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Specific angular momentum

Specific angular momentum, denoted as \vec{h}, is a fundamental vector quantity in classical mechanics and astrodynamics, representing the angular momentum of a body per unit mass and defined as the cross product of its position vector \vec{r} and velocity vector \vec{v}, yielding \vec{h} = \vec{r} \times \vec{v}.^[1] With SI units of square meters per second (m²/s), it quantifies the rotational motion intrinsic to the body's trajectory relative to a reference point, such as a central gravitational body.^[2] This quantity is particularly crucial in orbital contexts, where its magnitude h = r v \sin \phi (with \phi the angle between \vec{r} and \vec{v}) determines key orbital characteristics without dependence on the body's total mass.^[3] In the two-body problem, specific angular momentum is conserved due to the central nature of the gravitational force, which exerts no torque on the orbiting body, ensuring \frac{d\vec{h}}{dt} = 0.^[1] The vector \vec{h} is always perpendicular to the plane of motion defined by \vec{r} and \vec{v}, thereby fixing the orbital plane and orientation in space.^[2] This conservation property underpins Kepler's second law of planetary motion, where the areal velocity \frac{dA}{dt} = \frac{h}{2} remains constant, meaning equal areas are swept by the position vector in equal times.^[1] Perturbations, such as atmospheric drag or non-spherical gravitational fields, can alter \vec{h}, but in ideal central-force scenarios, it remains invariant throughout the orbit.^[4] Specific angular momentum plays a pivotal role in characterizing orbital shapes and parameters within conic section trajectories, including ellipses, parabolas, and hyperbolas.^[1] For elliptical orbits, h relates directly to the semi-major axis a and eccentricity e via the vis-viva equation and specific energy considerations, enabling predictions of perigee and apogee distances.^[2] Its magnitude influences the flight path angle and radial velocity components, which adjust dynamically to maintain constancy as the orbit progresses.^[3] In broader astrophysical contexts, such as planetary formation or binary star systems, variations in specific angular momentum help model disk evolution and transfer processes.^[5] Applications of specific angular momentum extend to spacecraft mission design, where it is used to compute transfer orbits, such as Hohmann transfers, by matching \vec{h} at key points like perigee.^[1] In satellite operations, monitoring changes in h detects perturbations, aiding in orbit maintenance and collision avoidance.^[2] Furthermore, in cometary and asteroid dynamics, low specific angular momentum values indicate capture origins, while higher values suggest hyperbolic escapes from solar systems.^[6]

Basic Concepts

Definition

Specific angular momentum, denoted as \mathbf{h}, is defined as the angular momentum vector \mathbf{L} of a body divided by its mass m, such that \mathbf{h} = \frac{\mathbf{L}}{m}.^[4] This quantity represents the angular momentum on a per-unit-mass basis, commonly used in classical mechanics to describe the rotational dynamics of particles or extended bodies relative to a reference point.^[3] Physically, specific angular momentum quantifies the degree of rotational motion per unit mass about the chosen origin, capturing how "far" and "fast" the motion is from pure radial translation.^[4] In isolated systems subject to no external torques, \mathbf{h} is conserved, mirroring the conservation of total angular momentum but independent of the system's total mass.^[7] As a vector, \mathbf{h} points perpendicular to the plane containing the position and velocity vectors, thereby defining the plane of motion for orbital or rotational trajectories.^[3] In the International System of Units (SI), specific angular momentum has dimensions of length squared per time, or [L² T⁻¹], corresponding to square meters per second (m²/s).^[4] Unlike total angular momentum, which scales with mass and has units of kilogram square meters per second (kg·m²/s), the specific form is mass-invariant, making it particularly advantageous for analyzing systems with variable mass—such as rockets—or for comparing motions on a per-particle basis without scaling effects from differing masses.^[3]

Mathematical Formulation

The specific angular momentum \mathbf{h} is a vector quantity defined as the cross product of the position vector \mathbf{r} from a chosen reference point (typically the center of mass or the primary body) and the velocity vector \mathbf{v} of the body, given by

\mathbf{h} = \mathbf{r} \times \mathbf{v}.

