Fact-checked by Grok 2 weeks ago

Elliptic orbit

An elliptic orbit is a curved trajectory followed by a smaller celestial body, such as a planet or satellite, under the gravitational attraction of a more massive central body, like the Sun or Earth, where the path forms an ellipse with the central body positioned at one of the two foci.^[1] This orbit is characterized by an eccentricity e satisfying $0 < e < 1, which measures the degree of elongation from a perfect circle (e = 0); as e approaches 1, the ellipse becomes more elongated, approaching but not reaching a parabolic shape.^[2] The defining geometric property is that the sum of the distances from any point on the ellipse to the two foci remains constant and equal to twice the semi-major axis a, the average distance from the center to the orbit.^[3] The concept of elliptic orbits was empirically established by Johannes Kepler in the early 17th century through his analysis of planetary motions, particularly Mars, leading to his three laws that govern such paths.^[1] Kepler's first law states that planets orbit the Sun in ellipses with the Sun at one focus, explaining the varying distances observed.^[1] Kepler's second law, the law of equal areas, describes how a line connecting the orbiting body to the central focus sweeps out equal areas in equal times, implying the body moves fastest at periapsis (closest point to the focus) and slowest at apoapsis (farthest point).^[1] Kepler's third law relates the orbital period T to the semi-major axis via T^2 \propto a^3, a relationship that holds for all elliptic orbits around the same central body and is derived from Newtonian gravity.^[1] These laws apply universally to bound orbits with negative total energy, where the kinetic energy decreases as potential energy increases with distance.^[4] In the solar system, all eight planets follow elliptic orbits around the Sun, with eccentricities ranging from nearly circular (Earth's e \approx 0.017) to more pronounced (Mercury's e \approx 0.206); the dwarf planet Pluto follows a more eccentric orbit with e \approx 0.249.^[1]^[5] Artificial satellites in elliptic orbits, such as geosynchronous transfer orbits, leverage these properties for efficient launches, with parameters like inclination (tilt relative to the reference plane) and argument of periapsis (orientation of the ellipse) defining the full orbital elements alongside a and e.^[2] The mathematical description of motion in an elliptic orbit uses the polar equation r = \frac{a(1 - e^2)}{1 + e \cos \theta}, where r is the radial distance and \theta is the true anomaly from periapsis, enabling precise predictions in astrodynamics.^[3]

Fundamentals

Definition and Characteristics

An elliptic orbit describes the trajectory of a smaller body, such as a planet or satellite, moving under the gravitational influence of a much more massive central body in a two-body system governed by Newton's law of universal gravitation. This path forms a closed ellipse with the central body located at one of the two foci, ensuring bounded motion that repeats periodically without escaping to infinity.^[6] This geometric configuration arises directly from Kepler's first law of planetary motion, which posits that the orbit of every planet is an ellipse with the Sun at one focus, a principle derived empirically from observations and later explained through Newtonian mechanics. In polar coordinates centered at the focus, the radial distance r from the central body to the orbiting body as a function of the true anomaly \theta is given by

r = \frac{a(1 - e^2)}{1 + e \cos \theta},

where a is the semi-major axis representing the average size of the orbit and e (with $0 \leq e < 1) is the eccentricity quantifying the deviation from a perfect circle.^[7]^[8] Physically, the elliptic orbit exhibits key implications from conservation laws in the two-body problem. Conservation of angular momentum dictates that the orbiting body's speed varies inversely with its distance from the focus, being maximum at periapsis (closest point) and minimum at apoapsis (farthest point), which sweeps out equal areas in equal times as per Kepler's second law. Additionally, the total mechanical energy of the system is negative, distinguishing elliptic orbits as bound states where the kinetic energy cannot overcome the gravitational potential energy, preventing escape.^[9]^[10]

Comparison to Other Conic Sections

In orbital mechanics, the trajectories of bodies under a central inverse-square force, such as gravity, are conic sections formed by the intersection of a plane with a right circular cone.^[11] This geometric property arises from the conservation of energy and angular momentum in the two-body problem, with the shape determined by the eccentricity e, a dimensionless parameter ranging from 0 to infinity.^[11] Parabolic orbits occur when e = 1, corresponding to zero total mechanical energy, where the orbiting body achieves exactly the escape velocity from the central body.^[12] These trajectories represent marginal escape paths that extend to infinite range, allowing the body to approach from infinity, swing around the central attractor, and recede back to infinity without being captured.^[13] Hyperbolic orbits arise when e > 1, characterized by positive total mechanical energy, indicating that the body possesses excess kinetic energy beyond escape requirements.^[12] Such paths describe unbound scattering encounters, where the orbiting body originates from and returns to infinity, deflected by the central force but not orbiting periodically.^[13] In contrast, elliptic orbits, with $0 \leq e < 1, feature negative total mechanical energy, which binds the body to a closed, periodic path around the central attractor, repeatedly traversing the same elliptical trajectory.^[12] This negative energy distinguishes them from the open, non-repeating paths of parabolic and hyperbolic orbits.^[13] The transition between conic types in the two-body problem occurs at specific energy thresholds: orbits shift from elliptic to parabolic at zero total energy and to hyperbolic above it, with eccentricity serving as the parameter that continuously varies the shape across these regimes.^[12]

