The Radial Velocity Equation almostfinal
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a detailed derivation of
The Radial Velocity Equation
Kelsey I. Clubb
ABSTRACT
Of the over 300 extrasolar planets discovered to date, the vast majority have been
foundusing theRADIALVELOCITYMETHOD (alsoknownasDOPPLERSPECTROSCOPYor the DOPPLER METHOD). The purpose of this paper is to derive the theoreticalequation that is associated with the variation over time of a star’s velocity along an
observer’s line‐of‐sight – a quantity that can be precisely measured with an opticaltelescope–fromfirstprinciples.Inotherwords,wewillbeginwiththeseeminglysimplesituationofaplanetorbitingastar(oftenreferredtoasthe“parentstar”or,theterm
thatwillbeused throughout thispaper, the“hoststar”)and thenstartapplyingbasicprinciplesofphysicsandutilizemathematicaltoolsinordertoanalyzeanddescribethestar‐planet system. The end result will be a complete and detailed derivation of the
RADIALVELOCITYEQUATION.
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2 THERADIALVELOCITYEQUATION
PREFACE “The most incomprehensible thing about the universe is that it is comprehensible”
– Albert Einstein Thediscoveryofextrasolarplanetshasprofoundimplicationsnotonlyforthescientiststhatfindthem,butfortheentirepopulationofEarth.Forthisreason,Ihaveattemptedtomakethisderivationaccessibletothelayperson;however,someknowledgeofadvancedmathematics,physics,andastronomyisneededinordertocompletelyfolloweverystep.Iwouldliketothankmyadviser,DebraFischer,forsuggestingthisderivationasoneofmyfirstintroductionstoextrasolarplanetresearch.Theentireprocesshashelpedmeimmenselytonotonlyunderstandthedelicateintricaciesoftheradialvelocitymethod,butalsotobeabletoseethebiggerpictureofhowresearchmakesuseofallthephysics,astronomy,andmathI’velearned.Thankyou!AsmuchasIwouldlikethisderivationtobewithouterror,nothinginlifeisperfect.Withthatsaid,ifyouhappentocomeacrossanerror,typo,oranythingelsethatneedscorrectionI’dbehappytobenotified.Youcane‐[email protected],comments,orsuggestions.
Kelsey I. Clubb
Department of Physics & Astronomy San Francisco State University San Francisco, CA USA
August 2008
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THERADIALVELOCITYEQUATION 3
TableofContents
TheOrbitsofaPlanetanditsHostStar ....................................................... 4
AnalyzingTheStar‐PlanetSystem ............................................................... 6
TheCenterofMassFrameofReference...................................................... 7
TheLawsofIsaacNewton........................................................................... 8
TheLawsofJohannesKepler .................................................................... 10
AngularMomentum ................................................................................. 14
TheEquationofanEllipse ......................................................................... 15
PropertiesofanEllipse ............................................................................. 18
DefinitionsofOrbitalParameters.............................................................. 20
RadialVelocity .......................................................................................... 22
RadialVelocitySemi‐Amplitude ................................................................ 25
AppendixA:Proofthatδ = 0..................................................................... 29AppendixB:TheExpressionforz.............................................................. 33ReferenceforCalculatingRVSemi‐Amplitude........................................... 34
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4 THERADIALVELOCITYEQUATION
THE ORBITS OF A PLANET AND ITS HOST STAR
Webeginourderivationwiththesimplesituationofaplanetorbitingitshoststar(showninFigure1).Inactuality,boththestarandplanetorbittheirmutualcenterofmass(henceforthabbreviatedCM),butbecausethestarismuchmoremassivethantheplanet,theCMofthestar‐planetsystemismuchclosertothestar.Infact,theCMisonlyslightlydisplacedfromthecenterofthestar.Nowweneedtosetupacoordinatesystem(showninFigure2)andreviewsomebasicsofvectors.
(a)
(b)
NOTTOSCALE
NOTTOSCALE
Figure1–Aplanetorbitingitshoststar.
Thedashedlinerepresentsthepathoftheplanet’sorbit.
(a)Topviewofthestar‐planetsystem.(b)Sideviewofthestar‐planetsystem.
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THERADIALVELOCITYEQUATION 5
SomeBasicsofVectors
• Vectorquantities(or“VECTORS”)havebothmagnitudeanddirection,unlikescalarquantities(or“SCALARS”)whichhaveonlymagnitude
• Vectorquantitiesareusuallydenotedbyplacinganarrowabovethesymbolrepresentingthevector(e.g.
€
F )and/orplacingthesymbolinboldface(e.g.F)
• TheMAGNITUDEOFAVECTORisascalarequaltothelengthofthevector(whichisalwaysapositivenumber)
• Themagnitude(orlength)ofavectorisusuallydenotedbyplacingbarsaroundthevector(e.g.
€
F )orjustthesymbolrepresentingthevectorinnormalfont(e.g.F)
• AUNITVECTORisavectorwhosemagnitudeisequalto1
• TheDIRECTIONOFAVECTORtellsyouwhichwaythevectorpoints
• Thedirectionofavectorisusuallydenotedbyplacingahatabovethesymbolrepresentingthevector(e.g.
€
ˆ F )
• TheNEGATIVEOFAVECTORisanothervectorthathasthesamemagnitudeastheoriginalvector,butpointsintheoppositedirection(e.g.if
€
F isavectorwitha
magnitudeequaltoFandadirectionpointingtotheright,then
€
− F isavectorwitha
magnitudeequaltoFandadirectionpointingtotheleft)
Figure2–Thecoordinatesystemwewillutilizewhenanalyzingthestar‐planetsystem.