This formulation arises in classical mechanics for point masses and is particularly relevant in orbital contexts where the reference point is fixed.

\] The vector $\mathbf{h}$ is [perpendicular](/page/Perpendicular) to both $\mathbf{r}$ and $\mathbf{v}$, thus lying normal to the plane of motion.\[

The magnitude of the specific angular momentum is h = |\mathbf{h}| = r v \sin \phi, where r = |\mathbf{r}|, v = |\mathbf{v}|, and \phi is the angle between \mathbf{r} and \mathbf{v}; this follows directly from the geometric definition of the cross product magnitude.

\] In polar coordinates for planar motion, where the position is described by radial distance $ r $ and angular position $ \theta $, the magnitude simplifies to $ h = r^2 \dot{\theta} $, with $ \dot{\theta} = d\theta/dt $ representing the angular speed.\[

The Cartesian components of \mathbf{h} are expressed as

h_x = y v_z - z v_y, \quad h_y = z v_x - x v_z, \quad h_z = x v_y - y v_x,

where (x, y, z) and (v_x, v_y, v_z) are the components of \mathbf{r} and \mathbf{v}, respectively.$$] In the context of Keplerian orbits, the magnitude h relates to the orbital parameters through h = \sqrt{\mu p}, where \mu = G(M + m) is the standard gravitational parameter (with G as the gravitational constant, M the mass of the primary body, and m the mass of the orbiting body) and p is the semi-latus rectum of the conic section orbit.[$$

Conservation Laws

In Central Force Fields

In central force fields, the force acting on a particle is directed along the position vector \mathbf{r} from the force center and depends only on the magnitude r = |\mathbf{r}|, expressed as \mathbf{F} = f(r) \hat{\mathbf{r}}, where \hat{\mathbf{r}} = \mathbf{r}/r and f(r) is a scalar function.^[8]^[9] The torque \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} vanishes because \mathbf{F} is parallel to \mathbf{r}, yielding \boldsymbol{\tau} = 0.^[8] This implies that the time derivative of the angular momentum \mathbf{L} = m \mathbf{r} \times \mathbf{v} is zero, \frac{d\mathbf{L}}{dt} = \boldsymbol{\tau} = 0, so \mathbf{L} is conserved; consequently, the specific angular momentum \mathbf{h} = \mathbf{r} \times \mathbf{v} (per unit mass) is also conserved, with \frac{d\mathbf{h}}{dt} = 0.^[9] A general proof follows from the definition:

\frac{d\mathbf{h}}{dt} = \frac{d}{dt} (\mathbf{r} \times \mathbf{v}) = \mathbf{v} \times \mathbf{v} + \mathbf{r} \times \mathbf{a} = \mathbf{r} \times \mathbf{a},

where \mathbf{a} = \mathbf{v}/dt = \mathbf{F}/m is parallel to \mathbf{r} for a central force, making \mathbf{r} \times \mathbf{a} = 0.^[10] Thus, \mathbf{h} remains constant in magnitude and direction. Conservation of \mathbf{h} confines the particle's motion to a plane perpendicular to \mathbf{h}, as any out-of-plane component would vary.^[8] Additionally, the areal velocity is constant: the rate at which area is swept by the position vector is dA/dt = |\mathbf{h}|/2, leading to equal areas in equal times for any central force.^[10]^[9] Examples include the gravitational force, \mathbf{F} = -GMm/r^2 \hat{\mathbf{r}} (an inverse-square law), and more generally, inverse-power-law forces \mathbf{F} \propto r^{-n} \hat{\mathbf{r}} (for n > 0), where the central symmetry ensures conservation holds.^[8] The isotropic harmonic oscillator, \mathbf{F} = -k \mathbf{r}, also exemplifies this for linear restoring forces.^[9]

