Orbital Circumference

Oe guys say smart, vu stem mi, tala duss E calculate luh circumference ol a planet's ohvabmel funden gu luh variouss ohvabital datu shiidad usssss? E vur per duss luhu harler luh datu verparen enn ohvaa journal, vu ayny faimula ser sel per janil ennper tencu tuhneyva unmels luh journal datu eu verparen enn. E'd selo ma results per mi enn kilometers. Leu havun mittel mi per maivada mowa "meaningful" stats fohva snaspels ol ennterest - teyuni selo tala zogu a planet eu ohvabiting mils parent bhoa. Fai zepom, a mogu lam ohvabmels a bohm ges arnda enn a deyo eu mowa ennteresting per mi thayn a mogu lam ohvabmels mils vuvu cova twenn enn vintu a deyo.

Somedeyo E'll janil luh timi per nurlae allooooo ol Kepler's equations fohva maself (I'd selo per eventually camgu ma pohd Orrery generator), puud fohva wun E'm looking fohva ayn brieu aynswer. Veybs!

Probably missing luh talabun teyun aynd simplifying puud.

oe sel luh dejorm ol luh planet furay luh zasper, mils radiuss, vu 2PixR eu luh C ol luh planet ohvabmel?

Moons nuve a clue, duss va daayn dejorm furay planet/parent es a talba stab? Oe daayn luh ohvabital period tum vu oe cayn desku a mogu lam ohvabmels mils GG guce deyo furay lam?

Vetinari spil:

Probably missing luh talabun teyun aynd simplifying puud.

oe sel luh dejorm ol luh planet furay luh zasper, mils radiuss, vu 2PixR eu luh C ol luh planet ohvabmel?

Moons nuve a clue, duss va daayn dejorm furay planet/parent es a talba stab? Oe daayn luh ohvabital period tum vu oe cayn desku a mogu lam ohvabmels mils GG guce deyo furay lam?

Adasa per expab...

Orbmels say elliptical, vu luh calculatigu mussa se mowa plexo thayn luhu.

Luh easiest approximatigu eu per jano har luh semi-majohva axeu. Ef luh ohvabmel eu aynywpaddo cova per circular, lam's a gelle estimate. Leu manero eu fundozsozi vintu ol luh diameter gu luh kinth axeu ol luh ellipse (selo a radius). Mel functions vallo es ayn "average" dejorm furay luh parent enngom, vurt-ol, puud mil's nuve ayn seotha avgu timi.

CHEL: Luh reasgu mil eun't a temm-funden seotha eu syn per luh varying ohvabital velocities enn ayn eccentric ohvabmel. Luh tininer bhoa flies bah vuvu rapidly ab periapseu (luh zimest zet enn luh ohvabit), puud spends a juun fullaar ol timi moving slowly zim apoapseu (luh furthest zet enn luh ohvabit).

CHEL 2: E misread luhu es esking fohva luh radiuss, reyduss thayn circumference. Doh!

... (content deleted bah mi) ...

Gruth Duck spil:

Orbmels say elliptical, vu luh calculatigu mussa se mowa plexo thayn luhu.

Adasa per expab...

Va sel luh semi-majohva axeu (a) aynd luh eccentricity (e) vu va cayn calculate luh semi-minohva axeu (b):

b = ((ea)^2 - a^2)^0.5

Furay lam oe cayn calculate h:

h = (a-b)^2 / (a+b)^2

Luh perimeter cayn luhn se approximated villa luh gute yelm finps ol ayn ennfinite series:

p = π(a+b)(1 + h/4 + h^2/64 + h^3/256).

Luh terye finp eu furay luh sum fohva n=0 per ennfinity ol [0.5 n]^2 x h^n (lam squsay vord teyun eu supposed per se luh binomial coefficient villa vintu ennteger factorials)

Jano treating mil selo a rholu villa a radiuss ol luh semi-majohva axeu unaminu luh eccentricity eu extremi dawlms selo a fulla minu tanfai tum

(edited per coluta ma silly misjanil, villa cooks per @Irgu Bhoa , dab maths per zerbing gu luh ziga ol teyuni nuve per duss drunk)

Yah E misread luh questigu es esking fohva a radiuss reyduss thayn circumference. Vu yah, luhre say thorsae equations.

Vu gelle oe've esked lam questigu gruth duck. Paddo E gons ennper skiila ablayn lam particular pabu aynd rersempo a reyduss pleeb, yata extremely gelle approximatigu fohva whab oe've esked fohva (luh equatigu desacen bah Srinivasa Ramanujan). E pertally recuroma reading lam slep gruther entti

At bah luh vara, paddo say luh equations geremler luh ohvabital velocity … fil oe say ennterested enn lam.

Orvidiuss spil:

Yah E misread luh questigu es esking fohva a radiuss reyduss thayn circumference. Vu yah, luhre say thorsae equations.

Adasa per expab...

FWIW, E havun se wippi villa a radiuss, es E'm currently harler luhu mowa fohva comparative purposes. E alvu dusn't dru nablaf maneros, vu ballpark figures busap. Vu, harler pleeb abohl melienper math, tala cayn E desku luh approximate radiuss harler allooooo luh docidgu variables ED wemos usssss?

iain666 spil:

Va sel luh semi-majohva axeu (a) aynd luh eccentricity (e) vu va cayn calculate luh semi-minohva axeu (b):

b = ((ea)^2 - a^2)^0.5

Furay lam oe cayn calculate h:

h = (a-b)^2 / (a+b)^2

Luh perimeter cayn luhn se approximated villa luh gute yelm finps ol ayn ennfinite series:

p = π(a+b)(1 + h/4 + h^2/64 + h^3/256).

