The Four Elements: A Quest for Atmosphere (continued)

This post is for those, who will be mod­el­ing their plan­et atmos­pheres in detail. If you have used Star­form (accrete) code to gen­er­ate your sys­tem and evolved the sys­tem with Mer­cury after­wards, using a cli­mate mod­el for your plan­et would be a good choice to con­tin­ue.

Before I will go into details with data prepa­ra­tion and actu­al mod­el­ing, there are mul­ti­ple issues with com­pil­ing some parts of the FORTRAN code, espe­cial­ly the ones con­tain­ing f90 or fIV/f66 fea­tures. For *nix sys­tems ifort and f90 com­pil­ers are the nec­es­sary choice.

Now, I know for sure f77, gfor­tran, g95 or FTN95 won’t be much help. (The lat­ter is Win­dows only; now with this one I’m not exact­ly sure, but I couldn’t make it com­pile with 8 byte-inte­ger AND real options, it allowed me one or anoth­er, and not both; even if it is pos­si­ble, it is NOT OBVIOUS HOW at all.) For Win­dows users issues are resolved with Com­paq Visu­al FORTRAN Pro­fes­sion­al v6.6, for exam­ple, or Intel’s ifort com­pil­er if you have one. I like Com­paq bet­ter because you don’t need to install Visu­al Stu­dio addi­tion­al­ly, and it han­dles pret­ty much every­thing. Please note that ver­sion 6.5 or low­er won’t do because of the integer/real intrin­sic data types sup­port (8 bytes are nec­es­sary).

Once you are tech­ni­cal­ly set, you can move onto the next phase of atmosphere/climate mod­el­ing – data prepa­ra­tion. It is also assumed that you have decid­ed on the atmos­pher­ic com­po­si­tion. I’m not an astro­physi­cist, so every­thing I do below this point is what I have came up so far by guess, tri­al and error method (yes, I’m THAT per­sis­tent) and is not claimed to be flaw­less.

Com­ments, sug­ges­tions and cor­rec­tions are wel­come.

Data prepa­ra­tion

I’ll start with the 1-D cou­pled mod­el and there are sev­er­al things to do before actu­al mod­el­ing.

Stel­lar spec­trum. If you are using BaSeL serv­er, your out­put is the flux moment giv­en in erg/cm^2/s/Hz/sr. To get this in ergs/s/cm^2/Hz you need to mul­ti­ply it by PI to get rid of stera­di­ans (sr). Then, flux Flamb­da, in units ergs/s/cm^2/Angstrom, can be cal­cu­lat­ed using the equa­tion Flambda=0.4*flux_moment*c/Lambda^2, where c=2.997925e17 is the veloc­i­ty of light and Lamb­da is the wave­length in nanome­ters. The numer­i­cal fac­tor of 0.4 (=4*0.1) in equa­tion above comes from the con­ver­sion of the flux moment into flux (*4) and from the con­ver­sion of flux per nanome­ter into flux per Angstrom (*0.1). How­ev­er Flamb­da still is a flux at the sur­face of a star.

The flux at the top of the atmos­phere is used in mod­els; the alti­tude of 100 km is tak­en as „the top”. For Earth, an alti­tude of 120 km marks the bound­ary where atmos­pher­ic effects become notice­able dur­ing space­craft re-entry. Most of the atmos­phere (99.9999 per­cent) is below 100 km, although in the rar­efied region above this there are auro­ras and oth­er atmos­pher­ic effects.

Now, flux, S, from a star drops off with increas­ing dis­tance. In fact, it decreas­es with the square of the radi­al dis­tance, r, from the star, as pro­por­tion­al to 1/r^2. You will need to pro­vide dis­tance in par­secs (see Stel­lar dis­tance d in pc below).

Stellar flux with distance
Stel­lar flux with dis­tance

There is also an option to obtain stel­lar spec­trum by using ATLAS9 (or ATLAS12) and SYNTHE pro­grams by Kurucz and Castel­li (or SPECTRUM by Richard O. Gray, needs Atlas or MARCS stel­lar atmos­phere mod­el any­way). Here you can be more detailed with your star, pro­vid­ing addi­tion­al ini­tial con­di­tions like micro­tur­bu­lent veloc­i­ty* and mix­ing length para­me­ter** into mod­el. Then, there is also Chris Sneden’s MOOG.

