The Nacreous Oughts

20 May 2014


The star now known as GJ 687, but in older lists of nearby stars BD+68°946 (also AOe 17415-6, Vyssotsky 322, & Ci[ncinnati 18h] 2354), in Draco, appears to have a Neptune-sized planet near the outer edge of its habitable zone. I am calling this one Cendrillon, for no very good reason, except that i've been reading multi-cultural versions of this fairy tale in some of my English tutoring. It is one of the closest stars to Earth, a M3.5 dwarf of either 0.21 or 0.413 times the mass of the Sun. At parallax "0.22084 & visual magnitude 9.15, its luminosity is 0.003907010002. For that spectral type, i would expect BC= -1.97, T= 3200 K, M= 0.32 (which gives a radius of 0.5), whereas the temperature given in the first, exoplanet discovery, source as 3413 K, corresponds to M0, BC= -1.12 & 0.50 times the solar mass (R= 0.3). The visual luminosity corresponds to a typical M1 star BC= -1.35 & 3350 K; its mass would be 0.45. They give the radius as 0.4183, & the best fit i can find is BC= -1.645 T= 3295 K (e.g. type M2.25--note that the B-V given as +1.5 in one of the sources is close to M2 standard) which gives a bolometric luminosity of 0.017776429 & a derived radius 0.413111619. (Not a bad fit for the larger mass.) These i will use.

If the stellar mass is 0.413 & the planet's period is 38.14 days, its semi-major axis works out to 0.165134804. The tidal effect is 15.145, so it is definitely trapped rotation. The blackbody temperature is 251 K, not much cooler than Earth's 279 K; trying some combinations of albedo, 0.19-0.36-0.51, with greenhouse effect 20-40-60, give average surface temperatures of -15, -8, -3. In short, this will be more "habitable" at the East Pole than anywhere near the terminator. There's a catch, though. Its mass is given as 18.394; thus there will be a strong tendency to hold molecular hydrogen at these temperatures (e.g. for any planet density greater than 0.336 times the Earth), & i expect this planet to have an ammonia-based biosphere, if any.

In Our Neighbor Stars (2012), Thomas William Hamilton writes: "The mass is 21% of the Sun's mass, but it has a much higher percentage of elements higher than helium than the Sun does [in Wikipedia i find: Fe/H +0.11; here +0.05]. This is generally taken to mean that the star is significantly younger than the Sun. ...It is a prolific producer of X-rays (also suggestive of youthfulness)..." Solstation has: "close binary? LHS 450" but no separate entry. Old books (ca. 1969) cite a dark companion, either 8-26 times the mass of Jupiter (1977), or 10-60. This is clearly another ghost planet, like those of Barnard's Star & 61 Cygni. Long ago i calculated Ci 2354 as a binary (period 26 years) with very high eccentricity (0.90). A separate, Earthlike planet close in is barely possible, but it does experience some wild gyrations of temperature. I wonder if everyone has forgotten this model?

(image via reneaigner on deviant art)


07 May 2014

A Dissent on Odette 

(via phl dot upr dot edu)

The planet Kepler-186f (in Cygnus, which is why i call it "Odette") is being touted as the latest, most likely Earth-analogue yet discovered. While i like for these exoplanets to capture public attention, albeit ill-informedly (there's already talk of "going there"--as if 3 quadrillion miles were a Sunday drive), i have only just made a few preliminary calculations, without having seen the actual scientific paper.

At 492 light years, it's not in the immediate neighborhood. That makes a parallax of something like ".00662; which with an apparent magnitude 14.625 gives a visual luminosity of 0.028038933 solar. This looks to me like a main sequence star of about K8 spectral type. (We'll come back to that question.) When i plug in K8's 3800 K stellar temperature & the corresponding bolometric correction of -0.91, i get a total luminosity of 0.064827829 solar. However, the radius produced by these numbers is 0.59315683, much different from the 0.472 given by my sources. (They say the temperature is 3788, actually.) For the mass given as 0.478, my models suggest a visual luminosity of only 0.008 solar (with that apparent magnitude, the distance would better be at 263 light years.); that should be type M0.5 with a temperature of 3375 K, BC of -1.235, & derived radius 0.87387727 (total luminosity 0.087555618). I would put the mass at 0.52-0.53 instead.

Let's assume the distance is correct. the visual luminosity suggests type K9 (3600 K) BC -1.01 which gives total luminosity 0.071167878 & stellar radius 0.692457296, which i will provisionally use. We know the planet's rotation period is 129.94598 days. When i calculate the planet's semi-major axis from mass 0.52, i get a= 0.403748549 (they say "0.36-0.41"--good agreement). This works out to a blackbody temperature of 227 K.

Finding out the planet's real temperature from there is a matter of further supposition. If it has the same albedo & greenhouse effect as Earth (A 0.36, G 38 C) the average surface temperature is -32 C. That's between Earth & Mars, as we might expect. Assuming a thicker atmosphere (A 0.4375 G 50, A 0.51 G 60) makes it -26 or -23 C. In other words, we can make it a little warmer, but not a lot.

We know a little more; two things cause a reevaluation, to my mind, of our whole picture of this planet. First, the metallicity (log Fe/H) is given as -0.28, or 52% solar. I have long used a rule of thumb, not based on actual physical processes, but which makes it possible to assign a rocky-planet assumed density per the stellar metals. (A star of lower metallicity might produce less dense planets, all other things being equal.) If i adopt, by my rule, a planetary density of 0.616595001 times Earth's, with the observed radius 1.11 the mass will be only 0.84 & the gravity 0.68. So in effect it's a smaller world... Trying a more realistic A 0.2775, G 30 gives the average temperature of -33 C. This is distinctly "Barsoom-like".

It's being assumed too blithely that the rotation of this planet would be synchronous. But its tidal effect, 3.25, comes between Venus & Mercury, neither of which is captured (--Venus if anything is synched to Earth!); some of my calculations provide a likely rotation period of 1/10th its revolution period e.g. 13 days. So a "day" is a week & a "night" is a week. I estimate, with the lesser atmosphere (the actual amount of water being an extreme unknown factor), the diurnal temperature variation being on the order of 43 C either way, from 10 C to -76 C. In sum, a fairly slow rotator, heavy ice caps north & south, considerable day-to-night temperature extremes, with a big (1.7 solar apparent size) red sun in the sky... Odette.

(via kickass.to)

(Another formula i found gives, irrespective of stellar (& perhaps planetary) metallicity, the radius of a rocky planet is proportional to its mass by the power of 0.26 or 0.27; the derived mass for Odette becomes 1.482620868; then i get a density of 1.0840796 times Earth's, & gravity 1.203328357. Which means the albedo & greenhouse effect, & thus temperature, might be more like the exact earth analogue mentioned previously. Though the diurnal temperature variation would then be less, than the final number i came up with.)

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