Orbital Parameters

Semi-Major Axis and Eccentricity

The semi-major axis, denoted a, is half the length of the major axis of the elliptic orbit, equivalent to the mean distance of the orbiting body from the central focus. This parameter sets the scale of the orbit and is intrinsically tied to its total mechanical energy, with larger values of a corresponding to orbits of higher energy (less negative specific mechanical energy) while remaining bound. In the context of Keplerian orbits, a also governs the timescale of motion through its relation in Kepler's third law, where the orbital period scales with a^{3/2} for a given central mass.^[14]^[15]^[1] The eccentricity, denoted e, quantifies the deviation of the orbit from circularity and ranges from 0 (perfect circle) to values approaching but less than 1 (highly elongated ellipse). Defined as the ratio of the focal distance c to the semi-major axis, e = c / a, it describes the degree of flattening, with the distance from focus to center given by c = a e. Higher eccentricity results in greater variation in radial distance, affecting the dynamics near periapsis and apoapsis. The true anomaly \theta, measured from periapsis at the focus, relates to the radial distance r via the polar equation:

r = \frac{a (1 - e^2)}{1 + e \cos \theta}

This equation illustrates how e modulates r as a function of angular position, with minimum r at \theta = 0^\circ and maximum at \theta = 180^\circ.^[8]^[1]^[16] Together, a and e uniquely determine the size and shape of the ellipse in the two-body central force problem, fixing the geometric path without reference to orientation or timing elements. For bound orbits, a > 0 and $0 \leq e < 1 ensure closure and stability under inverse-square gravitation. In perturbed multi-body systems, such as planetary configurations, low eccentricity orbits can still be unstable if they enter mean-motion resonances, where perturbations excite eccentricity and risk close encounters or ejections.^[17] Observationally, a and e are determined by fitting Keplerian models to time-series data of the orbiting body's position and velocity. Astrometric measurements of angular separation across multiple epochs, as in visual binary stars or asteroid tracking, yield relative orbital parameters through least-squares adjustment, directly estimating a (scaled by distance) and e from the fitted ellipse. Radial velocity observations, via Doppler shifts, complement this by revealing e through the non-sinusoidal variation in line-of-sight speed and constraining a via the velocity semi-amplitude, particularly effective for spectroscopic binaries or exoplanets. These methods achieve high precision with sufficient data coverage, often combining ground- and space-based telescopes.^[18]^[19]

Periapsis and Apoapsis

In an elliptic orbit, the periapsis represents the point of closest approach to the central gravitating body, where the orbiting object attains its minimum radial distance and maximum orbital speed. For orbits centered on Earth, this point is termed the perigee. The periapsis distance r_p is calculated as r_p = a(1 - e), where a denotes the semi-major axis and e the eccentricity, both parameters that shape the orbit's overall size and elongation.^[20] Conversely, the apoapsis marks the farthest point from the central body, featuring the maximum radial distance and minimum orbital speed. Around Earth, it is known as the apogee. The apoapsis distance r_a follows r_a = a(1 + e), again depending on the semi-major axis and eccentricity.^[20] Conservation of angular momentum governs motion at these extrema, as the velocity vector aligns perpendicular to the radius vector, ensuring the product of radial distance and tangential speed remains constant throughout the orbit; this results in the highest speed at periapsis and the lowest at apoapsis. Relatedly, the slower speeds near apoapsis mean the orbiting body spends disproportionately more time in that region compared to near periapsis, a consequence of Kepler's second law stating that equal areas are swept in equal times.^[2] These points hold significant practical implications in orbital dynamics. Tidal forces, which scale inversely with the cube of distance, peak at periapsis, heightening risks of structural disruption for loosely bound objects like comets during solar approaches.^[21] For comets in highly eccentric elliptic orbits, periapsis proximity to the Sun (perihelion) triggers intense sublimation of ices, forming prominent dust and ion tails as solar radiation and wind interact with released material.^[22] In satellite design, elliptic orbits exploit apoapsis dwell time for extended coverage; for instance, Molniya orbits, with perigee near 500 km and apogee around 40,000 km, position communication satellites over high-latitude regions for up to eight hours per 12-hour period to serve polar areas effectively.^[23]