(NotethatthelocationoftheCMinthisfigureisexaggeratedinordertomoreeasilyillustratethevectorsthatdependonit)
NOTTOSCALE
distance between the star and planet
r
êr
r1
r2
CM
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6 THERADIALVELOCITYEQUATION
ANALYZING THE STAR-PLANET SYSTEM
UsingFigure2andwhatweknowaboutvectors,wecandefinethefollowing:
€
ˆ e r ≡unitvectoralongthelineconnectingthestarandplanetthatpointstotheright
€
r ≡ vectorpointingfromstartoplanet(right)
€
r 1 ≡vectorpointingfromCMtostar(left)
€
r 2 ≡ vectorpointingfromCMtoplanet(right)
€
r ≡ magnitudeof
€
r =distancebetweenthestarandplanet
€
r1 ≡magnitudeof
€
r 1 =distancebetweentheCMandstar
€
r2 ≡ magnitudeof
€
r 2 =distancebetweentheCMandplanet
€
ˆ r ≡directionof
€
r =
€
+ ˆ e r
€
ˆ r 1 ≡directionof
€
r 1 =
€
− ˆ e r
€
ˆ r 2 ≡ directionof
€
r 2 =
€
+ ˆ e r Usingthedefinitionsaboveandthefactthatavectorhasbothamagnitudeandadirection,wefind:
€
r ≡ r ˆ r = r (+ ˆ e r) r 1 ≡ r1 ˆ r 1 = r1 (− ˆ e r ) r 2 ≡ r2 ˆ r 2 = r2 (+ ˆ e r)
Thus,
€
r = r ˆ e r r 1 = −r1 ˆ e r r 2 = r2 ˆ e r
(Eq1)
UsingFigure2andEq1,wecansee:
€
r = − r 1 + r 2
= − −r1 ˆ e r( ) + r2 ˆ e r( )= r1 ˆ e r + r2 ˆ e r= r1 + r2( ) ˆ e r
Therefore,themagnitude(orlength)of
€
r is
€
r = r1 + r2 and:
€
r = r 2 − r 1 (Eq2)
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THERADIALVELOCITYEQUATION 7
THE CENTER OF MASS FRAME OF REFERENCE
Thegeneraltwo‐bodyequationforthecenterofmassis:
€
R = m1
r 1 + m2 r 2
m1 + m2
wherem1≡massofthefirstbody(which,inthisderivation,isthestar)
m2≡massofthesecondbody(which,inthisderivation,istheplanet)
€
R ≡vectorpointingfromspecifiedorigintoCM
€
r 1 ≡vectorpointingfromspecifiedorigintofirstbody
€
r 2 ≡ vectorpointingfromspecifiedorigintosecondbody Inthecenterofmassreferenceframe,thelocationoftheCMisdesignatedasthelocationoftheoriginandthefollowingsubstitutionsthenfollow:
€
R = 0
€
r 1 =vectorpointingfromCMtofirstbody
€
r 2 =vectorpointingfromCMtosecondbodyThegeneralequationthenbecomes:
€
0 =m1 r 1 + m2
r 2m1 + m2
€
0 = m1 r 1 + m2
r 2 Therefore,
€
m1 r 1 = −m2
r 2 (Eq3)Usingtheexpressionsfor
€
r 1 and
€
r 2 inEq1andsubstitutingthemintoEq3,wefind:
€
m1 r 1 = −m2
r 2m1(−r1 ˆ e r ) = −m2(r2 ˆ e r )−m1r1 ˆ e r = −m2r2 ˆ e r
Therefore,
€
m1r1 = m2r2 (Eq4)
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8 THERADIALVELOCITYEQUATION
THE LAWS OF ISAAC NEWTON
Newton’sLawofUniversalGravitationtellsusthestarexertsagravitationalforceontheplanetandtheplanetexertsagravitationalforceonthestar.Bothforcesareequalinmagnitude,oppositeindirection,andactalongthelineconnectingthestarandplanet(showninFigure3).
Notethat,inFigure3:
€
F 12 ≡ Forceonobject1(thestar)exertedbyobject2(theplanet)
€
F 21 ≡Forceonobject2(theplanet)exertedbyobject1(thestar)
Themagnitudeofanygravitationalforcebetweentwoobjectsisdirectlyproportionaltotheproductofthemassesofeachobjectandinverselyproportionaltothedistancebetweenthemsquared:
€
Fg =G m1m2
r2
TheconstantofproportionalityiscalledNewton’sgravitationalconstant:
€
G = 6.67 ×10−11 m3
kg s2
€
F 12 and
€
F 21thushavethesamemagnitude
€
Fg andonlydifferindirection:
€
F 12 = G m1m2
r2 + ˆ e r( ) F 21 = G m1m2
r2 − ˆ e r( )
Therefore,
€
F 12 = G m1m2
r2ˆ e r (EQ5)
€
F 21 = −G m1m2
r2ˆ e r (EQ6)
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THERADIALVELOCITYEQUATION 9
Now,Newton’s2ndLawtellsus:
€
F = m a ∑
Foreachbody,theonlyforceexertedonitisthegravitationalforce,so:
€
forbody1(thestar) : F = F 12∑
Thus,
(Eq7)
And,
€
forbody2(theplanet) : F = F 21∑
Thus,
(Eq8)
Usingtheexpressionsfor
€
F 12 and
€
F 21inEq5andEq6andrewriting
€
a as
€
d2 r dt 2
,Eq7andEq8become:
€
G m1m2
r2ˆ e r = m1
d2 r 1dt 2
(Eq9)
and
€
−G m1m2
r2ˆ e r = m2
d2 r 2dt 2
(Eq10)
SubtractingEq9fromEq10gives:
€
−Gm1
r2ˆ e r
−
Gm2
r2ˆ e r
=
d2 r 2dt 2
−
d2 r 1dt 2
−G m1 + m2( )
r2ˆ e r =
d2
dt 2 r 2 − r 1( )
€
F 12 = m1
a 1
€
F 21 = m2
a 2
€
Gm2
r2ˆ e r =
d2 r 1dt 2
€
−Gm1
r2ˆ e r =
d2 r 2dt 2
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10 THERADIALVELOCITYEQUATION
Now,let’smakethefollowingdefinition:
(Eq11)
UsingEq11andEq2,wehave:
€
−GMr2
ˆ e r =d2
dt 2 r ( )
Using
€
d2 r dt 2 ≡
˙ ̇ r and
€
r ≡ r ˆ e r ⇒ ˆ e r = r r,wehave:
(Eq12)
(Eq13)
THE LAWS OF JOHANNES KEPLER
Asthestarandplanetorbittheirmutualcenterofmass,thedistancebetweenthem,r,changes.AccordingtoKepler’s1stLaw(andextendingthetreatmenttoextrasolarplanets),planetsrevolvearoundtheirhoststarinanellipticalorbitwiththestaratonefocusoftheellipse(showninFigure4).
