ecuacion de velocidad radial

7/17/2019 Ecuacion de Velocidad Radial

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a detailed derivation ofThe Radial Velocity Equation

Kelsey I. Clubb

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Kelsey I. Clubb

Department of Physics & AstronomySan Francisco State UniversitySan Francisco, CA USA

August 2008



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1+ e cos"

dA

dt =

1

2r2"

A

P=

1

2r2"

A = " a21# e

2

r "

=

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r P



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r P

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%

&

'

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(

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*

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a 1$ e2( )

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= r " e sin"

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#

$ %

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=

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P 1* e2

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$ %

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e sin "

1+ e cos"

#

$ %

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A%:.5

<JB __?

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=

2$ a 1+ e cos" ( ) cos " +# ( )

P 1% e2

+

2$ a e sin " sin " +# ( )

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r " =2# a 1+ e cos" ( )

P 1$ e2

r =

2" a e sin#

P 1$ e2

V r=

2" a sin i

P 1# e2

cos $ +% ( )+ e cos$ cos $ +% ( )+ e sin$ sin $ +% ( )[ ]



$0, %!()!* +,*-&)$. ,23!$)-4 5X

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= cos" cos# $ sin" sin# + e cos# cos2" + sin

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RADIAL VELOCITY SEMI-AMPLITUDE

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V r=

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P 1# e2

cos $ +% ( )+ e cos% [ ]

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P 1$ e2

V r (star )

=

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cos $ +% ( )+ e cos% [ ]

V r ( planet ) =2" a2 sin i

P 1# e2

cos $ +% ( )+ e cos% [ ]

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a2

3

=

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2( ) P2

4" 2

P2= 4"

2

G m1+ m

2( ) a

2

3



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sin i

1" e2

8# 3

G m1+ m

2( ) P2

4# 2

P3

$

% &

'

( )

1

3

=

m2

m1

sin i

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2( )P

$

% &

'

( )

1

3

=

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P

*

+ ,

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m1

m1+ m

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a1=

m2

m1

a2

a2=

G m1+ m

2( ) P2

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#

$ %

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K 1=

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m2

m1

a2

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( )

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P 1# e2

m2

m1

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P

#

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m1

m1( )

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1) e2

=

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#

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$0, %!()!* +,*-&)$. ,23!$)-4 5[

APPENDIX A: PROOF THAT = 0

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h2=

A

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A cos " #$ ( ) +GM

h2

a 1# e( ) =1

A cos 0 #$ ( ) + A

e

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e=

1

a 1# e( )

A cos #$ ( ) =1

a 1# e( ) # A

e

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$0, %!()!* +,*-&)$. ,23!$)-4 77

APPENDIX B: THE EXPRESSION FOR z

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sin " +# ( ) =bold length( )

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sin i = z

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REFERENCE FOR CALCULATING RV SEMI-AMPLITUDE

F# 8': =1'> $%& 1:6&)34-/ (-/:& #') &-4% '# $%& #'//'>31; 0-)-6&$&). #') - 0/-1&$ -12 3$. %'.$ .$-)5

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! M

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K star = 0.6395 m s"1 1 day

P

#

$ %

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1

3 M planet sin i

M Earth

#

$ %

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M Sun

M star

#

$ %

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2 / 3

1

1" e2

K star = 0.0895 m s"1 1 yr

P

#

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1

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M Earth

#

$ %

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M Sun

M star

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K star = 203.255 m s"1 1 day

P

#

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M Jupiter

#

$ % %

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M Sun

M star

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2 / 3

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K star = 28.435 m s"1 1 yr

P

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