Equations of Motion

Position as a Function of Time

In elliptic orbits, the position of a body as a function of time is obtained by relating the elapsed time to angular parameters known as anomalies, which parameterize the body's location along the orbital ellipse. The mean anomaly M, defined as M = n(t - \tau) where n = \sqrt{\mu / a^3} is the mean motion, \mu = GM is the standard gravitational parameter of the central body with mass M, G is the gravitational constant, a is the semi-major axis, and \tau is the time of periapsis passage, advances uniformly with time and serves as the primary time-dependent input. This is connected to the geometric position through the eccentric anomaly E, which describes the body's projection onto an auxiliary circle of radius equal to the semi-major axis a. The relationship between the mean and eccentric anomalies is given by Kepler's equation:

M = E - e \sin E,

where e is the orbital eccentricity ($0 < e < 1 for elliptic orbits). This transcendental equation, which links linear time progression to the nonlinear motion along the ellipse, was first derived by Johannes Kepler in Chapter 60 of his 1609 work Astronomia Nova based on observations of Mars.^[24] No closed-form solution exists for E in terms of M, necessitating iterative numerical methods for computation.^[25] A widely used approach to solve Kepler's equation is the Newton-Raphson method, an iterative root-finding technique that applies the recurrence relation

E_{k+1} = E_k - \frac{E_k - e \sin E_k - M}{1 - e \cos E_k},

starting from an initial guess such as E_0 = M or E_0 = \pi for near-apoapsis cases. This method exhibits quadratic convergence for e < 1, typically requiring only a few iterations (e.g., 3–5) to achieve high precision in orbital predictions. Enhanced variants, such as second-order formulations, further optimize convergence for computational efficiency in space mission analysis.^[25] Once E is determined, the true anomaly \theta, which measures the angular position from periapsis in the orbital plane, is computed via the conversion formula

\tan\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + e}{1 - e}} \tan\left(\frac{E}{2}\right).

The radial distance r follows as r = a(1 - e \cos E), or equivalently in polar form r = \frac{a(1 - e^2)}{1 + e \cos \theta}. The position coordinates in the perifocal frame are then \mathbf{r} = r (\cos \theta \, \hat{p} + \sin \theta \, \hat{q}), where \hat{p} and \hat{q} are unit vectors along the periapsis and orthogonal directions, respectively. This yields the full time-dependent position \mathbf{r}(t). To initialize this framework for a specific orbit, the orbital elements (including a and e) must first be derived from the initial position vector \mathbf{r}_0 and velocity vector \mathbf{v}_0 at a reference epoch. The specific angular momentum vector is \mathbf{h} = \mathbf{r}_0 \times \mathbf{v}_0, and the eccentricity vector is \mathbf{e} = \frac{1}{\mu} \left[ (|\mathbf{v}_0|^2 - \frac{\mu}{r_0}) \mathbf{r}_0 - (\mathbf{r}_0 \cdot \mathbf{v}_0) \mathbf{v}_0 \right], where \mu is the standard gravitational parameter and r_0 = |\mathbf{r}_0|. The eccentricity is e = |\mathbf{e}|, and the semi-major axis is obtained from the vis-viva relation as a = \left( \frac{2}{r_0} - \frac{|\mathbf{v}_0|^2}{\mu} \right)^{-1}. Additional elements like inclination and argument of periapsis follow from \mathbf{h} and \mathbf{e}, enabling subsequent position propagation via the anomalies.^[26]