Proof in the Two-Body Problem

In the two-body problem, two point masses m_1 and m_2 interact solely through their mutual gravitational attraction, with no external forces acting on the system.^[11] The dynamics can be reduced to an equivalent one-body problem by introducing the relative position vector \mathbf{r} = \mathbf{r}_1 - \mathbf{r}_2, where \mathbf{r}_1 and \mathbf{r}_2 are the position vectors of the two masses relative to an inertial frame.^[12] The reduced mass is defined as \mu = \frac{m_1 m_2}{m_1 + m_2}, which governs the motion of this effective single body in the relative coordinate system.^[11] The gravitational force between the masses is central and directed along the line joining them, given by \mathbf{F} = -\frac{[G](/page/G) m_1 m_2}{r^2} \hat{\mathbf{r}} = -\frac{[G](/page/G) m_1 m_2 \mathbf{r}}{r^3}, where [G](/page/G) is the gravitational constant, r = |\mathbf{r}|, and \hat{\mathbf{r}} = \mathbf{r}/r.^[12] This force law ensures that the acceleration of the relative vector is \ddot{\mathbf{r}} = -\frac{[G](/page/G) (m_1 + m_2)}{r^3} \mathbf{r}, as derived from Newton's second law applied to the reduced mass system: \mu \ddot{\mathbf{r}} = \mathbf{F}.^[11] The specific angular momentum \mathbf{h} for the relative motion is defined as \mathbf{h} = \mathbf{r} \times \mathbf{v}, where \mathbf{v} = \dot{\mathbf{r}} is the relative velocity; this quantity has units of area per unit time and is independent of the reduced mass.^[13] To prove its constancy, compute the time derivative:

\frac{d\mathbf{h}}{dt} = \frac{d}{dt} (\mathbf{r} \times \mathbf{v}) = \dot{\mathbf{r}} \times \mathbf{v} + \mathbf{r} \times \dot{\mathbf{v}} = \mathbf{v} \times \mathbf{v} + \mathbf{r} \times \mathbf{a} = \mathbf{r} \times \mathbf{a},

where \mathbf{a} = \ddot{\mathbf{r}} is the relative acceleration.^[12] Substituting the gravitational acceleration yields

\mathbf{r} \times \mathbf{a} = \mathbf{r} \times \left( -\frac{G (m_1 + m_2)}{r^3} \mathbf{r} \right) = -\frac{G (m_1 + m_2)}{r^3} (\mathbf{r} \times \mathbf{r}) = \mathbf{0},

since the cross product of any vector with itself is zero.^[11] Thus, \frac{d\mathbf{h}}{dt} = \mathbf{0}, confirming that \mathbf{h} is constant in both magnitude and direction.^[12] This proof holds under the assumptions of an inertial reference frame, absence of external forces or torques on the system, and treatment of the masses as point particles (or spherically symmetric bodies, ensuring the gravitational force remains central).^[11] For an orbiting body where one mass dominates (e.g., m_2 \ll m_1), the specific angular momentum approximates \mathbf{h} \approx \mathbf{r}_2 \times \mathbf{v}_2 relative to the primary body, retaining its constant magnitude h = r v_\perp (with v_\perp the perpendicular velocity component) and direction perpendicular to the orbital plane.^[13]

