as 3004 stellar dynamics energy of orbit energy of orbit is e = t+w; (ke + pe) –where v is the...

AS 3004 Stellar Dynamics

Energy of Orbit

• Energy of orbit is E = T+W; (KE + PE)

– where V is the speed in the

relative orbit

Hence the total Energy is

is always negative.

Binding Energy = -E > 0

KE T 12m1v1

2 12m2v2

2 m1m2

2(m1 m2 )V 2

V 2 G(m1 m2 )2r 1a

PE W Gm1m2

r 2r

Gm1m2

r

E T W Gm1m2

2a


Angular momentum of the orbit

• Angular momentum vector J, defines orbital plane– J = m1L1 + m2L2 and L2 =k l = G(m1 + m2 )a(1-e2)

and L12 =k l = Gm2

3 /(m1 + m2 ) 2 a1 (1-e2)

and a1/a = m2 /(m1 + m2 )

– same for L2

– hence

therefore

and the final expression for J is

)1()(

2

)1()(

)( ;

)(

2

21

212

2

21

22

212

221

21

2221

22

1

emmP

mmaJ

eamm

mGmJ

Lmm

mLL

mm

mL


Orbital Angular momentum

• Given masses m1,m2 and Energy E,– the angular momentum J determines the shape of the orbit

– ie the eccentricity (or the conic section parameter l)

• For given E, – circular orbits have maximum J

– J decreases as e 1

– orbit becomes rectilinear ellipse

• relation between E, and J very important indetermining when systems interact

mass exchange and orbital evolution


Orbit in Space

– N: ascending node. projection of orbit onto sky at place of maximum receding velocity

– angles Ø is the true anomoly, longitude of periastron and , the longitude of the ascending node


Elements of the orbit

• using angles i, inclination and , we have

• Lx = Lsin(i)sin(), Ly = -Lsin(i)cos(), Lz = Lcos(i)

• hence, to define orbit in plane of sky, we have– quantities (a,e,i,,,T) are called the elements of the ellipse

– provide size, shape, and orientation of the orbit in space

and time!

• N.B. difference between barycentre and relativeorbits

– radial velocity variations give information in the barycentre

orbits

– light curves give information in terms of the relative orbit.


Applications to Spectroscopic Binary Systems

• star at position P2, polar coords are (r, +)– project along line of nodes: r cos(+) and

– Perpendicular to line of nodes, r sin(+)

– project this along line of sight: z = r sin(+) sin(i)

• radial velocities along line-of sight is then

using

and Kepler’s 2nd Law

)cos()cos(1

sin2

/)1(2r

)cos1/()sin()cos1/()1(

)cos()sin(sin

2

21222

2

eeP

iaV

Pea

erereear

rrizV

rad

rad


Spectroscopic Orbital Velocities

• The radial velocity is usually expressed in the form

– where, K, is the semi-amplitude of the velocity

defined as

K has maximum and minim values at ascending and descending nodes when() = 0 and () = , hence

– if e=0, Vrad is a cosine curve

– as e > 0, velocity becomes skewed.

Vrad K cos( ) ecos

K (2asin i) (P 1 e2 )

AK ecos 1 ;B K ecos 1 K (A B) / 2; A 0; B 0


Radial Velocities

• Radial velocity for two stars in circular orbit– with K1=100 km/s and K2 = 200 km/s q=m2/m1 = 0.5


Radial Velocities

• Radial velocity for two stars in eccentric orbit– with e=0.1, = 45o


Radial Velocities

• Radial velocity for two stars in eccentric orbit– with e=0.3 , = 0o


Radial Velocities



Minimum masses– From Ki (2ai sin i) (P 1 e2 )

ai sin i 1 e2

2PK i

m1a1 m2a2 ; and G(m1 m2 ) 4 2a3 / P2

m1 m1a1a2

4 2a3

GP2

m1 sin3 i 4 2

GP2

a3 sin3 i

1 a1 sin ia2 sin i

m1 sin3 i 12G 1 e2 3

2 K1 K2 2K2P


Minimum masses II• In typical astronomical units

– solar masses, km/s for velocities, and days for periods

• a1,2 sin(i) = 1.3751 x 104 (1-e2) 1/2 K1,2 P km– projected semi-major axes

• m1,2 sin3(i) = 1.0385 x 10-7 (1-e2)3/2 (K1+K2)2 K2,1 P solar masses

• minimum mass only attainable for double-lined spectroscopic binaries (know both K1 and K2 )

– known as SB2s– otherwise have mass function

1/(2G) = 1.0385 x 10-7 when measuring in solar masses

PKeGmm

immf 1

23

22

21

332 1

2

1

)(

sin)(

as 3004 stellar dynamics energy of orbit energy of orbit is e = t+w; (ke + pe) –where v is the...

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