Designing a Planetary System: Output

All planets, side view

While pro­cras­ti­nat­ing and enjoy­ing my week­end I decid­ed to go berserk on a full scale sys­tem build­ing spree. This is the sec­ond part of the plan­e­tary sys­tem build­ing series. The first part can be found here. I have updat­ed it and made some cor­rec­tions.

If you decid­ed to make plan­ets of your own after all, I sug­gest read­ing these papers: A Hybrid Sym­plec­tic Inte­gra­tor that Per­mits Close Encoun­ters between Mas­sive Bod­ies, J. E. Cham­bers, 1998; Mak­ing More Ter­res­tri­al Plan­ets, J. E. Cham­bers, 2001; Build­ing the ter­res­tri­al plan­ets: Con­strained accre­tion in the inner Solar Sys­tem, Ray­mond et al. 2009.

Now, let’s get down to our busi­ness.

In the pre­vi­ous post I promised to make a sam­ple sys­tem. Here it is; you can use the fol­low­ing data as you like since I don’t plan to use it any­where in my sto­ries, but give me cred­it (or at least drop a com­ment here).

I have based my sys­tem on exist­ing stars; this bina­ry star sys­tem age esti­ma­tions range from 6 to 8 Gyr (we’ll start some­where at the begin­ning of the main sequence). The dis­tance between the pri­ma­ry and the sec­ondary is short­ened to as close as 55 AU (the orig­i­nal dis­tance is 1061 AU).

The pri­ma­ry has lit­tle or no vari­abil­i­ty and only low emis­sion from its chro­mos­phere.

Some notes on stel­lar evo­lu­tion

You can build a sys­tem the same way I did; you can also use stel­lar mod­el grids (as men­tioned in this post, or the ones here and here, for exam­ple); or you can use this awe­some Stel­lar Evo­lu­tion Sim­u­la­tor to cal­cu­late para­me­ters of your star’s evo­lu­tion. You can also plot your per­son­al H-R dia­gram after the sim­u­la­tion run.

When stars are stud­ied spec­tro­scop­i­cal­ly it is found that most stars are com­posed of around 70% hydro­gen and 28% heli­um by mass, very sim­i­lar to what we see in the Sun. The frac­tion of all oth­er ele­ments is small and varies con­sid­er­ably from 2 or 3 % by mass in Sun-like stars (pop­u­la­tion I) to 0.1 to 0.01 per­cent by mass in stars found in glob­u­lar clus­ters (pop­u­la­tion II). The Sun has about 1.8% heavy ele­ments by mass.

Among the Solar-type stars observed in the Galaxy, many appear to be met­al-rich rel­a­tive to the Sun. The case of plan­ets hosts is par­tic­u­lar­ly inter­est­ing in that respect since they present, on aver­age, an over­metal­lic­i­ty [Fe/H] of 0.2 dex. This metal­lic­i­ty is prob­a­bly orig­i­nal, from the pro­to­stel­lar neb­u­la, but it could also have been increased by accre­tion of hydro­gen-poor mate­r­i­al dur­ing the ear­ly stage of plan­e­tary for­ma­tion.

Once nuclear fusion of hydro­gen becomes the dom­i­nant ener­gy pro­duc­tion process and the excess ener­gy gained from grav­i­ta­tion­al con­trac­tion has been lost, the star lies along a stan­dard main sequence curve on the HR dia­gram. Astronomers will some­times refer to this stage as “zero age main sequence”, or ZAMS. The ZAMS curve can be cal­cu­lat­ed using com­put­er mod­els of stel­lar prop­er­ties at the point when stars begin hydro­gen fusion. From this point, the bright­ness and sur­face tem­per­a­ture of stars typ­i­cal­ly increase with age.

Pre-main sequence to ZAMS stellar evolution (masses 0.1 to 6 solar). Credit: The Formation of Stars. Steven W. Stahler and Francesco Palla, 2005
Pre-main sequence to ZAMS stel­lar evo­lu­tion (mass­es 0.1 to 6 solar). Cred­it: The For­ma­tion of Stars. Steven W. Stahler and Francesco Pal­la, 2005

A star’s posi­tion on the ZAMS depends on both its mass and its ini­tial heli­um abun­dance.

The mass frac­tions in H, He, and all ele­ments heav­ier than He (“met­als”) are labeled by the cap­i­tal­ized let­ters X, Y, and Z, respec­tive­ly. They are relat­ed by: X + Y + Z = 1.

Often (Z/X) ratios are quot­ed, so
X = (1 + Y)/(1 + (Z/X))
Y = 1 — Z — Z/(Z/X)
Z = (1 — Y)/(1 + 1/(Z/X)).

