Developing a Workflow – Part III: Orbital Positions

I’m going to base my statistical distribution of orbital intervals on planet and moon data from Evolution of the Solar System by Hannes Afvén and Gustaf Arrhenius(alternate link). Unlike my previous effort in this vein, instead of basing my distribution on a list of values of Ratio of semimajor axes:qn = an+1 / an, directly, I will determine the mean and standard deviation of log(qn - 1). The mean is equal to -0.746371, and the standard deviation is equal to 1.20553. Using random numbers, X, based on this distribution, I can create a table of intervals qn, by observing that each qn = 1 + exp(X).

Looking at a chart of generated values versus the values from Afvén and Arrhenius, I see that the long-tail values are a bit more common than they really should be, but it’s good enough for me. Sometime in the (near?) future, I’ll publish a cleaned-up pdf of the work I did in Mathematica, which may explain my work better.

Now, the usual method would be to select an innermost orbit radius and, in sequence, multiply by the generated values for qn to determine each successive orbital radius. Since I want to assure that Yaccatrice is somewhere in the habitable region, I’m going to modify this a bit.

First thing is, I want to approximate a desirable planetary temperature for Yaccatrice. Now, the final temperature will take into account the different albedo and greenhouse effect of the planet based on information I do not yet possess. If I wanted an exact temperature, I’d have to change up the order of operations here a bit to determine the planetary and atmospheric attributes responsible for calculating albedo and greenhouse effect early in the sequence and work from that. As it is, I just want to come up with a good approximate set of conditions, so for now I’ll just assume Yaccatrice is, other than temperature and its star, a perfect twin for Earth. I don’t really have too much of an Idea for the temperature of Yaccatrice, so I randomly select something between -10° C and 40° C. That came out to 20°C, or 293 K. Given that, and assuming that Earth’s mean surface temperature is 15° C or 288 K, then the (approximate) temperature for Yaccatrice can be calculated as T4 = (288 K)4*L / D2 or D2 = (288 K / T)4*L, where L is in units of Solar luminosity, D is in astronomical units and T is in Kelvins. The semimajor axis for Yaccatrice then comes out to 0.1151 a.u. or 17,223,000 km. For later reference, Cintila will appear to be about 2.5 times as big in the sky of Yaccatrice as Sol appears from Earth. Tidal forces on Yaccatrice from Cintila will be about 197 times what Earth experiences from the Sun. I was considering having Yaccatrice face-locked as a moon to a giant planet to avoid face-locking to its star, but it feels like this planet would be torn to shreds in the tug-of-war between giant planet and star. I think I’ll go for an orbit-rotation resonance instead. Oh well. That wasn’t a primary parameter anyway. The tidal force also makes it unlikely that a planet of anything like Earthlike size could hold onto a satellite for geological time periods, so no moons either. Thus, no months to add interest to our calendar. Bugger!

I suspect there aren’t any planets closer to Cintila than Yaccatrice, but let’s try for an inner planet. For an inner planet, we divide the known orbit by the ratio of orbital radii. So, in this case, the inner orbit will be 0.1151 divided by 1.96, or 0.059 a.u. From my previous post on this subject, and GURPS Traveller First-In, any orbit within the greater of

R = 0.2 x Mstar/MSol or

R = 0.0088 x Sqrt(Lstar/LSol)

will be considered untenable. For Cintila these radii will be 0.06 a.u. and 0.001048 a.u. As it turns out, an inner planet was not entirely impossible nor even terribly unlikely. Any ratio less than 1.92 would have been sufficient. If I understand the cumulative distribution function correctly, that’s close to a 71% probability with this distribution.

Running this thing through Mathematica, I get the following set of orbital radii with periods determined as described in An Explosion of Equations(Orbital Edition): (P/PEarth)^2 = (a/aEarth)^3 / (m1/mSol)

1 – Yaccatrice – 0.1151 a.u. (Orbital period: 0.0713 years = 26.0 days)

2 — 0.2034 a.u. (Orbital period: 0.167 years = 61.2 days)

3 — 0.2260 a.u. (Orbital period: 0.196 years = 71.6 days)*

4 — 0.2611 a.u. (Orbital period: 0.244 years = 89.0 days)*

5 — 0.5098 a.u. (Orbital period: 0.655 years = 243 days)

