Jenab2
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The time required for the observation of three successive transits of an exoplanet in front of its primary star is the minimum time necessary for the confirmation of the exoplanet. Three transits, not two, are needed to establish the periodicity of the transits. Given only two dips in brightness, you don't know whether both of them were caused by the same transiting object, or by two different transiting objects.
The distance of an Earth-like exoplanet from its star is proportional to the square root of the ratio of the star's bolometric luminosity to the sun's bolometric luminosity.
R/au = (L/L๏)⁰·⁵
The mass-luminosity relationship at the lower end of the main sequence is, roughly,
L/L๏ = (M/M๏)³·²
So...
R/au = (M/M๏)¹·⁶
The Newtonian form of Kepler's Third Law,
(P/yr)² = (R/au)³/(M/M๏)
Doing the algebra,
P/yr = {(R/au)³/(M/M๏)}⁰·⁵
P/yr = { (M/M๏)⁴·⁸/(M/M๏) }⁰·⁵
P/yr = {(M/M๏)³·⁸}⁰·⁵
P/yr = (M/M๏)¹·⁹
M/M๏ = (P/yr)⁰·⁵²⁶³²
If the interval of time devoted to observation of a patch of sky is 27 days, then the largest period for an exoplanet, for which we might observe three transits within that interval, is 9 days.
If P=9 days, then
P/yr = 0.02464
M/M๏ = 0.1424 ← for luminosity class V, this is a spectral type M6 star.
That's a maximum mass for a star having an Earthlike planet that can be confirmed by TESS.
TESS' mission design does not seem adequate to its intended purpose. For stars of spectral types G and K, TESS will be able to find only hot Jupiters and other planets that are too near the star to be habitable.
Dr. Thomas Barclay was recently quoted in a Deep Astronomy video (posted on YouTube) as saying that TESS was intended to find only planets with 10-day orbits or less, and that it was the intention all along to find habitable planets only around red dwarf stars. I don't think that the public was aware of that. Besides that, we were told repeatedly that TESS was going to look at the "brighter stars." These red dwarfs aren't among the brighter stars.
Later in the same video, we are told that TESS wasn't optimized for exoplanets in the habitable zone. Too right, that! For the brighter stars, TESS just plain won't discover any exoplanets except for the very hot ones. You might think that the discoveries of lots of hot Jupiters can be handed over to the users of the JWST, so that it can find transiting exoplanets that are in bigger-than-9-day orbits, but this is a dubious assumption. Remember that the odds for transit diminish with increasing orbital distance for exoplanets. Just because you can see that very close, very hot exoplanet does not mean that you can also see a more temperate exoplanet orbiting in the same exo-ecliptic plane further out.
Probability of transit = (2/π) arcsin(R/a)
Where (R) is the star's radius and (a) is the planet's orbital radius. Although the relationship isn't quite linear, it is almost so, and, roughly speaking, when you double the exoplanet's orbital radius, you halve the odds for transit.
For exoplanets in 9-day orbits: Star Mass, Star spec, orbit distance (AU), Equilibrium bb Temp (K)
0.100 , M7 , 0.03930 , 237.7
0.200 , M4 , 0.04952 , 335.5
0.300 , M3 , 0.05669 , 344.8
0.400 , M3 , 0.06239 , 374.7
0.500 , M2 , 0.06721 , 452.8
0.600 , K9 , 0.07142 , 534.2
0.700 , K5 , 0.07519 , 633.2
0.800 , K2 , 0.07861 , 740.6
0.900 , G9 , 0.08176 , 846.1
1.000 , G2 , 0.08468 , 956.4
1.100 , G0 , 0.08741 , 1050.9
1.200 , F7 , 0.08998 , 1142.0
What would be ideal is for there to be an improved TESS orbiting the sun at 19.2 AU, probably in the Sun-Uranus L4 and L5 points, where it could devote 2562 days to each of 12 rows of scanning sectors, with each row stretching from the North Celestial Pole to the South Celestial Pole. After 84 years, it would have found every transiting exoplanet orbiting within the habitable zones of every star (having a mass of 1.2 suns or less) for which the S/N ratio was acceptably high, in addition to those hotter exoplanets to which the current TESS is sadly limited. If we didn't want to wait 84 years, we could launch twelve of those improved TESS probes and do the job in seven years.