Mazz­itel­li (1989) gives stel­lar evo­lu­tion mod­el with heli­um mass frac­tion depend­ing on metal­lic­i­ty, in which heli­um con­tent Y for stars is giv­en by the empir­i­cal rela­tion Y=0.243+(dY/dZ)*Z and where 0.243 is the pri­mor­dial heli­um mass frac­tion. The rea­son­able val­ue for dY/dZ is con­sid­ered 2.0, though it is a very uncer­tain quan­ti­ty and varies from object to object, e.g. the mod­el with orig­i­nal solar val­ues, Y = 0.267 and Z = 0.0188, would have a dif­fer­ent dY/dZ val­ue (note that X+Y+Z=1 must hold). An increase in Y decreas­es the main-sequence life­time, and an increase in Z increas­es the main-sequence life­time. The hab­it­abil­i­ty for plan­e­tary sys­tems is thus robust under changes in stel­lar mod­el para­me­ters (Jones et al. 2005).

HZs for a 0.9 solar mass star with the original values Y = 0.303, Z=0.0298 following the above-mentioned relation (black lines) and Y = 0.269 (solar value), Z = 0.0317 (increased metallicity) (grey lines). Credit: Jones et al, 2005.
HZs for a 0.9 solar mass star with the orig­i­nal val­ues Y = 0.303, Z=0.0298 fol­low­ing the above-men­tioned rela­tion (black lines) and Y = 0.269 (solar val­ue), Z = 0.0317 (increased metal­lic­i­ty) (grey lines). Cred­it: Jones et al, 2005.

Stars with high­er ini­tial heli­um abun­dances have high­er lumi­nosi­ties and effec­tive tem­per­a­tures. This is pre­dict­ed by homol­o­gy; homol­o­gous stars are built with the assump­tion that star with mass M1 will just be a scaled ver­sion of a star with mass M0, because the physics which deter­mines the struc­ture of main sequence stars does not change rapid­ly with mass. Thus the high­er mean mol­e­c­u­lar weight trans­lates into low­er core pres­sures. Heli­um rich stars there­fore are more con­densed, which mean they have high­er core tem­per­a­tures and larg­er nuclear reac­tion rates.

Changes in metal­lic­i­ty shift the loca­tion of the ZAMS; the met­al-poor main sequence is blue­ward of the solar metal­lic­i­ty main sequence. This is pri­mar­i­ly due to the reduced amount of bound-free absorp­tion through­out the star (which only comes from met­als). The small­er opac­i­ty of met­al poor stars allows the ener­gy to escape more eas­i­ly, and thus increas­es the lumi­nos­i­ty.

Typ­i­cal­ly the por­tion of heavy ele­ments is mea­sured in terms of the iron con­tent of the stel­lar atmos­phere, as iron is a com­mon ele­ment and its absorp­tion lines are rel­a­tive­ly easy to mea­sure. Because the mol­e­c­u­lar clouds where stars form are steadi­ly enriched by heav­ier ele­ments from super­novae explo­sions, a mea­sure­ment of the chem­i­cal com­po­si­tion of a star can be used to infer its age.

My Pri­ma­ry is more enriched in heavy ele­ments than the Sun, with almost two times solar abun­dance of iron; it is there­fore clas­si­fied as a rare “super met­al-rich” (SMR) star. This abun­dance of met­al makes esti­mat­ing the star’s age and mass dif­fi­cult, as evo­lu­tion­ary mod­els are less well defined for such stars.

If you’re into stel­lar astro­physics, you might want to try and write your own code to sim­u­late stel­lar struc­tures and evo­lu­tion. Ref­er­ences: Stel­lar struc­ture mod­el, Stel­lar Mod­el­ing, The ATON 3.1 stel­lar evo­lu­tion­ary code and the ATON 3.1 stel­lar evo­lu­tion­ary code, a ver­sion for aster­o­seis­mol­o­gy.

The mod­el

Data Box 1 — Orig­i­nal stel­lar data

Star A, yel­low dwarf, the pri­ma­ry
Spec­tral class: G8V
Mass: 0.95 solar mass­es
Radius: 1.152 solar radii
Lumi­nos­i­ty (bolo­met­ric): 0.63 solar
Tem­per­a­ture: 5373 K
Appar­ent mag­ni­tude: 5.95
Absolute mag­ni­tude: 5.46
U−B col­or index : 0.65
B−V col­or index: 0.86
Metal­lic­i­ty: [Fe/H] = 0.31 (that makes Z = 0.038, greater than solar)

Star B, red dwarf, the com­pan­ion
Spec­tral class: M3.5–4V
Mass: 0.13 solar mass­es
Radius: 0.30 solar radii
Lumi­nos­i­ty (bolo­met­ric): 0.0076 solar
Tem­per­a­ture: 3134 K (cal­cu­lat­ed)
Appar­ent mag­ni­tude: 13.15
Absolute mag­ni­tude: 12.66
U−B col­or index : 1.66
B−V col­or index: 1.21

