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Overview of the solution of 2-body gravity problem.
● Reduced-mass system → 1-body gravity problem;
● Conservations: E, P, L
● Three orbits: elliptical, parabolic, hyperbolic;
● Kepler’s law (= the orbits of the planets in the solar system) is a special case of the 2-body problem;
● Orbital Elements
Kepler’s law and Astrometric Orbit Determination
Kepler’s Laws of Planetary Motion
Sun
Planet
Kepler’s Laws of Planetary Motion
“Reminder of the Gravity”
Summation over all bodies in a given system
Momentum conservation
Angular Momentum conservation
Energy conservation
Note the vector direction of the torque and angular momentum
Linear Momentum Conservation
Gravity General
Two-body Gravity Problems in Astronomy
Critical to understand binary stars (including black holes & neutron stars), planets, galaxy mergers, etc
Antennae Galaxies
(Galaxy Merger)
Black Hole
(X-ray Binary)
● Generally, the Reduced-mass Solution (one moving body)
● If M1 << M2 (and bound) Kepler’s Law (one moving body problem)
Gravity General
From van Kerkwijk
“Reduced-mass Solution”
Gravity General Let’s change the 2-body problem to 1-body problem around the center of mass!
Describable by rμ & vμ
L & Econservation
Equation of a conic section.
Three solutions for rfor different e.
Ellipse!
Eccentricity
Parabola!
Hyperbola!
“Reduced-mass Solution”
Let’s change the 2-body problem to 1-body problem around the center of mass!
Gravity General
Conic Sections
Parabola: The set of all points in the plane whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are always equal.
Hyperbola: The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
Ellipse: A curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve.
Conic Sections
“Conic Sections”2-body gravity system predicts orbits of conic sections
Gravity General
● Three different orbits are possible: elliptical, parabolic, & hyperbolic;● Elliptical for bound systems (E < 0; e.g., solar system planets);parabolic for E = 0; hyperbolic for open (E > 0; unbound) systems;● “Kepler’s laws represent the case for bound systems (motions around CM)”
Why couldn’t Kepler know the existence of the parabolic and/or hyperbolic orbits?
Where is the CM of the Solar system?
Ellipse!
Parabola!
Hyperbola!
Newton’s form of Kepler’s 3rd law (see next slides)
General solution of bound 2-body orbit
“Reduced-mass Solution”
Let’s change the 2-body problem to 1-body problem around the center of mass!
Gravity General
2-Body Problem
F1 = m1v12 / r1 = 42m1r1 / P2 ( v1 = 2r1/P)
F2 = m2v22 / r2 = 42m2r2 / P2
r1 / r2 = m2 / m1 & a = r1 + r2
r1 = m2a / (m1 + m2)or r2 = m1a / (m1 + m2)
Fgrav = F1 = F2 = Gm1m2 / a2
Gravity General
Newton’s form of Kepler’s 3rd law
You can drive Kepler’s 3rd law directly from the gravity.
For a bound system:
Hyperbola!
Orbits of 2-body gravity problemCircular Velocity Escape Velocity
• Circular orbit is the special case of the elliptic orbits.
• v < vE → Ellipse
• v = vE → Parabola
• v > vE → Hyperbola
Gravity General
The dynamics of many important astronomical objects can be treated as 2-body (gravity) problem.
Examples: binary stars, planets rotating around a star, galaxy collisions, etc.
Gravity General
“Elliptical orbit for a bound 2-body system”
Two bodies with similar mass orbiting around a common barycenter with elliptic orbits.
Example: 2-body orbit
Gravity General
Example: 2-body orbit
“Comets have elliptical or parabolic/hyperbolic orbits”
Orbits of Comet Kohoutek (red) and Earth (blue), illustrating the
high eccentricity of the orbit and more rapid motion when closer
to the Sun.
Comets of hyperbolic orbits will leave the Solar system at the end (cf: non-periodic comets)
Gravity General
From van Kerkwijk
“Planets & Satellites”Example: 2-body orbit
Gravity General
Solar System Planets (in elliptical orbits)
Planetary orbits are inclined.
Orbital Elements
So how many parameters (= orbital elements) do we need to describe a planetary orbit?
Orbital Elements
So how many parameters (= orbital elements) do we need to describe a planetary orbit?
Reference plane (e.g., ecliptic, equatorial)
Planetary orbital plane
Planetary orbital planeReference plane (e.g., ecliptic, equatorial)
Orbital Elements
So how many parameters (= orbital elements) do we need to describe a planetary orbit?
Reference plane (e.g., ecliptic, equatorial)
Planetary orbital plane
Planetary orbital planeReference plane (e.g., ecliptic, equatorial)
1. We need to know the size and shape of the elliptical orbit itself.
2. We need to know the orientations of the orbit.
3. We need a reference time.
Orbital Elements
Vernal point Ascending node
1. Semi-major axis (a) and eccentricity (e).
2. Inclination (i), longitude of ascending node (), argument of perihelion (): 3 angles; and are measured in the counter-clockwise direction and show the location of the node and rotation of the plane, respectively.
3. Epoch of perihelion ().
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