*Micro­tubu­lent veloc­i­ty. This is one of the fun­da­men­tal stel­lar prop­er­ties, a stan­dard para­me­ter in one-dimen­sion­al analy­ses of solar-type stars. For ATLAS mod­els the val­ue of micro­tur­bu­lent veloc­i­ty, ξ, can be cho­sen [among 0, 1, 2, 4, etc. km/s] appro­pri­ate for the star’s sur­face grav­i­ty between the two veloc­i­ties that brack­et the micro­tur­bu­lent veloc­i­ty. HOWEVER, the rela­tion giv­en by Kir­by (Eq. 2 of Kir­by et al. 2009, also here): ξ (km/s) = (2.13 ± 0.05) − (0.23 ± 0.03)*log g is only appro­pri­ate for giants, not sub-giants or dwarfs. Of course, for dwarfs there also exists a cor­re­la­tion between the lumi­nos­i­ty class and the sur­face grav­i­ty (log g), the micro­tur­bu­lent veloc­i­ty (t), and the metal­lic­i­ty ([M/H]) (Gray et al, 2000).

Dwarfs (V class) from F to G have the small­est micro­tur­bu­lent veloc­i­ty, thus, tech­ni­cal­ly, the high­er the log g, the low­er the micro­tur­bu­lent veloc­i­ty. For pre­cise results the tech­nique of par­tial cor­re­la­tion can be employed, since the rela­tion­ships between three or more ran­dom vari­ables are being inves­ti­gat­ed. The trends for K stars (at least up to mid-K) are prob­a­bly main­ly sim­i­lar to G dwarfs (Spec­tral stud­ies of K dwarfs, Spec­tro­scop­ic Prop­er­ties of Cool Stars).

**Mix­ing length (ML) para­me­ter. It is the alpha (a = l/Hp) para­me­ter used in stel­lar mod­el, which rep­re­sents the ratio between the mean free path of a con­vec­tive ele­ment (l) and the pres­sure scale height (Hp). The vari­a­tions of this para­me­ter strong­ly affect the struc­ture of the out­er enve­lope (i.e. radius and tem­per­a­ture). In fact, this para­me­ter deter­mines the effi­cien­cy of ener­gy trans­port by con­vec­tion in the out­er­most lay­er of a star: for a giv­en stel­lar lumi­nos­i­ty, it fix­es the radius of the star, hence its tem­per­a­ture and col­or. In case of a real star, evo­lu­tion­ary tracks must then be cal­i­brat­ed by com­par­i­son with stel­lar radii and/or tem­per­a­tures derived from obser­va­tions. The most obvi­ous ML cal­i­bra­tor is the Sun. The ML para­me­ter can be fixed by con­strain­ing the­o­ret­i­cal solar mod­els (i.e. with solar mass, age and chem­i­cal com­po­si­tion) to repro­duce the solar radius. In low-mass stel­lar mod­els the derived stel­lar radii depend on the opac­i­ty which sig­nif­i­cant­ly con­tributes in deter­min­ing the tem­per­a­ture gra­di­ent in their tur­bu­lent exter­nal lay­ers. Hence, stel­lar mod­els, based on dif­fer­ent opac­i­ty tables, could require dif­fer­ent val­ues of alpha. It means that a homo­ge­neous dataset of stel­lar tem­per­a­tures at dif­fer­ent metal­lic­i­ties to prop­er­ly cal­i­brate the ML para­me­ter is urgent­ly need­ed before any fur­ther attempt to use evo­lu­tion­ary mod­els to derive rel­e­vant prop­er­ties of stel­lar pop­u­la­tions (Fer­raro et al. 2006).