Pre­sum­ably the region this sys­tem formed in was dense and met­al-rich. So I decid­ed to go with six plan­ets (and don’t for­get the extra star). Three of them are ter­res­tri­als and the rest are gas and ice giants. Now, before you start argu­ing about var­i­ous pos­si­ble issues with this con­fig­u­ra­tion, the plau­si­bil­i­ty of such sys­tem will be dis­cussed in the upcom­ing Exten­sion post, ded­i­cat­ed to bina­ry and mul­ti­ple star sys­tems.

Data Box 2 – Plan­e­tary sys­tem data

Open/download all as text file.
Down­load archived files of the run and of the star evo­lu­tion tracks.

The life­time of the pri­ma­ry on the main sequence is about at least 11.37 Gyr (the great­est esti­mate was 14.6 Gyr). The com­pan­ion star will have a very long life, at least twice that of the pri­ma­ry. So there is plen­ty of time to live.

A brief cal­cu­la­tion gave me cur­rent water & UV hab­it­able zones for the pri­ma­ry between 0.8753 AU and 1.3574 AU, and the frost line at 2.436 AU. I have mod­eled prob­a­ble evo­lu­tion of the main star with Stel­lar Evo­lu­tion Sim­u­la­tor and here is how the hab­zone might evolve.

Continuous HZ of star A
Con­tin­u­ous HZ of star A

Plan­et E might (or even­tu­al­ly will) be hab­it­able; I have no infor­ma­tion on its atmos­phere and leave it to your imag­i­na­tion. Even if it is tru­ly ‘earth­like’, it will still be noth­ing like we know.

Plan­et F is the local “Jupiter”. It’s HUGE and it rotates faster than our Jupiter does. It prob­a­bly has sim­i­lar atmos­phere, but with even more spec­tac­u­lar bands and local “giant spots”.

This is the chart of the sys­tem at the end of the final inte­gra­tion run.

Full sys­tem, front & side views:

Full system, head on view
Full sys­tem, head on view
All planets, side view
All plan­ets, side view

All plan­ets, front & side views:

Planet system, head-on view
Plan­et sys­tem, head-on view
All planets, side view
All plan­ets, side view

Four inner plan­ets, front & side views:

4 inner planets, head-on view
4 inner plan­ets, head-on view
4 inner planets, side view
4 inner plan­ets, side view

Ter­res­tri­al group, front & side views:

Terrestrials, head-on view
Ter­res­tri­als, head-on view
Terrestrials, side view
Ter­res­tri­als, side view

All ephemerids can serve as a good basis for you fic­tion­al cal­en­dar lat­er. If you are a Celes­tia user, then you can eas­i­ly make all these num­bers into visu­al­ly pret­ty 3D.

Non­ame sys­tem dynam­ics and sta­bil­i­ty

I did 100 000 year inte­gra­tion run to see if the sys­tem is sta­ble at least that long. With the time step of 8 days for mod­er­ate accu­ra­cy it takes around 1 hour to do this on my spare machine (its CPU ticks at 789 MHz). One tenth of a mil­lion years is too lit­tle to see what’s real­ly hap­pen­ing in this sys­tem and how it will evolve. Its com­po­nents bare­ly start­ed to adjust to their would-be reg­u­lar paths. Some of the orbits that might sur­vive for 1–20 Myrs would very prob­a­bly not sur­vive for 1 Gyr.

Here is a bunch of plots of orbital para­me­ters from the final run (see ini­tial data for it in Box 2).

Semi­ma­jor axes from full to detailed view:

Semimajor Axes, 0 - 100 000 years
Semi­ma­jor Axes, 0 — 100 000 years
Semimajor Axes, 0 - 10 000 years
Semi­ma­jor Axes, 0 — 10 000 years
Semimajor Axes closeup, 0 - 3000 years
Semi­ma­jor Axes close­up, 0 — 3000 years
Semimajor Axes, Inner planets C, D and E. 0 — 3000 years
Semi­ma­jor Axes, Inner plan­ets C, D and E. 0 — 3000 years

Incli­na­tions of all major objects in the sys­tem:

Inclination i
Incli­na­tion i

Eccen­tric­i­ties of all sys­tem com­po­nents:

Eccentricity of planet C, 0 - 100000 years
Eccen­tric­i­ty of plan­et C, 0 — 100000 years
Eccentricity of planet D, 0 - 100000 years
Eccen­tric­i­ty of plan­et D, 0 — 100000 years
Eccentricity of planet E, 0 - 100000 years
Eccen­tric­i­ty of plan­et E, 0 — 100000 years
Eccentricity of planet F, 0 - 100000 years
Eccen­tric­i­ty of plan­et F, 0 — 100000 years
Eccentricity of planet G, 0 - 100000 years
Eccen­tric­i­ty of plan­et G, 0 — 100000 years
Eccentricity of planet H, 0 - 100000 years
Eccen­tric­i­ty of plan­et H, 0 — 100000 years
Eccentricity of companion B, 0 - 100000 years
Eccen­tric­i­ty of com­pan­ion B, 0 — 100000 years

At first I have used incli­na­tion of 25 degrees, eccen­tric­i­ty of 0.055 and two dif­fer­ent dis­tances for star B. This had some strong effects on the plan­et C (clos­est to the pri­ma­ry): with com­pan­ion B at a dis­tance of 920 AU plan­et C was eject­ed into space some­where at 54 000 years of inte­gra­tion; with com­pan­ion B at a dis­tance of 220 plan­et C became tilt­ed at 100 degrees by the end of inte­gra­tion. The rest of the sys­tem remained rel­a­tive­ly unaf­fect­ed; e.g. the would-be hab­it­able plan­et E didn’t have any scary shifts in eccen­tric­i­ty or obliq­ui­ty (yet). Plan­e­tary rota­tion rates must be tak­en into account as well: they are less than 20 hours and this may give plan­e­tary axes addi­tion­al sta­bil­i­ty. I didn’t include any moons into the sys­tem.

The “calm belt” is locat­ed between 0.4 and 5 AU as seen from orbital dynam­ics of plan­ets D, E, and F. Plan­et F is the third mas­sive body in the sys­tem after the pri­ma­ry and its com­pan­ion. Plan­et C is strong­ly per­turbed by star A; plan­ets G and H are per­turbed by both stars. If the sys­tem was real, the plan­ets on G and H orbits may have nev­er been formed.

Cur­rent­ly the sys­tem is wide­ly chaot­ic; acci­dents might hap­pen in the future. Plan­et C is prob­a­bly the first can­di­date to fly away. Or maybe not.

What to watch for

1. Eccen­tric­i­ty (cou­pled with the dis­tance from the pri­ma­ry)

Why? The orbital eccen­tric­i­ty of a hab­it­able world might gen­er­al­ly increase under the influ­ence of oth­er sys­tem bod­ies and might rise to the point where the plan­et is car­ried out­side the HZ for a sig­nif­i­cant frac­tion of its orbital peri­od. Whether a plan­et could be hab­it­able in such a case depends on the response time of the atmos­phere-ocean sys­tem; Williams & Pol­lard (2002) con­clude that a plan­et like the Earth prob­a­bly could.

2. Close encoun­ters

The sys­tem bod­ies might per­turb the orbit of the plan­et to the point it will be kicked out of the hab­it­able orbital course or com­plete­ly out of the sys­tem for good. Or even sent on a col­li­sion course with some­thing not quite small (like a star or anoth­er plan­et; heck, even the moon will be enough). I don’t know what is worse in this case: being molten or frozen to death. In case of sim­ply being kicked out of the sys­tem sur­vival might still be pos­si­ble with the help of tech­nol­o­gy, though.

3. Obliq­ui­ty (axi­al tilt)

This is the most inter­est­ing part, since alto­geth­er with orbital para­me­ters it gov­erns inso­la­tion and cli­mate of the plan­et. I will dis­cuss this in detail in the appro­pri­ate post lat­er on.

Here are the obliq­ui­ties of plan­ets in the non­ame sys­tem. The axi­al tilt of the earth­like world expe­ri­ences quite large but not too severe vari­a­tions (yet).

Obliq­ui­ties:

Obliquities, 0 - 100 000 years
Obliq­ui­ties, 0 — 100 000 years
Obliquity of planet E, 0 - 100 000 years
Obliq­ui­ty of plan­et E, 0 — 100 000 years

Oth­er exper­i­ments

Here is also an inter­est­ing sim­u­la­tion of the sce­nario when a plan­et insert­ed between the orbits of Mars and Jupiter. Mul­ti­ple sim­u­la­tions were run, vary­ing the eccen­tric­i­ty, orbital semi-major axis and mass of the insert­ed plan­et, and the result­ing data ana­lyzed with the help of visu­al data plots.

Jeno Marz
JENO MARZ is a science fiction writer from Latvia, Northern Europe, with background in electronics engineering and computer science. She is the author of two serial novels, Falaha’s Journey: A Spacegirl’s Account in Three Movements and Falaha’s Journey into Pleasure. Marz is current at work on a new SF trilogy. All her fiction is aimed at an adult audience.

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