Velocity and Acceleration

In an elliptic orbit, the velocity vector is naturally expressed in polar coordinates centered at the focus, where the position is defined by the radial distance r and the true anomaly \theta. The velocity \mathbf{v} decomposes into radial and tangential components: the radial component is \dot{r}, representing the rate of change of distance from the focus, and the tangential component is r \dot{\theta}, perpendicular to the radius vector and related to the angular momentum. ^[27] ^[26] These components arise from the differentiation of the position vector in polar form, \mathbf{r} = r \hat{r}, yielding \mathbf{v} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}, where \hat{r} and \hat{\theta} are the unit vectors in the radial and tangential directions, respectively. ^[28] The magnitude of the velocity v = |\mathbf{v}| at any point in the orbit is given by the vis-viva equation, v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right), where \mu = GM is the standard gravitational parameter for the central body of mass M, r is the instantaneous radial distance, and a is the semi-major axis. ^[26] This equation derives from conservation of energy in the two-body problem and provides a direct way to compute speed without resolving components, highlighting how velocity increases as the orbiting body approaches periapsis and decreases toward apoapsis. ^[26] The acceleration \mathbf{a} in an elliptic orbit is governed by the two-body equation of motion, \mathbf{a} = -\frac{\mu}{r^3} \mathbf{r} , which points toward the focus and combines gravitational attraction with the effective centripetal acceleration required for curved motion. ^[28] In polar coordinates, this manifests as a radial acceleration component \ddot{r} - r \dot{\theta}^2 = -\frac{\mu}{r^2}, where -r \dot{\theta}^2 is the centripetal term balancing part of the gravitational pull, and a tangential component r \ddot{\theta} + 2 \dot{r} \dot{\theta} = 0, ensuring conservation of angular momentum with no net tangential force. ^[28] ^[26] The purely radial direction of acceleration underscores the central nature of the gravitational force in producing the elliptic trajectory. To determine the full trajectory from initial conditions, the position vector \mathbf{r}_0 and velocity vector \mathbf{v}_0 at a given epoch are used to compute the orbital elements, such as the semi-major axis and eccentricity, via vector cross products and scalar products (e.g., angular momentum \mathbf{h} = \mathbf{r}_0 \times \mathbf{v}_0). ^[26] These elements then allow analytic integration of the equations of motion using Keplerian formulas, or numerical propagation by solving the differential equations \ddot{\mathbf{r}} = -\frac{\mu}{r^3} \mathbf{r} forward in time. ^[26] Polar coordinates are preferred for analytic derivations in orbital mechanics due to the central force symmetry, simplifying expressions for conserved quantities like angular momentum, whereas Cartesian coordinates facilitate numerical integration of trajectories, as the equations become linear in components and are easier to implement in computational algorithms for long-term predictions. ^[26] ^[29]

Dynamic Properties

Orbital Period

The orbital period T of an elliptic orbit is the duration required for a celestial body to complete one full revolution around the central attracting body under the influence of gravity. In Newtonian mechanics, this period is governed by Kepler's third law, which states that T^2 = \frac{4\pi^2 a^3}{\mu}, where a is the semi-major axis of the orbit and \mu = GM is the standard gravitational parameter, with G the gravitational constant and M the mass of the central body.^[26] This relationship applies to all bound elliptic orbits produced by a central inverse-square gravitational force and notably does not depend on the orbit's eccentricity.^[26] Distinctions exist between the sidereal orbital period, which measures the time for one complete orbit relative to the fixed stars and represents the intrinsic dynamical period, and the synodic period, which is the time between successive alignments as viewed from another orbiting body, such as Earth relative to the Sun.^[30] For example, the sidereal year for Earth is approximately 365.256 days, while synodic considerations affect observed planetary cycles.^[30] Eccentricity affects the temporal distribution within the orbit, even though the total period remains unchanged; the orbiting body travels slower near apoapsis due to lower gravitational pull and conservation of angular momentum, thus spending proportionally more time in the outer portion of the ellipse compared to the inner portion near periapsis.^[31] This periodicity finds practical use in space operations, such as predicting when low-Earth-orbit satellites will pass over specific ground locations to facilitate data downlink, and in astronomy for calculating planetary orbital years, like Jupiter's approximately 11.86 Earth years.^[32]^[1]