Implications for Keplerian Orbits

Second Law: Equal Areas

Kepler's second law, also known as the law of equal areas, states that a line segment joining a planet to the Sun sweeps out equal areas during equal intervals of time. This empirical observation, derived by Johannes Kepler from Tycho Brahe's precise astronomical data in the early 17th century, was later explained by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (1687) as a direct consequence of angular momentum conservation under a central gravitational force.^[14] In orbital mechanics, the specific angular momentum \mathbf{h} = \mathbf{r} \times \mathbf{v} remains constant for motion in a central force field, as proven for the two-body problem. The magnitude h = |\mathbf{h}| determines the areal velocity, the rate at which area is swept by the radius vector \mathbf{r}. The infinitesimal area dA swept in time dt is given by dA = \frac{1}{2} r^2 d[\theta](/page/Theta), where \theta is the polar angle, leading to the areal velocity \frac{dA}{dt} = \frac{1}{2} r^2 \dot{[\theta](/page/Theta)}. Since the tangential component relates to h = r^2 \dot{\theta} (or more generally \frac{dA}{dt} = \frac{1}{2} |\mathbf{r} \times \mathbf{v}| = \frac{h}{2}), the areal velocity is constant at \frac{h}{2}.^[15]^[16] To derive this explicitly in polar coordinates, consider the transverse equation of motion for a central force, which yields \frac{d}{dt}(r^2 \dot{\theta}) = 0, implying r^2 \dot{\theta} = h (constant). Thus, \dot{\theta} = \frac{h}{r^2}, and substituting into the area element gives \frac{dA}{dt} = \frac{1}{2} h. Integrating over a finite time interval \Delta t results in the total area swept \Delta A = \frac{h}{2} \Delta t, demonstrating that equal time intervals correspond to equal areas regardless of the planet's varying speed.^[16]^[15] Geometrically, this conservation implies that the radius vector accelerates near the central body (where r is small, so \dot{\theta} increases to maintain h) but slows farther away, yet the net area swept remains uniform over time, providing a kinematic signature of central force dynamics without dependence on the force law's specifics.^[14]

First Law: Elliptical Orbits

In the context of Keplerian orbits under an inverse-square central force, the shape of the trajectory is determined by combining the conservation of specific angular momentum h with the conservation of total mechanical energy \mathcal{E}. The specific angular momentum h = r^2 \dot{\theta} remains constant, allowing the radial equation of motion to be expressed in terms of the polar angle \theta. By introducing the substitution u = 1/r (Binet's transformation), the radial dynamics reduce to a second-order differential equation: \frac{d^2 u}{d\theta^2} + u = \frac{\mu}{h^2}, where \mu is the standard gravitational parameter (product of the gravitational constant and central mass).^[17] The general solution is u = \frac{\mu}{h^2} + A \cos(\theta - \theta_0), which inverts to the polar orbit equation r(\theta) = \frac{h^2 / \mu}{1 + e \cos(\theta - \theta_0)}, where e is the eccentricity and \theta_0 sets the orientation of the periapsis.^[10] This equation describes conic sections with the central body at one focus, classified by the eccentricity e: for e < 1, the orbit is an ellipse (bound trajectory); e = 0 yields a circle (special elliptical case); e = 1 a parabola (marginally unbound); and e > 1 a hyperbola (unbound scattering orbit).^[10] The parameter p = h^2 / \mu is the semi-latus rectum, representing the radial distance at \theta = 90^\circ from the focus, which scales the overall size of the conic. The eccentricity e emerges from the energy conservation, specifically e = \sqrt{1 + \frac{2 \mathcal{E} h^2}{\mu^2}}, linking the orbit's shape directly to the specific values of h and \mathcal{E}.^[17] The role of specific angular momentum h is pivotal in shaping elliptical orbits: for a fixed total energy \mathcal{E} < 0 (bound motion), larger h reduces e, resulting in more nearly circular orbits by increasing the semi-latus rectum p and widening the radial excursion.^[18] This can be understood through the effective potential framework, where the radial motion is governed by an effective potential V_{\text{eff}}(r) = -\frac{\mu}{r} + \frac{h^2}{2 r^2}. The centrifugal term \frac{h^2}{2 r^2} creates an angular momentum barrier that prevents collapse to the center, confining bound orbits between turning points r_{\min} and r_{\max} for \mathcal{E} < 0, with the barrier's strength scaling as h^2.^[18] Thus, h not only determines the angular scale of the orbit but also stabilizes elliptical paths against radial perturbations.