€
M ≡ m1+m2
€
˙ ̇ r = − GMr2
ˆ e r
€
˙ ̇ r = − GMr3 r
r
θ
Figure4–Astheplanetorbitsitshoststar,thedistancebetweenthestarandplanetchanges.(Nottoscale!)
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THERADIALVELOCITYEQUATION 11
Itisusefultoexpressrinpolarcoordinatesand,thus,asafunctionofθ.Recallthat
€
r = r ˆ e r .Now,takethederivativeof
€
r withrespecttotime:
(Eq14)
€
ˆ e r isthedirectionofthevectorpointingfromthestartotheplanetanditshouldbeunderstoodthatthisdirectionchangeswithtime.Inordertocalculatethederivativeof
€
ˆ e r withrespecttotime,wemustfirstexpress
€
ˆ e r intermsofCartesiancoordinates(thederivationofthisexpressioncanbelookedupelsewhere):
€
ˆ e r = cosθ ˆ x + sin θ ˆ y Now,wecaneasilydeterminethederivativeof
€
ˆ e r withrespecttotime:
€
ddt
ˆ e r( ) =ddt
cos θ ˆ x + sinθ ˆ y ( )
= cosθ ddt
ˆ x ( ) + ˆ x ddt
cosθ( ) + sinθ ddt
ˆ y ( ) + ˆ y ddt
sin θ( )
= 0 + ˆ x ddt
cos θ( ) + 0 + ˆ y ddt
sinθ( )
Wewereabletodothelaststepbecause
€
ˆ x and
€
ˆ y areindependentoftime(and,thus,takingthederivativeofeitherwithrespecttotimegiveszero).However,θdoesdependontime,sowemusttakethisintoaccountwhentakingthetimederivativeofsinθandcosθ:
€
ddt
ˆ e r( ) = ˆ x ddt
cos θ( ) + ˆ y ddt
sinθ( )
= ˆ x −sin θ ˙ θ ( ) + ˆ y cosθ ˙ θ ( )= − ˙ θ sinθ ˆ x + ˙ θ cos θ ˆ y
= ˙ θ −sin θ ˆ x + cosθ ˆ y ( )
TheexpressioninparenthesesisEqualto
€
ˆ e θ expressedintermsofCartesiancoordinates(thederivationofthisexpressioncanbelookedupinthesame“elsewhere”).Inotherwords:
€
ˆ e θ = −sinθ ˆ x + cos θ ˆ y Andso,
(Eq15)
€
ddt r ( ) =
ddt
r ˆ e r( ) = r ddt
ˆ e r( ) + ˆ e rddt
r( )
€
ddt
ˆ e r( ) = ˙ θ ˆ e θ
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12 THERADIALVELOCITYEQUATION
SubstitutingEq15intoEq14gives:
€
ddt r ( ) = r ˙ θ ˆ e θ( ) + ˆ e r ˙ r
Using
€
d r dt≡ ˙ r ,wehave:
(Eq16)
Now,takethederivativeof
€
˙ r withrespecttotime:
€
ddt ˙ r ( ) =
ddt
˙ r ˆ e r + r ˙ θ ˆ e θ( )
= ˙ r ddt
ˆ e r( ) + ˆ e rddt
˙ r ( ) + r ddt
˙ θ ˆ e θ( ) + ˙ θ ˆ e θddt
r( )
= ˙ r ˙ θ ˆ e θ( ) + ˆ e r ˙ ̇ r ( ) + r ˙ θ ddt
ˆ e θ( ) + ˆ e θddt
˙ θ ( )
+ ˙ θ ˆ e θ ˙ r
(Eq17)
Now,wemusttakethederivativeof
€
ˆ e θ withrespecttotime:
€
ddt
ˆ e θ( ) =ddt
−sin θ ˆ x + cosθ ˆ y ( )
= −sinθ ddt
ˆ x ( ) + ˆ x ddt
−sin θ( ) + cos θ ddt
ˆ y ( ) + ˆ y ddt
cos θ( )
= 0 + ˆ x −cos θ ˙ θ ( ) + 0 + ˆ y −sinθ ˙ θ ( )= − ˙ θ cosθ ˆ x − ˙ θ sin θ ˆ y
= − ˙ θ cos θ ˆ x + sinθ ˆ y ( )= − ˙ θ ˆ e r( )
Andso,
(Eq18)
SubstitutingEq18intoEq17gives:
€
ddt ˙ r ( ) = ˙ r ˙ θ ˆ e θ + ˙ ̇ r ˆ e r + r ˙ θ − ˙ θ ˆ e r( ) + r ˙ ̇ θ ˆ e θ + ˙ r ˙ θ ˆ e θ
= 2 ˙ r ˙ θ ˆ e θ + ˙ ̇ r ˆ e r − r ˙ θ 2 ˆ e r + r ˙ ̇ θ ˆ e θ= ˙ ̇ r − r ˙ θ 2( ) ˆ e r + r ˙ ̇ θ + 2 ˙ r ˙ θ ( ) ˆ e θ
€
˙ r = ˙ r ˆ e r + r ˙ θ ˆ e θ
€
ddt ˙ r ( ) = ˙ r ˙ θ ˆ e θ + ˙ ̇ r ˆ e r + r ˙ θ
ddt
ˆ e θ( ) + r ˙ ̇ θ ˆ e θ + ˙ r ˙ θ ˆ e θ
€
ddt
ˆ e θ( ) = − ˙ θ ˆ e r
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THERADIALVELOCITYEQUATION 13
Therefore,
(Eq19)
SubstitutingEq19intoEq12gives:
€
˙ ̇ r − r ˙ θ 2( ) ˆ e r + r ˙ ̇ θ + 2 ˙ r ˙ θ ( ) ˆ e θ = −GMr2
ˆ e r
Theonlywayfortheaboveequationtobetrueisif:
(Eq20)
and,
(Eq21)
ToextractanimportantpieceofinformationfromEq21,recallthis:
€
ddt
r2 ˙ θ ( ) = r2 ddt
˙ θ ( ) + ˙ θ ddt
r2( )= r2 ˙ ̇ θ + ˙ θ 2r˙ r ( )= r2 ˙ ̇ θ + 2r˙ r ˙ θ
= r r ˙ ̇ θ + 2˙ r ˙ θ ( )
So,theleft‐handsideofEq21isalsoequalto
€
1rddt
r2 ˙ θ ( ) andthus,
€
1rddt
r2 ˙ θ ( ) = 0
fromwhich,follows:
(Eq22)
Thequantity
€
r2 ˙ θ doesnotchangewithtime,sothequantityissaidtobeconserved.Infact,
€
r2 ˙ θ isawell‐knownquantity,asweshallnowsee.