The distance of an Earth-like exoplanet from its star is proportional to the square root of the ratio of the star's bolometric luminosity to the sun's bolometric luminosity.
R/au = (L/L๏)⁰·⁵
The mass-luminosity relationship at the lower end of the main sequence is, roughly,
L/L๏ = (M/M๏)³·²
So...
R/au = (M/M๏)¹·⁶
The Newtonian form of Kepler's Third Law,
(P/yr)² = (R/au)³/(M/M๏)
Doing the algebra,
P/yr = {(R/au)³/(M/M๏)}⁰·⁵
P/yr = { (M/M๏)⁴·⁸/(M/M๏) }⁰·⁵
P/yr = {(M/M๏)³·⁸}⁰·⁵
P/yr = (M/M๏)¹·⁹
M/M๏ = (P/yr)⁰·⁵²⁶³²
If the interval of time devoted to observation of a patch of sky is 27 days, then the largest period for an exoplanet, for which we might observe three transits within that interval, is 9 days.
If P=9 days, then
P/yr = 0.02464
M/M๏ = 0.1424 ← for luminosity class V, this is a spectral type M6 star.
That's a maximum mass for a star having an Earthlike planet that can be confirmed by TESS.
TESS' mission design does not seem adequate to its intended purpose. For stars of spectral types G and K, TESS will be able to find only hot Jupiters and other planets that are too near the star to be habitable.
Dr. Thomas Barclay was recently quoted in a Deep Astronomy video (posted on YouTube) as saying that TESS was intended to find only planets with 10-day orbits or less, and that it was the intention all along to find habitable planets only around red dwarf stars. I don't think that the public was aware of that. Besides that, we were told repeatedly that TESS was going to look at the "brighter stars." These red dwarfs aren't among the brighter stars.
Later in the same video, we are told that TESS wasn't optimized for exoplanets in the habitable zone. Too right, that! For the brighter stars, TESS just plain won't discover any exoplanets except for the very hot ones. You might think that the discoveries of lots of hot Jupiters can be handed over to the users of the JWST, so that it can find transiting exoplanets that are in bigger-than-9-day orbits, but this is a dubious assumption. Remember that the odds for transit diminish with increasing orbital distance for exoplanets. Just because you can see that very close, very hot exoplanet does not mean that you can also see a more temperate exoplanet orbiting in the same exo-ecliptic plane further out.
Probability of transit = (2/π) arcsin(R/a)
Where (R) is the star's radius and (a) is the planet's orbital radius. Although the relationship isn't quite linear, it is almost so, and, roughly speaking, when you double the exoplanet's orbital radius, you halve the odds for transit.
For exoplanets in 9-day orbits: Star Mass, Star spec, orbit distance (AU), Equilibrium bb Temp (K)
0.100 , M7 , 0.03930 , 237.7
0.200 , M4 , 0.04952 , 335.5
0.300 , M3 , 0.05669 , 344.8
0.400 , M3 , 0.06239 , 374.7
0.500 , M2 , 0.06721 , 452.8
0.600 , K9 , 0.07142 , 534.2
0.700 , K5 , 0.07519 , 633.2
0.800 , K2 , 0.07861 , 740.6
0.900 , G9 , 0.08176 , 846.1
1.000 , G2 , 0.08468 , 956.4
1.100 , G0 , 0.08741 , 1050.9
1.200 , F7 , 0.08998 , 1142.0
What would be ideal is for there to be an improved TESS orbiting the sun at 19.2 AU, probably in the Sun-Uranus L4 and L5 points, where it could devote 2562 days to each of 12 rows of scanning sectors, with each row stretching from the North Celestial Pole to the South Celestial Pole. After 84 years, it would have found every transiting exoplanet orbiting within the habitable zones of every star (having a mass of 1.2 suns or less) for which the S/N ratio was acceptably high, in addition to those hotter exoplanets to which the current TESS is sadly limited. If we didn't want to wait 84 years, we could launch twelve of those improved TESS probes and do the job in seven years.
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