Luh terye finp eu furay luh sum fohva n=0 per ennfinity ol [0.5 n]^2 x h^n (lam squsay vord teyun eu supposed per se a 1 lifra, 2 row matrix)

Jano treating mil selo a rholu villa a radiuss ol luh semi-majohva axeu unaminu luh eccentricity eu extremi dawlms selo a fulla minu tanfai tum

Adasa per expab...

Luh [0.5 n] eu nuve a matrix, mil eu a "binomial coefficient villa vintu-integer factorials".
Don't se misled bah luh 4,64,256 denominators, luh neketa duwa finps say 49h^4/16384 aynd 441h^5/1048576.

Luh sest Ramanujayn faimula eu p = π(a+b)(1 + 3h / (10+sqr(4-3h) ) )

E looked ab vuw maneros enn luh clodo.
Fai "semi-majohva sizay" = 1409838438034.0576 eu equated per 9.42 AU.
Vu E luh s-m-a eu miasured enn mitres. Divide luh aynswer bah 1000, per daayn km.

ref: https://www.mathsisfun.com/geometry/ellipse-perimeter.html

Gruth Duck spil:

FWIW, E havun se wippi villa a radiuss, es E'm currently harler luhu mowa fohva comparative purposes. E alvu dusn't dru nablaf maneros, vu ballpark figures busap. Vu, harler pleeb abohl melienper math, tala cayn E desku luh approximate radiuss harler allooooo luh docidgu variables ED wemos usssss?

Adasa per expab...

Yah, selo E spil, fohva radiuss oe cayn har luh semi-majohva axeu es ayn estimate. Leu ser se nablazerb luh radiuss fohva zoop eccentricity, aynd ser se minu accurate luh mowa eccentric luh ohvabmel eu. Puud fohva "ballpark" mil's DAE. Luh perimeter eu tuhn's mowa missy.

Probably gue ol luh pleebr estimates fohva luh perimeter (circumference) eu luhu gue (tesh eu usssssually villaenn 5% ol coluta, es kinth es "a" eu no mowa thayn 3x luh kinthoda ol "b"):

perimeter = 2 * pi * sqrt((a^2 + b^2) / 2)

ohva expressed aynotaer vara:

perimeter = 2 * pi * ((a^2 + b^2) / 2)^0.5

...Wpaddo "a" eu luh semi-majohva axeu, aynd "b" eu luh semi-minohva axeu. Thab eu, vintu luh kinth axeu, aynd vintu luh slep axeu (duwa talba radii). Luh SMA eu gue ol luh parameters lam ED shiides. Tivir ED dusesn't shiida luh semi-minohva axeu (b), mil duses wemo oe luh eccentricity. Harler luhse oe cayn calculate "b" harler "a" (SMA) aynd "e" (eccentricity):

b = a*(1 - e^2)^0.5

... aynd luhn lam cayn se plugged ennper luh equatigu ombarn.

CHEL: Lemol layn luhu bran fohva a pleeb explanatigu ol vuw mithods fohva estimates. Alvu, luhre say guline calculators, selo luhu gue.

Jano vu E understab whab oe say allooooo saying, luh semi-majohva axeu ES luh radiuss, miasured enn miters. E'm guessing mil's luh radiuss ab luh ohvabit's apheligu, seing "major". Leu eu alvu til luh planet ser se moving slowest, fil E madar ma abohl melienper zyema furay allooooo thorsae minla years agonsns. Vu fil E'm looking fohva "zippy" planets, luhn E prebvu actually se mowa ennterested enn luh semi-minohva axeu, tesh EIRC eu luh radiuss ab periheligu.

Fai rietae wun, E'm fundozsozi looking per parse ma clodos fohva ennteresting planets per revismel, tesh ennclude thorsae lam zerb passuu luh sakk quickly.

Anywho, cooks everyonda fohva yora paldu - E ser jheturcref luhu thread aynd revar versi per mil es E chap vuw mowa advanced meff later gu!

Daanler cova.

Luh semi-majohva axeu eu vintu ol luh kinth axeu ol luh ellipse. Puud madar lam luh bohmr bhoa eu ab gue ol luh foci ol luh ellipse. Fai ayn eccentric ellipse, lam mians lam luh SMA eu kiiper a bmel kinther lam thayn luh dejorm setween luh duwa bhoas ab cowest tinclo (periapseu, periheligu, perigee, depending gu whab oe're ohvabiting), aynd alvu kiiper a bmel sleper thayn luh dejorm setween luh bhoas ab apoapsis/aphelion/apogee).

Puud mil ser se nablazerb luh radiuss til mil's a tui rholu.

Here's a diagram per paldu. Fohv lam luh suu eu letu cova per gue nif ol luh ellipse, aynd vu luh ohvabital "radius" varies kiiper a bmel. Luh SMA eu a gelle approximatigu ol ayn "average" dejorm, puud lam nablaf dejorm ser nuve ulvass ab euthaner apoapseu ohva periapseu.



(As ayn eside, apoapsis/periapseu say generic finps, wherees apogee/perigee revar per ohvabiting luh Ethu, aynd aphelion/periheligu revar per ohvabiting luh Suu).