Lyman alpha flux at the plan­et (xLy). Lyman-alpha line is the bright­est emis­sion line of neu­tral hydro­gen at the wave­length of 1215.67 A (121.567 nm) in the spec­trum (in some papers 1215.7 A or 1216 A is men­tioned). Stel­lar Lya emis­sion lines are impor­tant spec­tral fea­tures in the con­text of exo­plan­et stel­lar envi­ron­ment and stel­lar physics. The Lya line is used as a proxy for deter­min­ing the tem­per­a­ture and pres­sure pro­files of upper stel­lar atmos­pheres. Lya flux is extreme­ly vari­able with time. To a first approx­i­ma­tion the solar La flux is com­posed of a qui­et and of an active com­po­nent. The Sun’s active com­po­nent changes with the 27 days peri­od (1 rota­tion around its axis); the qui­et one with the 11 year solar cycle. For the Sun, the inte­grat­ed Lya line flux may change by 37% dur­ing one rota­tion and up to 50% over a cou­ple of years.

These flux­es can be obtained from the cat­a­logue; in case of a syn­thet­ic star Lyman-alpha flux is the flux inte­grat­ed over the whole Lya emis­sion line of the syn­thet­ic spec­trum. The eas­i­er way around is to com­pute a syn­thet­ic stel­lar Lya pro­file of a giv­en linewidth, apply­ing a sim­ple lin­ear rescal­ing in wave­length to the solar Lya pro­file. Real stel­lar pro­files may be poor­ly rep­re­sent­ed by such a sim­ple rescal­ing of the solar pro­file.

Math Box 1 – Some inter­est­ing data about your star 

If you made your­self a syn­thet­ic star, some things are not giv­en, but can be obtained through cal­cu­la­tion. Now I was think­ing if I need to include this in one of my posts, but heck, maybe it can be use­ful to some­one.

If you know B-V col­or (from syn­thet­ic spec­trum or from stel­lar data), then use it. Oth­er­wise it can be cal­cu­lat­ed as B-V = (-3.684*LOG(Teff,K))+14.555. You can com­pare this data with the one from BaSeL result, for exam­ple, and make nec­es­sary adjust­ments.

Stel­lar rota­tion (vsin i)

Now, this one is tied to B-V and stel­lar age. I picked up for­mu­las from (Barnes, 2007). Gyrochronol­o­gy per­mits the deriva­tion of ages for solar- and late-type main sequence stars using only their rota­tion peri­ods and col­ors. If you know the age of your star and B-V col­or, you can go the oth­er way around.

f(B-V) =0.7725*(((B-V)-0.4)^0.601), where B-V is the col­or;

g(t) = T^0.5119, where T is star age in Myr;

Rota­tion­al peri­od P, days, is found as P = g(t)*f(B-V);

And final­ly, stel­lar equa­to­r­i­al rota­tion­al veloc­i­ty, vsi­ni, km/s, is found from the rota­tion­al peri­od:

vsi­ni = ((2*PI()*(R*2))/P)*(1/86400), where R is stel­lar radius and P is rota­tion­al peri­od in days.

Stel­lar activ­i­ty cycle

Approx­i­mate stel­lar cycle P, yrs, can be cal­cu­lat­ed as P = -1.22+(14.14*(B-V)). This is how long it takes your star to go from the min­i­mum to max­i­mum stel­lar activ­i­ty. The com­plete cycle is, obvi­ous­ly, dou­ble of that.

Stel­lar dis­tance d in pc. For your syn­thet­ic star you can take one of the pro­vid­ed dis­tances (and man­u­al­ly scale the spec­trum) for com­par­i­son with the VPL spec­tral data (I used 3.2 par­secs). If you used a real star, the real dis­tance from Earth must be pro­vid­ed. It is need­ed for con­ver­sion to val­ues expect­ed for an Earth-like plan­et (see The Afac cor­rec­tion para­me­ter). The flux will be scaled accord­ing to this dis­tance.