Third Law: Harmonic Periods

Kepler's third law relates the orbital period T to the semi-major axis a of an elliptical orbit around a central mass, stating that T^2 \propto a^3. In the Newtonian two-body problem, this takes the precise form T = 2\pi \sqrt{\frac{a^3}{\mu}}, where \mu = G(M + m) is the standard gravitational parameter, with G the gravitational constant, M the central mass (typically much larger than the orbiting mass m), and a the semi-major axis representing the average scale of the orbit.^[19] This formula arises from the balance of gravitational attraction and the orbital dynamics under inverse-square law forces.^[3] The specific angular momentum h, conserved in central force fields, links to Kepler's third law through the geometry of the elliptical orbit. For an ellipse, h = \sqrt{\mu p}, where p is the semi-latus rectum, and p = a(1 - e^2) with e the eccentricity; thus, h^2 = \mu a (1 - e^2). The period T depends solely on a and \mu, independent of e, so h influences T indirectly by determining the orbit's shape via e. As detailed in the first law, elliptical parameters like a and e define the conic section under gravitational forces.^[3] A derivation of the period formula integrates the orbital motion using the vis-viva equation and specific angular momentum. The vis-viva equation, from energy conservation, gives the speed as v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right), relating instantaneous velocity to radial distance r. Combined with h = r^2 \dot{\theta} (angular momentum conservation), the time element is dt = \frac{r^2}{h} d\theta. Substituting the conic orbit equation r = \frac{h^2 / \mu}{1 + e \cos \theta} and integrating over one full revolution (\theta from 0 to $2\pi) yields T = 2\pi \sqrt{\frac{a^3}{\mu}}, confirming the third law without reliance on area rates.^[4] This relation generalizes to any conic-section orbit in a central gravitational field, but the orbital period T is defined only for bound elliptical orbits (e < 1); for parabolic (e = 1) or hyperbolic (e > 1) trajectories, no periodic closure occurs.^[20]

Modern Applications

Spacecraft Trajectories

In spacecraft trajectory design, specific angular momentum plays a crucial role in determining orbital parameters and enabling efficient maneuvers. For instance, the Hohmann transfer orbit, an elliptical path used to shift a spacecraft between two coplanar circular orbits, relies on impulsive delta-v burns at perigee and apogee to alter the specific angular momentum magnitude while conserving its direction. This method minimizes propellant consumption by achieving the transfer with the least energy, as the semi-major axis of the transfer ellipse is the average of the initial and final orbital radii, directly influencing the required change in specific angular momentum.^[5]^[21] The vector components of specific angular momentum are integral to defining key orbital elements in spacecraft mission planning. Its direction, perpendicular to the orbital plane, specifies the inclination—the angle between the orbital plane and the reference equatorial plane—and the right ascension of the ascending node, which locates the point where the orbit crosses the equator from south to north. Meanwhile, the magnitude of specific angular momentum, for a given semi-major axis, governs the orbit's eccentricity, quantifying its deviation from circularity; higher magnitudes correspond to more circular orbits, while lower ones yield more elliptical paths. These relationships allow engineers to predict and adjust spacecraft orientation relative to Earth or other central bodies during orbit insertion or correction phases. Real-world spacecraft orbits experience perturbations that gradually modify specific angular momentum, necessitating active control. In low Earth orbit, atmospheric drag exerts a tangential force that primarily reduces the semi-major axis and eccentricity but affects specific angular momentum magnitude slowly over multiple orbits, as the drag acceleration is on the order of micro-g. Similarly, Earth's J2 oblateness perturbation, arising from its equatorial bulge, induces precession in the orbital plane and node regression, altering the direction of specific angular momentum at rates up to several degrees per day for inclined low-altitude orbits. To counteract these effects and maintain desired trajectories, spacecraft employ station-keeping maneuvers, such as periodic thruster firings, which restore specific angular momentum to nominal values and ensure long-term stability.^[22]^[23] A practical example is geostationary satellites, which orbit Earth at an altitude of approximately 35,786 km to match the planet's sidereal rotation period of 23 hours 56 minutes. These satellites require a specific angular momentum magnitude of about 130,000 km²/s to achieve the necessary circular equatorial orbit with zero inclination, ensuring they remain fixed over a single longitude on Earth's surface for continuous communication coverage. The vector's alignment with Earth's equatorial plane is critical, as any deviation would cause apparent motion relative to ground stations, requiring corrective delta-v to preserve the geostationary configuration.^[24] For trajectory optimization, numerical tools like NASA's General Mission Analysis Tool (GMAT) and AGI's Systems Tool Kit (STK) incorporate specific angular momentum computations to simulate and refine spacecraft paths. GMAT, an open-source platform, models two-body dynamics and perturbations to propagate orbits and optimize delta-v sequences, outputting specific angular momentum vectors for evaluating maneuver efficiency in interplanetary transfers or Earth-orbit adjustments. STK complements this by providing visualization and multi-body propagation, allowing analysts to iterate on specific angular momentum profiles for fuel-optimal trajectories in complex mission scenarios.^[25]^[26]

