Download The Two-Body Problem

The Two-Body Problem
Barbieri Chapter 12
Kathy Geise
12.1 Barycentric Treatment
• The gravitational force between two particles is
directed along the line joining them and has an
intensity that depends only on their relative
distance.
• Inertial reference frame
• Central force and conservative field
• Force derivable from potential function
• Then,
– F = G m1m2/|r1-r2|2
– U(r1,r2)=-Gm1m2/|r1-r2|
Relative Position*
• The gravitational potential, U, is a
conservative field.
• U is independent of the direction of (r1-r2)
• U depends on the magnitude |r1-r2|
• Relative Position
r=r -r
1
r=r1-r2
*Taylor, Chapter 8.1, page 294
2
r1
r2
Potential Energy
• U=U(r) is the potential energy of the
system
• U(r) is a scalar
• The potential energy depends only upon
the magnitude, r, of the relative position, r.
Barycenter, B
• Barycenter, B, is the center of mass of the
system.
• For a two particle system, it is a point
always on the line joining the two particles.
• The barycenter is closer to the heavier of
the two masses.
Reduced Mass, μ
• m1r1+m2r2=0
 Place B at the origin of
the system
m1
CM
X
m2
 (12.4)
R
r1
r2
• r2 = r * m1/(m1+m2)
• From Taylor, CM,
position R
 R=m1r1+m2r2/M
 Reduced mass
 μ = m1m2/m1+m2
 (8.11)
Lagrange Function
• The Lagrange function is
the difference between
the kinetic energy and
potential energy
• L=T-U
• In the barycentric system
the Lagrange function is
that of a single particle of
mass m in an external
field U(r) (B. pg 201)

1 2 1
2
L  m1 r 1  m2 r 2  U (r )
2
2
Barbieri page 200
 2
1
1 2
L  M R   r  U (r )
2
2
Taylor page 297
Effective Potential Energy
• The radial movement (of
the barycenter) takes
place as if the effective
potential was the sum of
that due to an external
force plus a centrifugal
potential Ucf(r).
• The motion of the relative
coordinate, with given
angular momentum, is
equivalent to the motion
of a particle in one
(radial) dimension.
2
U eff
K
 U (r ) 
2
2mr
Barbieri page 203
U eff  U (r ) 
l
2
2r
2
Taylor page 320
12.2 Gravitational Attraction
• In this case, the
U0
U0
U (r )  
, F (r )  2 ,U 0  Gm1m2
potential energy is
r
r
gravitational potential
energy
2 EK 2
e  1
• Eccentricity, e
2
mU 0
• Trajectory, r(φ)
• The sign of the total
1 1
energy, E, determines
 [1  e cos(  0 )]
r p
the value of the
eccentricity
Kepler Orbits
•
•
•
•
If E<0 then 0<=e<1 the orbit is an ellipse
If E=0 then e=1 the orbit is a parabola
If E>0 then e>1 the orbit is an hyperbole
When e=0 the orbit is a circle
• This applet illustrates circle versus ellipse
• http://physics.syr.edu/courses/java/mc_html/kepler_frame.html
• This applet show energy
• http://www.phys.hawaii.edu/~teb/java/ntnujava/Kepler/Kepler.html
Intermediate Polar Binary System
DQ Hercules
http://www.britastro.org/vss/00191a.html
http://antwrp.gsfc.nasa.gov/apod/ap031110.html
•
First slide: http://antwrp.gsfc.nasa.gov/apod/ap050830.html
 Albireo as seen through a small telescope. Albireo is a binary star system. The
brighter yellow star is also a binary. The stars are far from each other and take
about 75,000 years to complete a single orbit.