€
r ˙ ̇ θ + 2 ˙ r ˙ θ = 0
€
˙ ̇ r = ˙ ̇ r − r ˙ θ 2( ) ˆ e r + r ˙ ̇ θ + 2 ˙ r ˙ θ ( ) ˆ e θ
€
˙ ̇ r − r ˙ θ 2 = −GMr 2
€
ddt
r 2 ˙ θ ( ) = 0
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14 THERADIALVELOCITYEQUATION
ANGULAR MOMENTUM
Recallthegeneralequationfortotalangularmomentum:
€
L = r × p
€
where : L = total angular momentum r = displacement vector p = total linear momentum = m v
Foraplanetorbitingitshoststar,
€
r and
€
v arealwaysperpendicular(showninFigure5).
So,
€
r × p = r ×m v = m r × v ( ) = m rv sin 90°( )[ ] = m rv 1( )[ ] = mrv andthus,
€
L ≡ L = mvr
Now,
€
LmisdefinedtobetheSPECIFICANGULARMOMENTUMandisdenotedbythesymbolh.
Using
€
v =ω r and
€
ω ≡dθdt
= ˙ θ ,sothat
€
v = ˙ θ r,wehave
€
h ≡ Lm
=mvrm
= vr = ˙ θ r( )r = r2 ˙ θ ,therefore,
(Eq23)
Andso,Eq21tellsusthatthespecificangularmomentumforoursituationisaconservedquantity.
€
h = r2 ˙ θ
Figure5–Astheplanetorbitsitshoststar,itspositionvector
€
r andvelocityvector
€
v arealwaysperpendicular.(Nottoscale!)
r
v
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THERADIALVELOCITYEQUATION 15
THE EQUATION OF AN ELLIPSE
NowwearereadytoutilizeEq20forthepurposeoffindingrasafunctionofθ.First,abrieftrick.Let’sexpress
€
r ˙ θ 2 intermsofh:
€
h = r2 ˙ θ ⇒ h2 = r4 ˙ θ 2 = r3 r ˙ θ 2( ) ⇒ r ˙ θ 2 =h2
r3
SubstitutingthisresultintoEq20gives:
€
˙ ̇ r − h2
r3 = −GMr2
Or,solvingfor
€
˙ ̇ r :
(Eq24)
Next,wemustremovethetimedependenceofr.Todothis,weinvokeasubstitution:
€
u ≡ 1r
⇒ r =1u
= u−1
Takingthederivativeofrwithrespecttotimegives:
€
ddt
r( ) =ddt
u−1( ) = −u−2 dudt
Using
€
dudt
=dudθ
dθdt
=dudθ
˙ θ ,theaboveequationbecomes:
€
ddt
r( ) = − u−2( ) dudθ˙ θ = − r2( ) dudθ
˙ θ = − r2 ˙ θ ( ) dudθ = − h( ) dudθ
Therefore,
(Eq25)
Takingthederivativeof
€
˙ r withrespecttotime(andrememberingthathisindependentoftime)gives:
€
ddt
˙ r ( ) =ddt
− h dudθ
= − h d
dtdudθ
€
˙ ̇ r = h2
r 3 −GMr 2
€
˙ r = − h dudθ
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16 THERADIALVELOCITYEQUATION
Using
€
ddt
dudθ
=
ddθ
dθdt
dudθ
=d2udθ 2
dθdt
=d2udθ 2
˙ θ ,wefind:
€
ddt
˙ r ( ) = − h d2udθ 2
˙ θ
Therefore,
(Eq26)
SubstitutingEq26intoEq24gives:
€
− h ˙ θ d2udθ 2 =
h2
r3 −GMr2
Replacingrwith
€
1u,usingEq23,andsolvingfor
€
d2udθ 2
gives:
€
d2udθ 2 = −
1h ˙ θ
h2u3 −GMu2( )
= −h( ) u3
˙ θ +GMu2
h( ) ˙ θ
= −r2 ˙ θ ( ) u3
˙ θ +GMu2
r2 ˙ θ ( ) ˙ θ
= − r2( ) u3 +GMr2 ˙ θ 2
u2( )
= −1u2
u3 +
GMr2 ˙ θ 2
1r2
= − u +GMr4 ˙ θ 2( )
= − u +GMh2( )
Therefore,
(Eq27)
Eq27isadifferentialequationwiththefollowingsolution:
€
˙ ̇ r = − h ˙ θ d2udθ 2
€
d2udθ 2 = −u +
GMh2
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THERADIALVELOCITYEQUATION 17
(Eq28)
€
where : A ≡ constantδ ≡ phase shift
both will be determinedby initial conditions
Changingvariablesbacktor,Eq28becomes:
€
1r θ( )
= A cos θ −δ( ) +GMh2
So,
(Eq29)
Multiplyingtheright‐handsideofEq29by
€
h2
h2gives:
€
r θ( ) =h2
h2A cos θ −δ( ) +GM
Dividingtheright‐handsideoftheaboveequationbyGMgives:
(Eq30)
Thegeneralequationforanellipseis:
(Eq31)
€
where : a ≡ semi -major axis of the ellipse
e ≡ eccentricity of the ellipse
ComparingEq30toEq31,wecanimmediatelysee:
(Eq32)
€
u θ( ) = A cos θ −δ( ) +GMh2
€
r θ( ) =1
A cos θ −δ( ) +GMh2
€
r θ( ) =
h2
GMh2A cos θ −δ( )
GM+1
€
r θ( ) =a 1− e2( )1+ e cosθ
€
e =h2AGM
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18 THERADIALVELOCITYEQUATION
Or,
€
h2
GM=eAandthus,
€
a 1− e2( ) =h2
GM=eA.Sothat,
(Eq33)
Therefore,theunknownconstantAinEq28intermsofknownvariablesis:
(Eq34)
and,
€
δ = 0 (aproofofthisisprovidedinAppendixA).