Astrophysical Systems

In protoplanetary disks, the specific angular momentum of infalling gas parcels is conserved during the collapse of molecular cloud cores, resulting in a radial distribution that establishes Keplerian rotation profiles where the orbital velocity scales as v \propto r^{-1/2}. This conservation drives the flattening of the collapsing material into a disk, with the disk's size determined by the initial specific angular momentum of the core, typically on the order of $10^{20} to $10^{21} cm² s⁻¹ for solar-mass systems. Torques from gravitational instabilities, magnetic fields, or interactions with the central star can alter this specific angular momentum, facilitating planetary migration and the formation of gaps in the disk that influence planetesimal growth.^[27] In binary star systems, the specific angular momentum is defined for each star relative to the system's center of mass, contributing to the total orbital angular momentum J = \mu \sqrt{G (M_1 + M_2) a (1 - e^2)}, where \mu is the reduced mass, a the semi-major axis, and e the eccentricity. During Roche lobe overflow, mass transfer from the donor star carries away specific angular momentum, often comparable to that of the donor's orbit, which can widen or shrink the binary separation depending on whether the transferred material is isotropic or carries excess angular momentum from the donor. This process is critical in close binaries, such as those evolving into cataclysmic variables, where angular momentum loss via gravitational waves or magnetic braking further tightens the orbit and influences the stability of mass transfer.^[28] Accretion disks around compact objects, such as black holes or neutron stars, rely on the transport of specific angular momentum outward through viscous processes to enable inward radial infall of material. In the Shakura-Sunyaev α-disk model, the viscosity parameter \alpha parameterizes turbulent stresses that redistribute angular momentum, creating a gradient where inner regions lose specific angular momentum (typically Keplerian, h \approx \sqrt{G M r}) to outer parts, driving accretion rates up to \dot{M} \sim 10^{-8} M_\odot yr⁻¹ in active galactic nuclei. This mechanism ensures steady-state disk structure, with the inner edge truncated at the innermost stable circular orbit, beyond which angular momentum conservation prevents further infall without additional dissipation.^[29] In galactic dynamics, the specific angular momentum of stars orbiting the galactic center is conserved in axisymmetric potentials, analogous to central force fields, allowing stars to maintain nearly circular orbits in the disk while precessing slowly due to non-spherical perturbations. This conservation, quantified as h_z = R v_\phi where R is the cylindrical radius and v_\phi the azimuthal velocity, underpins the stability of galactic disks, with typical values around $10^{3} kpc km s⁻¹ for Milky Way stars, enabling the formation of spiral arms through density waves that temporarily alter radial positions without changing h_z. Deviations from axisymmetry, such as bars, can redistribute this angular momentum, driving secular evolution and fueling central bulges.^[30]^[31] Specific angular momentum in astrophysical systems is often measured through spectroscopy, particularly radial velocity observations of exoplanet host stars, which yield orbital periods, semi-major axes, and eccentricities to compute h = \sqrt{G M_* a (1 - e^2)} per planet, constraining formation scenarios like disk migration versus in-situ growth. For instance, hot Jupiters with relatively low specific angular momentum (~10^{15} m² s⁻¹) suggest inward migration from outer disk regions, while higher values in temperate giants align with protoplanetary disk models incorporating torques. These measurements, combined with Gaia astrometry for inclination, refine angular momentum distributions across exoplanetary systems, revealing correlations with host star metallicity and multiplicity.^[32]

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