruments are operating will also be highly significant, since many planets "shine" more brightly when viewed in the mid- to high infrared part of the spectrum compared to visual wavelengths. * * * * * End of Paper * * * * * A Tiny Ray of Hope in the Eternal Darkness... Successfully locating an Earth-like planet in the habitable zone around one of the two principal 'suns' of the Alpha Centauri system will surely rank as one of the greatest discoveries in the entire history of science. Such a discovery would indeed be a 'revelation' and far outweigh all the extrasolar planets logged in all the world's scientific journals to date put together! A thousand Jupiters discovered circling in sub-Mercury orbits around red supergiants hundreds of light years away from Earth hardly stirs the imagination... Yet a single discovery of just *one* Earth-sized planet located within the habitable zones on the nearest cosmic shores beyond our solar system will revolutionise our science forever. There is, however, one scary thought in all of this: what if the shores of Alpha Centauri turn out to be empty? How much hope would humanity lose as a result knowing that the next nearest planetary system that's looking even vaguely hospitable to us may be hopelessly out of our reach forever? And even if the nearest shores in the Alpha Centauri system are not empty, what if it turns out that they contain nothing more than gas giants like Jupiter or Saturn sitting inside the habitable zones? Well, fortunately we have not one but *two* habitable zones to look at, so the odds are stacked very much in our favour. It is also perfectly within the realm of possibility that New Earth could in fact turn out to be not a planet at all but instead a large *moon* of one of those gas giants! After all, God does work in mysterious ways... and if He gave us stars in the night sky to serve as anything more than celestial "light shows" merely for our amusement, then we may yet consider ourselves lucky and have hope. 
AA Institute of Space Science & Technology
R E F E R E N C E S [1]NASA Planetary Factsheets [2]Alpha Centauri system analysis [3]Terrestrial Planet Finder Mission [4]Bolometric corrections table by Kaler 1997, p. 263. A Manned Voyage to Alpha Centauri ! Copyright © 2004 by Abdul Ahad. All rights reserved. |