The Afac cor­rec­tion para­me­ter. To con­vert to val­ues expect­ed for an Earth-like plan­et, the mea­sured UV flux­es were mul­ti­plied by ((206265*d )^2)*(Lsun/Lstar)*Afac, here d is the dis­tance in par­secs (1 pc = 206264.806 =206205 AU), Lstar and Lsun are the respec­tive bolo­met­ric lumi­nosi­ties of the star and the Sun, and Afac is a cor­rec­tion fac­tor that accounts for the change in the planet’s albe­do with the wave­length of the inci­dent radi­a­tion. Val­ues of Afac of 1.11 for the F2V star and 0.95 for the K2V star were obtained by scal­ing the “water loss” Seff lim­its in Hab­it­able Zones around Main Sequence Stars”, Kast­ing, J.F., Whit­mire, D.P. & Reynolds, R.T. Icarus 101, 108–128 (1993), Table III, by the effec­tive radi­at­ing tem­per­a­ture of the stars, using a qua­drat­ic fit to the list­ed tem­per­a­tures. This pro­ce­dure ensures that hypo­thet­i­cal plan­ets would have the same sur­face tem­per­a­ture as the Earth (~288 K) if oth­er cli­mat­ic fac­tors (e.g. cloudi­ness and green­house gas con­cen­tra­tions) are the same.

I went the oth­er way around scal­ing giv­en Afac num­bers (x) to stel­lar tem­per­a­tures (y):

x y notes
0.9 3450 (M3.5V, AD Leo)
0.95 5084 (K2V, ε Eri­dani)
1 5778 (G2V, Sun)
1.11 6930 (F2V, σ Boo)

Using qua­drat­ic regres­sion I got coef­fi­cients for equa­tion y=ax^2+bx+c:

a = -70906.1520; b = 158654.0223; c = -86698.5002

For any giv­en tem­per­a­ture (K) with­in the range of [3450; 6930] you can solve the equa­tion to find the roots:

x1 = (-b+SQRT((b^2)-4ac))/2a; x2 = (-b-SQRT((b^2)-4ac))/2a

Use the root that fits the range of Afac [0,9; 1,11] — in all cas­es here it will be x1.

Height of the tropopause. In planet.dat file you will need to pro­vide the height of the tropopause. It is found to be strong­ly sen­si­tive to the tem­per­a­ture at the planet’s sur­face through changes in the mois­ture dis­tri­b­u­tion and its result­ing radia­tive effects. The tropopause height is less sen­si­tive to changes in the ozone dis­tri­b­u­tion and hard­ly sen­si­tive at all to mod­er­ate changes in the planet’s rota­tion rate (Thuburn & Craig, 1996).

Cou­pled pho­to­chem­i­cal and radiative/convective atmos­phere mod­el. When your data is ready, you can final­ly make exe­cuta­bles and run your mod­els. Make sure that every­thing in the files is set as you want it. And if you are run­ning a cou­pled mod­el, make sure your folder/file tree is put like it should be, and the cou­ple switch­es in the code are on. Then, com­pile and run.

Some more alter­na­tives

There is a very inter­est­ing mod­el suite called Most for atmos­phere and cli­mate. It includes Plan­et Sim­u­la­tor, PUMA (The Portable Uni­ver­si­ty Mod­el of the Atmos­phere) and SAM (The Shal­low Atmos­phere Mod­el) along with the Graph­i­cal User Inter­face, the Mod­el Starter (MoSt), the post­proces­sor Burn7 and all man­u­als.

NOTE: There is no cake for Cyg­win users here. You can get Most suite run­ning under Cyg­win (make sure X11 is prop­er­ly set up), but the GUI is remark­ably (read: awful­ly) slow­er than on Lin­ux. Dual boot (Win/Linux) or “vir­tu­al machine” is the sal­va­tion.

Anoth­er mod­el I’m going to men­tion here is called EPIC as in Explic­it Plan­e­tary Isen­trop­ic-Coor­di­nate gen­er­al cir­cu­la­tion atmos­pher­ic mod­el. This mod­el is imple­ment­ed in prin­ci­ple for all known atmos­pheres (one for all!); for ter­res­tri­al-plan­et appli­ca­tions the EPICwi­ki sug­gests using ver­sion 4.x. (I would love to try this one once I have free time.)

# Have fun digest­ing and rejoice — there will be no more mod­el­ing for now (unless I’ll run into a prop­er plan­e­tary inte­ri­or mod­el. Yum.)

## Also, this is prob­a­bly the last post this year but if you have ques­tions or want to dis­cuss some­thing, feel free to drop a line. HAPPY WINTER HOLIDAYS and come back in Jan­u­ary for more! 😉

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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