Specific Energy

In orbital mechanics, the specific mechanical energy \epsilon represents the total energy per unit mass of an orbiting body and is conserved in the two-body problem under central gravitational forces. It is given by the sum of specific kinetic energy and specific gravitational potential energy: \epsilon = \frac{v^2}{2} - \frac{\mu}{r}, where v is the speed, r is the radial distance from the central body, and \mu = GM is the standard gravitational parameter of the central body with mass M.^[33] This quantity remains constant throughout the orbit due to energy conservation, as derived by taking the dot product of the two-body equation of motion \mathbf{\ddot{r}} = -\frac{\mu}{r^3} \mathbf{r} with the velocity \mathbf{v} = \mathbf{\dot{r}}, yielding \frac{d}{dt} \left( \frac{v^2}{2} - \frac{\mu}{r} \right) = 0.^[33]^[34] For elliptic orbits, \epsilon is negative, distinguishing them as bound trajectories. The constant value is \epsilon = -\frac{\mu}{2a}, where a is the semi-major axis.^[34] This relation follows from the vis-viva equation, which expresses speed as v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right); substituting into the energy expression gives \epsilon = \frac{1}{2} \mu \left( \frac{2}{r} - \frac{1}{a} \right) - \frac{\mu}{r} = -\frac{\mu}{2a}, confirming the linkage between the conserved energy and the orbit's size parameter a.^[33]^[34] The kinetic component \frac{v^2}{2} and potential component -\frac{\mu}{r} vary inversely along the orbit: kinetic energy peaks at periapsis (minimum r, maximum v), while potential energy is least negative at apoapsis (maximum r, minimum v).^[34] Their sum \epsilon balances to maintain the closed elliptical path, with the negative total ensuring the body cannot escape to infinity.^[33] This conserved negative \epsilon determines the orbit type (elliptic versus unbound hyperbolic or parabolic paths) and fixes a = -\frac{\mu}{2\epsilon}; in unperturbed motion or under perturbations that preserve a, \epsilon remains invariant.^[34]^[35]

Flight Path Angle

The flight path angle, denoted as \phi, is defined as the angle between the velocity vector of a body in an elliptic orbit and the local horizontal, which is perpendicular to the position vector from the central body.^[36] This angle quantifies the inclination of the trajectory relative to the tangential direction at any point along the orbit.^[37] In elliptic orbits, the flight path angle is expressed as \phi = \arctan\left( \frac{e \sin \theta}{1 + e \cos \theta} \right), where e is the orbital eccentricity and \theta is the true anomaly.^[36] This formula arises from the geometry of the conic section and the conservation of angular momentum. The angle \phi varies continuously with \theta, reaching zero at periapsis (\theta = 0) and apoapsis (\theta = 180^\circ), where the velocity is purely tangential.^[37] The flight path angle relates directly to the radial and tangential components of velocity: \sin \phi = \frac{dr/dt}{v} and \cos \phi = \frac{r \, d\theta/dt}{v}, where dr/dt is the radial velocity, r \, d\theta/dt is the tangential velocity, and v is the total speed.^[36] These relations highlight how \phi captures the radial component of motion, with positive values indicating outward radial velocity (ascent phase after periapsis) and negative values indicating inward radial velocity (descent phase before apoapsis).^[37] The significance of the flight path angle lies in its role for identifying ascent and descent phases in the orbit, which is essential for trajectory planning and understanding directional changes in altitude.^[37] It is zero at the turning points (periapsis and apoapsis), where the orbit reverses radial direction. Computationally, \phi is obtained from the true anomaly \theta and eccentricity e using the arctan expression, often derived from orbital elements without needing time-dependent variables.^[36]

Applications and Examples

Trajectories in the Solar System

In the Solar System, the orbits of planets around the Sun are ellipses with low eccentricities, approximating near-circular paths for most bodies. Earth's orbit has an eccentricity of approximately 0.0167, resulting in a perihelion distance of about 0.983 AU and an aphelion of 1.017 AU, which causes minor seasonal variations in solar radiation received by the planet.^[38] Mercury exhibits the highest eccentricity among the planets at 0.2056, leading to a more pronounced variation in its distance from the Sun, ranging from 0.307 AU at perihelion to 0.467 AU at aphelion, influencing its orbital dynamics and surface temperatures.^[38] These parameters are derived from long-term observations and are consistent with Keplerian elliptic motion under the Sun's central gravity. Comets, in contrast, often follow highly eccentric elliptic orbits with periods ranging from years to millennia. Halley's Comet (1P/Halley), a short-period comet, has an eccentricity of 0.967 and a semi-major axis of approximately 17.8 AU, yielding a period of about 76 years and extreme distance variations from 0.586 AU at perihelion to nearly 35 AU at aphelion.^[39] This elongated path brings the comet close to the Sun periodically, sublimating its ices and producing its visible tail, before retreating to the outer Solar System. Such orbits distinguish comets from planets, with many long-period comets having eccentricities approaching but not reaching 1.0, keeping them bound to the Sun. Observational evidence for these elliptic trajectories comes from precise radar ranging and spacecraft flybys. Ground-based radar measurements, such as those from NASA's Goldstone Deep Space Communications Complex, have refined planetary ephemerides by providing sub-km accuracy in range and velocity, confirming the predicted elliptic positions of Venus and Mercury over decades. Spacecraft data from missions like Voyager 2, which encountered multiple planets along their forecasted elliptic paths in the 1970s and 1980s, further validate these orbits through direct imaging and ranging, aligning observed positions with precomputed elliptic models to within arcseconds. For comets, the ESA Giotto spacecraft's 1986 flyby of Halley's nucleus at 596 km confirmed the comet's position as predicted by its elliptic orbital elements, enhancing the accuracy of future apparition forecasts. While pure elliptic orbits assume a two-body central force, real Solar System trajectories include slight deviations due to non-central gravitational perturbations from other bodies. Planetary influences, particularly from Jupiter, cause secular changes in eccentricities and semi-major axes for both planets and comets; for instance, Mercury's orbit experiences precession attributable to perturbations from other planets, such as Venus and Earth, totaling about 532 arcseconds per century, with general relativity accounting for an additional 43 arcseconds per century. These perturbations are incorporated into numerical ephemerides like JPL's DE441, which model multi-body interactions to predict positions with uncertainties below 1 km over centuries, ensuring elliptic approximations remain highly accurate for most applications.