PROPERTIES OF AN ELLIPSE
Inasense,wehavethusfaranalyzedthestar‐planetsysteminonedimension(whenwecomputedthegravitationalforces)andintwodimensions(whenwedeterminedanexpressionforrasafunctionofθ)andournexttaskistodevelopthetoolsneededtofullyunderstandthissysteminthreedimensions.First,let’sreviewsomeimportantpropertiesofanellipse:Themajoraxisofanellipseisthestraightlinethatrunsbetweenthefocioftheellipseandextendstotheedgesoftheellipse(showninFigure6).One‐halfofthemajoraxis(aptlynamedthe“SEMI‐MAJORAXIS”)hasalengthdefinedtobeequaltoa,therefore,thelengthofthemajoraxisisequalto2a.
€
a =e
A 1− e2( )
€
A =GMeh2
Figure6–Themajoraxisofanellipse.
Focus Focus
length=2a
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THERADIALVELOCITYEQUATION 19
Theminoraxisofanellipseisthestraightlinethatliesperpendiculartothemajoraxisandpassesthroughthemidpointofthemajoraxis(showninFigure7).One‐halfoftheminoraxis(aptlynamedthe“SEMI‐MINORAXIS”)hasalengthdefinedtobeequaltob,therefore,thelengthoftheminoraxisisequalto2b.
PERIASTRONisdefinedtobethepositioninaplanet’sorbitarounditshoststarinwhichthedistancebetweenthestarandplanetisaminimumfortheorbit.Similarly,APASTRONisdefinedasthepositioninaplanet’sorbitarounditshoststarinwhichthedistancebetweenthestarandplanetisamaximumfortheorbit(showninFigure8).
Figure7–Theminoraxisofanellipse.
Focus Focus
length=2b
Figure8–Thepositionsofperiastronandapastroninaplanet’sorbitarounditshoststar.(Nottoscale!)
apastron
periastron
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20 THERADIALVELOCITYEQUATION
TheECCENTRICITYofanellipse,orhowelongatedtheellipseis,isdenotedbythesymboleandisdefinedintermsofthesemi‐majorandsemi‐minoraxes:
(Eq35)
Inthespecialcaseinwhich
€
a = b, e =1−1= 0andtheshapeofaplanet’sorbitarounditshoststarwouldbeaperfectcircle.Finally,theareaofanellipseisdefinedtobe:
(Eq36)
Noticethatforacircle
€
a = bandtheareaisequaltoπa2orπb2asonewouldexpect!
DEFINITIONS OF ORBITAL PARAMETERS
Now,let’sdefinetwoterms(showninFigure9):LINE‐OF‐SIGHT≡thestraightlineconnectinganobserverandtheobjectbeingorbited(inourcase,
wewilltakethistobethestar)PLANE‐OF‐SKY≡thetwodimensionalplanethatliesperpendiculartotheline‐of‐sightandpasses
throughtheobjectbeingorbited
€
e2 ≡ 1− b2
a2
€
Area = πab
line‐of‐sight
plane‐of‐sky(seen“edge‐on”)
plane‐of‐sky(seen“face‐on”)
Figure9–Anobserver’sline‐of‐sightandtheplane‐of‐skyseenboth“edge‐on”and“face‐on.”(Nottoscale!)
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THERADIALVELOCITYEQUATION 21
Wearenowreadytoanalyzethestar‐planetsysteminthreedimensions(showninFigure10):
TherearetwoseparateplanesinFigure10:theplane‐of‐sky(alreadydefined)andthePLANE‐OF‐ORBIT(thetwodimensionalplaneinwhichtheplanetorbitsitshoststar).Theplane‐of‐orbitistiltedwithrespecttotheplane‐of‐skybyacertainangle.WecallthistilttheORBITALINCLINATIONanddenotetheanglebythesymboli.Therearetwopointsintheplanet’sorbitthatpassthroughtheplane‐of‐sky.OneiscalledtheASCENDINGNODE(thepointatwhichtheplanetpassesfrom“below”theplane‐of‐skyto“above”theplane‐of‐sky)andtheotheriscalledtheDESCENDINGNODE(thepointatwhichtheplanetpassesfrom“above”theplane‐of‐skyto“below”it).ThelineconnectingthesetwopointsiscalledtheLINEOFNODES.Theangleθistheanglebetweentheperiastronandthepositionoftheplanet.It’sreferredtoastheTRUEANOMALYanditchangeswithtime.ThisisthesameθthatisfoundinFigure4,Eq28,&Eq31.Theangleωistheanglebetweenthelineofnodesandperiastron.It’sreferredtoastheARGUMENTOFPERIASTRONanditdoesnotchangewithtime.Bothθandωlieintheplane‐of‐orbit.
Figure10–Illustrationoforbitalparametersforthestar‐planetsystem.