6 — 0.8364 a.u. (Orbital period: 1.40 years = 510 days)#*530

7 — 1.363 a.u. (Orbital period: 2.91 years = 1060 days)g

8 — 1.574 a.u. (Orbital period: 3.61 years = 1320 days)*

9 — 1.815 a.u. (Orbital period: 4.46 years = 1630 days)g

10 — 2.053 a.u. (Orbital period: 5.37 years = 1960 days)*

11 — 2.097 a.u. (Orbital period: 5.54 years = 2030 days)*

12 — 6.314 a.u. (Orbital period: 29.0 years = 10600 days)

13 — 7.287 a.u. (Orbital period: 35.9 years = 13100 days)*

14 — 20.93 a.u. (Orbital period: 175.0 years = 63900 days)g

Honestly, these things seem awfully packed both in terms of distance and in terms of orbital period. First step to thin things out is to remove resonances with Yaccatrice(which we know, by definition, survived the crapshoot of history). Resonances are a complicated issue. To simplify, I’m going to start by eliminating all planets whose orbital period is less than 1.5 times the orbital period of the last remaining planet. That eliminates #3, #4, #8, #10, #11 and #13.

Now I set a snow line(the distance at which volatiles are likely to be a major component in planetary accretion, a good distance beyond which to form gas giants. I base this on the formula from First-In, which is the same, I think, as the formula used by the Accrete/Starform program, five times the square root of stellar luminosity in solar units. For Cintilla, this comes to 0.596 a.u. Now I’m going to try to determine which planets are gas giants. For simplicity, I’ll figure a 5% chance for planets inside the snow line, 90% for the first planet beyond the snow line and each additional planet till one comes up as a gas giant, 80% for all other planets between the snow line and twice the snow line, and 60% for all planets orbiting at more than twice the snow line. The probabilities are arbitrary and probably bollux, but the results should look pretty good and this is a good sign of something I need to do a bit of research on. From that #7, #9 and #14 became gas giants. As per First-In, I could eliminate all planets beyond a distance of 40 a.u. times the stars mass in solar units, which would eliminate #14, but I’m not entirely sure I want to use that ultimately, so I’m going to ignore it here. Determination of the outermost planet was based entirely upon my own caprice.

Regardless of anything else, I’m going to place the largest gas giant in the system as the innermost gas giant. That is what I am still referring to as #7 although two of the planets inside its orbit have been eliminated already. Any planet with an orbital period close to 3/4, 2/3, 1/2, 1/3 or 1/4 of #7s orbit will be eliminated. For purposes of this article, I will define “close” subjectively. I decide that 1/2 of the period of #7, which is 530 days, is close enough to #6s orbital period of 510 days to eliminate #6. All the other planets survive this last culling.

The final system then is as follows:

1 – Yaccatrice – 0.1151 a.u. – Orbital period: 0.0713 years = 26.0 days. Although, Yaccatrice is not itself a gas giant, I’m considering making it a moon of a gas giant. Still thinking on that.

2 – Niteliting – 0.2034 a.u. – Orbital period: 0.167 years = 61.2 days

3 – Stoppen – 0.5098 a.u. – Orbital period: 0.655 years = 243 days

4 – Viggun – 1.363 a.u. – Orbital period: 2.91 years = 1060 days – Gas giant

5 – Minnimun – 1.815 a.u. – Orbital period: 4.46 years = 1630 days – Gas giant

6 – Aterice- 6.314 a.u. – Orbital period: 29.0 years = 10600 days

7 – Farkeld- 20.93 a.u. – Orbital period: 175.0 years = 63900 days – Gas giant

I’ve taken note of the lost historical reasons for these names on the off chance that they might be useful elsewhere. As it is, some of the names set bounds on the attributes I set for the worlds later.

In terms of developing a world generation system, I have run into a bit of a problem at this stage. While I am happy with a system that can generate wildly variable intervals between planetary orbits, I feel that this one generates low values too frequently. My attempt to compensate by removing some of the potential “resonances” is, I think, both too cumbersome and insufficiently realistic for general use. For a start I need a better distribution than an exponentiated normal distribution. I also think I might do something similar to GURPS Space 4th ed, by placing gas giants and removing orbits that are within a minimum ratio of the gas giant orbits.

I’m already finding fundamental inadequacies to my original workflow. The originally planned next step was a total mess. “Determine the Physical Parameters of the Planets,” is just to much of a gigantic catch-all cop-out of a step. In the case of Yaccatrice, since I’ve decided to make it the satellite of a gas giant planet, I need to determine the nature of this gas giant and Yaccatrice’s orbit of it. That could be a two step sidebar right there… Then I have to figure out what effect this has on the nature of Yaccatrice itself. That’s what I’ll be doing on the next installment in Developing a Workflow.

Thank you for your attention,

The Astrographer

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