Radial Elliptic Trajectories

Radial elliptic trajectories describe a limiting case of elliptic orbits in the two-body Kepler problem where the angular momentum approaches zero, causing the orbit to degenerate into a straight-line oscillation along the radial direction through the primary focus. In this configuration, the motion resembles one-dimensional oscillation, with the secondary body moving directly toward and away from the primary along a fixed line, passing through the focus. The eccentricity e approaches 1 from below, while the specific energy remains negative (E = -\frac{GM}{2a} < 0), ensuring the trajectory is bound, unlike the parabolic case where e = 1 and E = 0. This degenerate ellipse has a semi-minor axis of zero, collapsing the orbital plane into a line segment between periastron and apastron.^[40] The radial position r(t) is governed by the radial equation of motion derived from the inverse-square law:

\frac{d^2 r}{dt^2} = -\frac{GM}{r^2},

where M is the mass of the primary and G is the gravitational constant. Conservation of energy provides the radial velocity:

\left( \frac{dr}{dt} \right)^2 = 2GM \left( \frac{1}{r} - \frac{1}{2a} \right),

with turnaround points at r_{\min} = a(1 - e) \approx 0 and r_{\max} = a(1 + e) \approx 2a, where dr/dt = 0. The time evolution is obtained by integrating:

t - t_0 = \int_{r_{\min}}^{r} \frac{dr'}{\sqrt{2GM \left( \frac{1}{r'} - \frac{1}{2a} \right)}}.

This integral evaluates to a form involving inverse trigonometric functions in the degenerate limit, yielding a finite orbital period T = 2\pi \sqrt{\frac{a^3}{GM}} even as e \to 1^-, consistent with Kepler's third law. In the strict zero-angular-momentum case, the solution describes motion reaching the origin in finite time, mathematically extendable to oscillation by assuming passage through the focus.^[41]^[28] Physically, radial elliptic trajectories are rare and primarily theoretical constructs for point-mass systems, as any infinitesimal perturbation in three dimensions introduces angular momentum, destabilizing the orbit and transforming it into a non-degenerate ellipse or hyperbolic path. For realistic extended bodies, such as in binary systems, the trajectory would lead to collision before completing an oscillation, precluding stable radial motion. These configurations arise hypothetically in simplified one-dimensional gravitational models or as approximations during the final stages of binary mergers, where orbits evolve to high eccentricity and near-zero angular momentum before coalescence. For instance, in simulations of binary black hole inspirals, nearly radial plunges occur post the last stable orbit, driven by gravitational wave emission. This bound radial oscillation contrasts with radial parabolic trajectories, which represent marginal escape with zero energy and no return from infinity.^[42]^[43]