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22 THERADIALVELOCITYEQUATION
RADIAL VELOCITY
Ifwedesignatetheplane‐of‐skytobethex-yplane,thenthedirectiontotheobserveralongtheline‐of‐sightisthez‐direction.Intermsofθ,ω,andi,zcanbeexpressedas:
€
z = r sin θ +ω( ) sin i (AppendixBexplainshowthisisdetermined.)Rememberthatthisentirederivationisdevotedtofindinganexpressionfortheradialvelocity(denotedbyvr)andthattheradialvelocityisdefinedtobethevelocityalongtheobserver’slineofsight.Thus,sincezisthedirectionalongtheobserver’slineofsight,thederivativeofzwithrespecttotimeshallprovideuswiththeexpressionforradialvelocity!Recallingthatθandrchangewithtime,however,ωandidonot,wefind:
€
vr ≡ ˙ z = ddt
z( ) =ddt
r sin θ +ω( ) sin i( )
= r ddt
sin θ +ω( ) sin i( ) + sin θ +ω( ) sin i ddt
r( )
= r sin θ +ω( ) ddt
sin i( ) + sin i ddt
sin θ +ω( ){ }
+ sin θ +ω( ) sin i ˙ r
= r sin θ +ω( ) cos i didt
+ sin i cos θ +ω( ) ddt
θ +ω( )
+ ˙ r sin θ +ω( ) sin i
= r 0 + sin i cos θ +ω( ) ˙ θ [ ] + ˙ r sin θ +ω( ) sin i
Therefore,
(Eq37)
Now,wemustfindexpressionsfor
€
˙ r and
€
˙ θ inordertoridEq37ofthesetwoquantities.Tofind
€
˙ r ,wetakethederivativeofEq31withrespecttotime(aandearetime‐independent):
€
˙ r ≡ ddt
r( ) =ddt
a 1− e2( )1+ e cos θ
=1+ e cos θ[ ] d
dta 1− e2( )[ ] − a 1− e2( )[ ] d
dt1+ e cosθ[ ]
1+ e cosθ[ ]2
€
vr = r ˙ θ cos θ +ω( ) + ˙ r sin θ +ω( )[ ] sin i
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THERADIALVELOCITYEQUATION 23
€
=0 − a 1− e2( )[ ] − e ˙ θ sinθ[ ]
1+ e cos θ[ ]2
=a 1− e2( )
1+ e cosθe ˙ θ sin θ
1+ e cosθ
Thus,
(Eq38)
Tofind
€
˙ θ ,wemakeuseofKepler’s2ndLaw:
(Eq39)
whereAistheareaoftheellipse.Integratingtheleft‐handsideofEq39overoneperiod(thetimeittakestheplanettomakeonecompleterevolutionarounditshoststar)yields:
(Eq40)
wherePistheperiod.Weknowtheareaofanellipseis
€
A = πab (fromEq36),buttogettheareaintermsofaandeonly,weneedtosolveEq35intermsofb:
€
e2 ≡1− b2
a2⇒
b2
a2=1− e2 ⇒ b2 = a2 1− e2( )
Andso,
€
b = a 1− e2 andputtingthisintotheequationfortheareaofanellipsegives:
€
A = πab = πa a 1− e2( )
Therefore,
(Eq41)
SubstitutingEq41intoEq40gives:
€
πa2 1− e2
P=
12r2 ˙ θ
Insteadofsolvingtheaboveequationfor
€
˙ θ ,wewillsolveitfor
€
r ˙ θ sincethisquantityisfoundinEq37,whichwearetryingtosimplify:
(Eq42)
€
˙ r = r e ˙ θ sinθ1+ e cosθ
€
dAdt
=12r 2 ˙ θ
€
AP
=12r 2 ˙ θ
€
A = πa2 1− e2
€
r ˙ θ =2πa2 1− e2
r P
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24 THERADIALVELOCITYEQUATION
Now,wehavefoundexpressionsfor
€
˙ r (Eq38)and
€
r ˙ θ (Eq42),buttheexpressionfor
€
˙ r hasan
€
r ˙ θ initandtheexpressionfor
€
r ˙ θ hasanrinit.So,let’sridtheseexpressionsofanyror
€
˙ θ dependence:
€
r ˙ θ =2πa2 1− e2
r P
=1r
2πa2 1− e2
P
=1+ e cos θa 1− e2( )
2πa2 1− e2
P
Thus,
(Eq43)
And,
€
˙ r = r e ˙ θ sinθ1+ e cos θ
= r ˙ θ e sinθ
1+ e cos θ
=2πa 1+ e cosθ( )
P 1− e2
e sin θ1+ e cosθ
Thus,
(Eq44)
SubstitutingEq43andEq44intoEq37gives:
€
Vr = r ˙ θ cos θ +ω( ) + ˙ r sin θ +ω( )[ ] sin i
=2πa 1+ e cosθ( ) cos θ +ω( )
P 1− e2+
2πa e sin θ sin θ +ω( )P 1− e2
sin i
(Eq45)
Next,weneedtosimplifyEq45byutilizingthefollowingtwotrigonometricidentities:
€
cos θ +ω( ) = cos θ cosω − sin θ sinωsin θ +ω( ) = sin θ cosω + cosθ sinω
€
r ˙ θ =2πa 1+ e cosθ( )
P 1− e2
€
˙ r = 2πa e sinθP 1− e2
€
Vr =2πa sin iP 1− e2
cos θ +ω( ) + e cosθ cos θ +ω( ) + e sin θ sin θ +ω( )[ ]
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THERADIALVELOCITYEQUATION 25
TheexpressionwithinthebracketsinEq45becomes:
€
[ ] = cosθ cosω − sinθ sinω + e cos θ cos θ cosω − sin θ sinω( ) + e sin θ sin θ cosω + cosθ sinω( )= cosθ cosω − sinθ sinω + e cosω cos2 θ − e cos θ sinθ sinω + e cosω sin2 θ + e cos θ sin θ sinω
= cosθ cosω − sinθ sinω + e cosω cos2 θ + sin2 θ( )= cosθ cosω − sinθ sinω + e cosω
Weusedthetrigonometricidentity
€
cos2θ + sin2θ =1inthelaststepandwecannowrevert
€
cosθ cosω − sinθ sinω backto
€
cos θ +ω( ) foramorecompactexpression:
(Eq46)
Finally,wehavederivedtheequationforradialvelocity!Thus,theradialvelocityofthehoststar(withrespecttothecenterofmass)is:
(Eq47)
andtheradialvelocityoftheorbitingplanet(withrespecttothecenterofmass)is:
(Eq48)
RADIAL VELOCITY SEMI-AMPLITUDE
Wehaveonemorequantitytoprovideanameandsymbolfor:
(Eq49)
Theradialvelocitysemi‐amplitudeofthehoststaris:
(Eq50)
andtheradialvelocitysemi‐amplitudeoftheorbitingplanetis:
(Eq51)
€
Vr =2πa sin iP 1− e2
cos θ +ω( ) + e cosω[ ]
€
K ≡ radialvelocitysemi ‐ amplitude =2πa sin iP 1− e2
€
Vr (star ) =2πa1 sin iP 1− e2
cos θ +ω( ) + e cosω[ ]
€
Vr (planet ) =2πa2 sin iP 1− e2
cos θ +ω( ) + e cosω[ ]
€
K1 ≡ radialvelocitysemi ‐amplitude of host star =2πa1 sin iP 1− e2
€
K2 ≡ radialvelocitysemi ‐amplitude of orbiting planet =2πa2 sin iP 1− e2
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26 THERADIALVELOCITYEQUATION
Theradialvelocitymethodofdetectingextrasolarplanetsinvolvestakingprecisemeasurementsofastar’sradialvelocitywithanopticaltelescope.Eachmeasurementisassociatedwithaspecifictimeandaplotcanbecreatedshowingthestar’sradialvelocityasafunctionoftime.Ifapreviouslyundiscoveredplanetexistsinorbitaroundtheobservedstar,thedataintheplotwillshowarepeatedtrendandacurvecanbefittothedataconnectingeachoftheindividualradialvelocitydatapoints(showninFigure11).
