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 ().