Historical Context

Kepler's Contributions

Johannes Kepler, a German astronomer, inherited the extensive observational records of his mentor Tycho Brahe upon Brahe's death in 1601 and spent the subsequent years meticulously analyzing the data, particularly the positions of Mars observed between 1580 and 1600.^[1] This analysis, conducted primarily in Prague where Kepler served as Imperial Mathematician, spanned from 1601 to the publication of his key works, revealing discrepancies of up to eight minutes of arc in prior models that prompted a complete overhaul of planetary theory.^[44] Kepler's efforts culminated in the rejection of traditional circular orbits, as he found that no combination of circles and epicycles could accurately fit Brahe's data for Mars.^[45] In his seminal 1609 treatise Astronomia Nova, Kepler articulated his first law of planetary motion: planets orbit the Sun in elliptical paths with the Sun positioned at one focus of the ellipse.^[1] This discovery arose directly from plotting Mars's trajectory, where Kepler identified the ellipse as the precise geometric form that matched Brahe's observations, abandoning the long-held assumption of uniform circular motion.^[45] Complementing this, Kepler's second law, also published in Astronomia Nova, states that a line connecting a planet to the Sun sweeps out equal areas in equal intervals of time, implying that planetary speeds vary—accelerating as they approach the Sun and decelerating as they recede.^[1] This areal law provided an empirical description of the non-uniform motion inherent in elliptic orbits, derived solely from geometric considerations of Brahe's data without reference to underlying physical causes.^[45] Kepler extended his empirical findings a decade later in Harmonices Mundi (1619), introducing his third law: the square of a planet's orbital period T is proportional to the cube of the semi-major axis a of its orbit, expressed as T^2 \propto a^3.^[1] This harmonic relation, verified across multiple planets using Brahe's observations and Kepler's own calculations, held without invoking any theory of gravitational forces, relying instead on proportionalities observed in the data.^[46] Collectively, these laws marked a profound shift in astronomy, replacing the Ptolemaic system of nested epicycles with simple elliptic geometry centered on the Sun, laying the empirical groundwork for later mechanical interpretations of celestial motion.^[45]

Newtonian Derivation

In his Philosophiæ Naturalis Principia Mathematica published in 1687, Isaac Newton demonstrated that a central force varying inversely as the square of the distance from the force center—now known as the inverse-square law—results in orbital paths that are conic sections, with ellipses corresponding to bound orbits of negative total energy.^[47]^[48] Newton established this in Book 1, Section 3, particularly through Propositions 11–13, where he showed that for a body under such a force directed toward one focus of the conic, the trajectory is an ellipse if the angular momentum and energy conditions confine the motion to a closed path.^[47]^[48] The derivation begins by considering the general problem of motion under a central force, reducing it to solving the differential equation governing the radial and angular coordinates. For a force \mathbf{F} = -\frac{k}{r^2} \hat{r} (where k > 0 is a constant), the orbit equation in polar coordinates r(\theta) is obtained by integrating the Binet form:

\frac{d^2 u}{d\theta^2} + u = -\frac{m}{l^2 u^2} F\left(\frac{1}{u}\right),

with u = 1/r, m the mass, and l the specific angular momentum. Substituting the inverse-square force yields a solution of the form u = \frac{1 + e \cos(\theta - \theta_0)}{p}, where p = l^2 / (m k) is the semi-latus rectum and e the eccentricity; for $0 \leq e < 1, this describes an ellipse.^[49] This closed-form solution arises because the inverse-square law linearizes the differential equation, producing conic sections as the general orbits.^[49] Central to Newton's proofs are the conservation of angular momentum, which implies that the areal velocity \frac{1}{2} r^2 \dot{\theta} = constant (Proposition 1, equating to equal areas swept in equal times), and conservation of energy, ensuring bound motion for elliptic paths with total energy E = \frac{1}{2} m v^2 - \frac{k m}{r} < 0.^[47]^[48] These principles link directly to Kepler's laws: the elliptic shape and focus placement confirm the first law (Proposition 11, Corollary 1), the area law verifies the second, and the relation between periodic time and semi-major axis—T^2 \propto a^3 (Proposition 15)—establishes the third for confocal ellipses.^[47]^[48] Newton extended the analysis to the two-body problem in Section 11 of Book 1, reducing the mutual gravitational attraction between two bodies to an equivalent one-body problem by introducing the center-of-mass frame and reduced mass \mu = \frac{m_1 m_2}{m_1 + m_2}, where the relative motion follows the same inverse-square law as a single body orbiting a fixed center of mass.^[47] In multi-body systems, he accounted for perturbations by treating them as small deviations from the two-body elliptic solution, using series expansions to approximate effects like planetary interactions on primary orbits.^[47] Modern refinements incorporate general relativity, which modifies the Newtonian potential with post-Newtonian terms, leading to a precession of the elliptic orbit's perihelion. For Mercury, Newtonian theory predicts 532 arcseconds per century from planetary perturbations, but observations show 575 arcseconds per century; the relativistic correction adds approximately 43 arcseconds per century due to spacetime curvature, matching the data precisely.^[50] This effect arises from an additional term in the effective force, f \approx -\frac{GM}{r^2} \left(1 + \frac{3 v^2}{c^2}\right), causing the orbit to advance by \delta \psi \approx \frac{6\pi G M}{c^2 a (1 - e^2)} radians per revolution, where a is the semi-major axis and e the eccentricity.^[50]

