Specific angular momentum

Short description: Vector quantity in celestial mechanics

In celestial mechanics, the specific relative angular momentum (often denoted h→ or 𝐡) of a body is the angular momentum of that body divided by its mass.[1] In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum, divided by the mass of the body in question.

Specific relative angular momentum plays a pivotal role in the analysis of the two-body problem, as it remains constant for a given orbit under ideal conditions. "Specific" in this context indicates angular momentum per unit mass. The SI unit for specific relative angular momentum is square meter per second.

## Contents

* 1 Definition
* 2 Proof of constancy in the two body case
* 2.1 Proof
* 3 Kepler's laws of planetary motion
* 3.1 First law
* 3.2 Second law
* 3.3 Third law
* 4 See also
* 5 References

## Definition

The specific relative angular momentum is defined as the cross product of the relative position vector 𝐫 and the relative velocity vector 𝐯. 𝐡=𝐫×𝐯=𝐋m

where 𝐋 is the angular momentum vector, defined as 𝐫×m𝐯.

The 𝐡 vector is always perpendicular to the instantaneous osculating orbital plane, which coincides with the instantaneous perturbed orbit. It is not necessarily perpendicular to the average orbital plane over time.

## Proof of constancy in the two body case

Distance vector 𝐫, velocity vector 𝐯, true anomaly θ and flight path angle ϕ of m2 in orbit around m1. The most important measures of the ellipse are also depicted (among which, note that the true anomaly θ is labeled as ν).

Under certain conditions, it can be proven that the specific angular momentum is constant. The conditions for this proof include:

* The mass of one object is much greater than the mass of the other one. (m1≫m2)
* The coordinate system is inertial.
* Each object can be treated as a spherically symmetrical point mass.
* No other forces act on the system other than the gravitational force that connects the two bodies.

### Proof

The proof starts with the two body equation of motion, derived from Newton's law of universal gravitation:

𝐫¨+Gm1r2𝐫r=0

where:

* 𝐫 is the position vector from m1 to m2 with scalar magnitude r.
* 𝐫¨ is the second time derivative of 𝐫. (the acceleration)
* G is the Gravitational constant.

The cross product of the position vector with the equation of motion is:

𝐫×𝐫¨+𝐫×Gm1r2𝐫r=0

Because 𝐫×𝐫=0 the second term vanishes:

𝐫×𝐫¨=0

It can also be derived that: ddt(𝐫×𝐫˙)=𝐫˙×𝐫˙+𝐫×𝐫¨=𝐫×𝐫¨

Combining these two equations gives: ddt(𝐫×𝐫˙)=0

Since the time derivative is equal to zero, the quantity 𝐫×𝐫˙ is constant. Using the velocity vector 𝐯 in place of the rate of change of position, and 𝐡 for the specific angular momentum: 𝐡=𝐫×𝐯 is constant.

This is different from the normal construction of momentum, 𝐫×𝐩, because it does not include the mass of the object in question.

## Kepler's laws of planetary motion

Main article: Kepler's laws of planetary motion

Kepler's laws of planetary motion can be proved almost directly with the above relationships.

### First law

The proof starts again with the equation of the two-body problem. This time the cross product is multiplied with the specific relative angular momentum 𝐫¨×𝐡=−μr2𝐫r×𝐡

The left hand side is equal to the derivative ddt(𝐫˙×𝐡) because the angular momentum is constant.

After some steps (which includes using the vector triple product and defining the scalar r˙ to be the radial velocity, as opposed to the norm of the vector 𝐫˙) the right hand side becomes: −μr3(𝐫×𝐡)=−μr3((𝐫⋅𝐯)𝐫−r2𝐯)=−(μr2r˙𝐫−μr𝐯)=μddt(𝐫r)

Setting these two expression equal and integrating over time leads to (with the constant of integration 𝐂) 𝐫˙×𝐡=μ𝐫r+𝐂

Now this equation is multiplied (dot product) with 𝐫 and rearranged 𝐫⋅(𝐫˙×𝐡)=𝐫⋅(μ𝐫r+𝐂)⇒(𝐫×𝐫˙)⋅𝐡=μr+rCcos⁡θ⇒h2=μr+rCcos⁡θ

Finally one gets the orbit equation[1] r=h2μ1+Cμcos⁡θ

which is the equation of a conic section in polar coordinates with semi-latus rectum p=h2μ and eccentricity e=Cμ.

### Second law

The second law follows instantly from the second of the three equations to calculate the absolute value of the specific relative angular momentum.[1]

If one connects this form of the equation dt=r2hdθ with the relationship dA=r22dθ for the area of a sector with an infinitesimal small angle dθ (triangle with one very small side), the equation dt=2hdA

### Third law

Kepler's third is a direct consequence of the second law. Integrating over one revolution gives the orbital period[1] T=2πabh

for the area πab of an ellipse. Replacing the semi-minor axis with b=ap and the specific relative angular momentum with h=μp one gets T=2πa3μ

There is thus a relationship between the semi-major axis and the orbital period of a satellite that can be reduced to a constant of the central body.

## See also

* Specific orbital energy, another conserved quantity in the two-body problem.
* Classical central-force problem § Specific angular momentum

## References

* v
* t
* e

Gravitational orbits

Types|

| General|

* Box
* Capture
* Circular
* Elliptical / Highly elliptical
* Escape
* Horseshoe
* Hyperbolic trajectory
* Inclined / Non-inclined
* Lagrangian point
* Osculating
* Parabolic trajectory
* Parking
* Prograde / Retrograde
* Synchronous
* semi
* sub
* Transfer orbit

|

Geocentric|

* Geosynchronous
* Geostationary
* Geostationary transfer
* Graveyard
* High Earth
* Low Earth
* Medium Earth
* Molniya
* Near-equatorial
* Orbit of the Moon
* Polar
* Sun-synchronous
* Tundra

About other points|

* Mars
* Areosynchronous
* Areostationary
* Lagrange points
* Distant retrograde
* Halo
* Lissajous
* Lunar
* Sun
* Heliocentric
* Earth's orbit
* Heliosynchronous

Parameters|

* Shape
* Size

* e Eccentricity
* a Semi-major axis
* b Semi-minor axis
* Q, q Apsides

|

Orientation|

* i Inclination
* Ω Longitude of the ascending node
* ω Argument of periapsis
* ϖ Longitude of the periapsis

Position|

* M Mean anomaly
* ν, θ, f True anomaly
* E Eccentric anomaly
* L Mean longitude
* l True longitude

Variation|

* T Orbital period
* n Mean motion
* v Orbital speed
* t0 Epoch

Maneuvers|

* Bi-elliptic transfer
* Collision avoidance (spacecraft)
* Delta-v
* Delta-v budget
* Gravity assist
* Gravity turn
* Hohmann transfer
* Inclination change
* Low energy transfer
* Oberth effect
* Phasing
* Rocket equation
* Rendezvous
* Transposition, docking, and extraction

Orbital mechanics|

* Celestial coordinate system
* Characteristic energy
* Escape velocity
* Ephemeris
* Equatorial coordinate system
* Ground track
* Hill sphere
* Interplanetary Transport Network
* Kepler's laws of planetary motion
* Lagrangian point
* n-body problem
* Orbit equation
* Orbital state vectors
* Perturbation
* Retrograde motion
* Specific orbital energy
* Specific relative angular momentum
* Two-line elements

* List of orbits

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1. ↑ 1.0 1.1 1.2 1.3 Vallado, David A. (2001). Fundamentals of astrodynamics and applications (2nd ed.). Dordrecht: Kluwer Academic Publishers. pp. 20–30. ISBN 0-7923-6903-3.

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