Figure11–Plotofradialvelocityvs.timeforthehoststarindicatinghowtheperiod,P,andradialvelocitysemi‐amplitude,K,canbedeterminedfromthedata.(ImageCredit:PlanetarySystemsandtheOriginsofLife,CambridgeUniversityPress,2007)
Theradialvelocitysemi‐amplitudeofthehoststarcanthenbedeterminedfromtheplot.Itisequaltohalfofthetotalamplitudeofthefittedcurve(asseeninFigure11).Notethattheperiod,whichisthetotalamountoftimeelapsedbetweentwoconsecutivepeaksofthefittedcurve(andisthesameforboththehoststarandorbitingplanet),canbedeterminedfromtheplotaswell(asseeninFigure11).Ifpreferred,thedependenceona1inthequantityK1canberemoved.Thiswillbeournexttask.Kepler’s3rdLawrelatesthesemi‐majoraxisofaplanet’sorbittotheperiodofitsorbitasfollows:
(Eq52)
SolvingEq52fora2gives:
€
a23 =
G m1 + m2( ) P 2
4π 2
€
P 2 =4π 2
G m1 + m2( )a23
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THERADIALVELOCITYEQUATION 27
Therefore,
(Eq53)
Nowweneedtoreferallthewaybacktopage7andEq4.Withour“new”knowledgeofellipses,wecannowrecognizethatther1inEq4,whichisthemagnitudeofthevectorpointingfromtheCMtothestar,issimplythesemi‐majoraxisofthestar’sorbitaroundthemutualCM,a1.Withthesamereasoning,ther2inEq4,whichisthemagnitudeofthevectorpointingfromtheCMtotheplanet,isthesemi‐majoraxisoftheplanet’sorbitaroundthemutualCM,a2.Therefore,
€
m1a1 = m2a2 or,solvingfora1:
(Eq54)
SubstitutingEq54intoEq50gives:
(Eq55)
SubstitutingEq53intoEq55gives:
(Eq56)
TheprocessofsimplifyingEq56goessomethinglikethis:
€
K1 =m2
m1sin i1− e2
8π 3G m1 + m2( ) P 2
4π 2P 3
13
=m2
m1sin i1− e2
2πG m1 + m2( )P
13
=2πGP
13 m2
m1m1 + m2( )
13sin i1− e2
€
a1 =m2
m1a2
€
a2 =G m1+m2( ) P2
4π 2
13
€
K1 =2π sin iP 1− e2
m2
m1a2
€
K1 =2π sin iP 1− e2
m2
m1
G m1+m2( ) P2
4π 2
13
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28 THERADIALVELOCITYEQUATION
Thisisthepointatwhichweutilizethefactthat
€
m1 >> m2 andsoouranswerisnotsignificantlyaffectedifweusethefollowingapproximation:
€
m1 + m2 ≈ m1Nowwecancontinueoursimplification:
€
K1 =2πGP
13 m2
m1m1( )
13sin i1− e2
=2πGP
13 m1( )
13
m1m2 sin i1− e2
=2πGP
13m1( )−
23m2 sin i1− e2
Therefore,initsmostcommonform,theequationfortheradialvelocitysemi‐amplitudeofthestaris:
(Eq57)
€
K1 =2πGP
13 m2 sin im1
2 / 311− e2
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THERADIALVELOCITYEQUATION 29
APPENDIX A: PROOF THAT δ = 0
Toprovideproofthat
€
δ = 0 inEq28,wewillanalyzethespecialcaseinwhichtheplanetislocatedatPERIASTRON(thepositioninaplanet’sorbitarounditshoststarinwhichthedistancebetweenthestarandplanetisaminimumfortheorbit).Thepositionofperiastronisusefulfortworeasons:(1)weknowthevalueofθisequaltozeroforthispositioninaplanet’sorbitarounditshoststarand(2)weknowtheexpressionforr(θ),thedistancebetweentheplanetanditshoststar.Thefirstreasoncanbeunderstoodbyre‐examiningFigure10orre‐readingthedefinitionoftheTRUEANOMALY.Thesecondreasonfollowsfromthefactthat,foranyellipse,thedistancebetweenthecenteroftheellipseandeitherfocusoftheellipse(oneofwhichisalsothepositionofthehoststar)isequaltoae(showninFigure12).Forexample,inthespecialcaseofacircularorbite=0andthe“ellipse”(orcircle,whichisaspecialtypeofellipse)hasonlyonefocus(thelocationofwhichisthepositionofthehoststar).Thissinglefocusislocatedatthecenteroftheellipse,therefore,thedistancebetweenthefocusandthecenteroftheellipseiszerowhichagreeswithae=a(0) =0(showninFigure13).
Theexpressionforr(θ),thedistancebetweentheplanetanditshoststar,isequaltothedistancebetweentheplanetandthecenteroftheellipse(which,inthiscase,isthelengthofthesemi‐majoraxis,a)minusthedistancebetweenthestarandthecenteroftheellipse,ae(showninFigure14).Thus,theexpressionforr(θ),thedistancebetweentheplanetanditshoststar,equals:
€
r θ( ) = a − ae
or,aftersimplifying:
(EqA.1)
€
r θ( ) = a 1− e( )
Figure12–Thedistancebetweenthecenterofanellipseandeitherofitsfociisequaltotheproductofthelengthofthesemi‐majoraxisandtheeccentricityoftheellipse.
Figure13–Thebluedotrepresentsboththecenterofthecircle(aspecialtypeofellipse)andthesinglefocusoftheellipse.Inthisspecialcase,theproductaeisequaltozero.
ae ae
centerofellipse
Focus Focus
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30 THERADIALVELOCITYEQUATION
ThetaskathandistosolveEq29forδ.WecansubstituteEqA.1fortheleft‐handsideofEq29anduse
θ=0.WecanalsorearrangeEq34as
€
GMh2
=Ae.Theresultofthesesubstitutionsis:
€
r θ( ) =1
A cos θ −δ( ) +GMh2
a 1− e( ) =1
A cos 0 −δ( ) +Ae
A cos −δ( ) +Ae
=1
a 1− e( )
A cos −δ( ) =1
a 1− e( )−Ae
(EqA.2)
Figure14–Thedistancebetweentheplanetanditshoststar(r(θ),representedbytheblackarrow)isequaltothedistancebetweentheplanetandthecenteroftheellipse(a,representedbythepurplearrow,whichisalsothesemi‐
majoraxis;notethatthisarrowisnotdrawnattheexactlocationofthesemi‐majoraxisotherwiseitwouldoverlapwiththeotherarrows)minusthedistancebetweenthestarandthecenteroftheellipse(ae,representedbytheredarrow).
apastron
periastron
a
ae r(θ)
centerofellipse
€
cos −δ( ) =1
Aa 1− e( )−1e
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THERADIALVELOCITYEQUATION 31
NowrearrangingEq33as
€
A =e
a 1− e2( )andsubstitutingthisintoEqA.2,wefind:
€
cos −δ( ) =1
Aa 1− e( )−1e
=1
ea 1− e2( )
a 1− e( )[ ]−1e
=1
e 1− e( )1− e2( )
−1e
=1− e2
e 1− e( )−1e
=1e1− e2
1− e−1
=1e1− e2
1− e−1− e1− e
=1e1− e2 − (1− e)
1− e
=1e1− e2 −1+ e1− e
=1ee − e2
1− e
=1ee (1− e)1− e
=1ee( )
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32 THERADIALVELOCITYEQUATION
Thus,
€
cos − δ( ) =1Thecosineisanevenfunction,sowecanexploitthefollowingpropertyofevenfunctions:
foranyanglea:
€
cos a( ) = cos −a( ) Andso,
€
cos − δ( ) = cos δ( ) =1Thus,
€
cos δ( ) =1 ⇒ δ = cos−1 1( ) = 0 Therefore,
€
δ = 0.
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THERADIALVELOCITYEQUATION 33
APPENDIX B: THE EXPRESSION FOR z
Itmightbeabitdifficulttounderstandwheretheequationforzcomesfrom,soI’veincludedamoredetailedillustrationoftheanglesandlengthsinquestion(showninFigure15).
Partoftheplanet’sorbitarounditshoststarhasbeenmadeboldinthediagram.Wewillassumethistobeastraightlengthandrefertoitas(bold length)inthefollowingequations.Notethattheactualvalueforthelengthisnotneeded.ωistheanglebetweentheline‐of‐nodes(y‐axisinFigure15)andperiastron.θistheanglebetweenperiastronandtheplanet’spositionvector,
€
r (themagnitudeofwhichisr,thedistancebetweentheplanetanditshoststar).Theanglebetweenthey‐axisandtheplanet’spositionvectoris
€
θ +ω (I’llleaveittothereadertomakecertainhe/sheunderstandswhythisisthecase;butifyouaren’tcertain,myhintistotrymakingupnumbersforθandωormovingtheplanettoanotherpositioninitsorbit).Wenowhavethenecessaryinformationand,withtheaidoftrigonometry,wefind:
€
sin θ +ω( ) =bold length( )
r⇒ bold length( ) = r sin θ +ω( )
€
sin i =z
bold length( )⇒ z = bold length( ) sin i ∴ z = r sin θ +ω( ) sin i
Figure15–Adiagramspecificallycreatedtoshowtheimportantlengthsandanglesneededfordetermininganexpressionforz.
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34 THERADIALVELOCITYEQUATION
REFERENCE FOR CALCULATING RV SEMI-AMPLITUDE
Ifyouknowthenumericalvalueforeachofthefollowingparametersforaplanetanditshoststar,
P(theplanetorhoststar’sorbitalperiodindaysoryears)
€
Mplanet sin i (theplanet’sminimummassintermsofthemassofEarthorJupiter)
€
Mstar (themassofthehoststarintermsofthemassoftheSun)
e(theeccentricityoftheplanet’sorbitarounditshoststar–rangesfrom0‐1)thenenteringtheappropriatevalueintooneofthefollowingfourequationswillgiveyouthenumericalvalueofthestar’sradialvelocitysemi‐amplitude:
€
Kstar = 0.6395 m s−1 1 dayP
13 Mplanet sin i
MEarth
MSun
Mstar
2 / 311− e2
Kstar = 0.0895 m s−1 1 yrP
13 Mplanet sin i
MEarth
MSun
Mstar
2 / 311− e2
Kstar = 203.255 m s−1 1 dayP
13 Mplanet sin i
MJupiter
MSun
Mstar
2 / 311− e2
Kstar = 28.435 m s−1 1 yrP
13 Mplanet sin i
MJupiter
MSun
Mstar
2 / 311− e2