References

[1]
Orbits and Kepler's Laws - NASA Science
May 2, 2024 · An ellipse is defined by two points, each called a focus, and together called foci. The sum of the distances to the foci from any point on the ...Orbits And Kepler's Laws · Kepler And The Mars Problem · Here Are Kepler's Three Laws
[2]
Chapter 5: Planetary Orbits - NASA Science
Orbital Parameters and Elements · Semi-major axis and · Eccentricity, which together are the basic measurements of the size and shape of the orbit's ellipse ( ...Orbital Parameters And... · Geosynchronous Transfer... · Lagrange Points And ``halo''...
[3]
Ellipses and Elliptic Orbits - HyperPhysics
An ellipse is defined as the set of points that satisfies the equation In cartesian coordinates with the x-axis horizontal, the ellipse equation is. The ...
[4]
Elliptic Orbits: Paths to the Planets - Galileo
The total energy of a planet in an elliptical orbit depends only on the length a of the semimajor axis, not on the length of the minor axis: Etot=− ...Missing: NASA | Show results with:NASA
[5]
[PDF] Kepler's Laws for the 2-Body Problem - Robert Vanderbei
By deriving these equations we have shown that there are elliptical orbits that are solutions to the 2-body problem. This is the essence of Kepler's First Law ...
[6]
2.2: The Ellipse - Physics LibreTexts
Mar 10, 2024 · Polar Equation to the Ellipse. We shall obtain the Equation in polar coordinates to an ellipse whose focus is the pole of the polar coordinates ...Tangents to an Ellipse · Director Circle · Directrices
[7]
[PDF] Polar Form of an Ellipse—CE
In this document, I derive three useful results: the polar form of an ellipse, the relation between the semilatus rectum and the angular momentum, ...
[8]
phy105 - celestial mechanics - orbital motion - vik dhillon
Hence it can be seen that orbits with a negative total energy are either circular or elliptical in shape. These are known as bound orbits. An orbiting body ...
[9]
[PDF] Gravity 2 - Duke Physics
For negative energy the two turning points coincide and the ellipse becomes a circle of radius . There is a simple rule about the nature of the trajectories:.
[10]
Conic Section Orbits - Greg Egan
Mar 22, 2012 · Conic section orbits are ellipses, circles, parabolas, hyperbolas, or straight lines, resulting from bodies under an inverse-square force.
[11]
https://www.gregegan.net/SCIENCE/ConicSectionOrbits/ConicSectionOrbits.html
[12]
Orbital Energies - Richard Fitzpatrick
Elliptical orbits have negative total energies, parabolic orbits have zero, and hyperbolic orbits have positive total energies.
[13]
Chapter 3: Gravity & Mechanics - NASA Science
Jan 16, 2025 · Half the maximum diameter, the distance from the center of the ellipse to one end, is called the semi-major axis. The shape of an ellipse is ...
[14]
semi-major axis - Glossary - NASA
The semi-major axis is one half of the major axis of the elliptical orbit, and the mean distance from the Sun.
[15]
Eccentricity - JPL Solar System Dynamics
Definition. An orbital parameter describing the eccentricity of the orbit ellipse. Eccentricity e is the ratio of half the distance between the foci c to the ...
[16]
[PDF] Lecture 3: Planar Orbital Elements: True Anomaly, Eccentricity, and ...
In this Lecture, you will learn: Geometry of the orbit. • The polar equation. • The planar orbital elements. • A derivation of Kepler's 3 laws.
[17]
Predicting the long-term stability of compact multiplanet systems
Jul 16, 2020 · Our stability analysis provides significantly stronger eccentricity constraints than currently achievable through either radial velocity or ...
[18]
[PDF] The Observational Determination of Orbits
What are the semi-major axis (ar) and the eccentricity (er) of the spacecraft's transfer orbit in terms of V? What is the orbital period of the spacecraft ...
[19]
Orbit Determination - an overview | ScienceDirect Topics
Orbit determination is defined as the process of calculating the trajectory of an object in space using angular measurements taken over time, ...
[20]
Imagine the Universe! Dictionary - NASA
The periapsis and apoapsis distances can be calculated from the semimajor axis and the eccentricity by. rp = a(1-e) and ra = a(1+e). sensitivity. A measure of ...
[21]
[1309.2288] Will Comet ISON (C/2012 S1) Survive Perihelion? - arXiv
Sep 9, 2013 · We find tidal disruption to be unlikely unless other factors (eg, spin-up via torquing) affect ISON substantially.
[22]
Lecture 44: Comets
Nov 25, 2007 · Comet Tails Comets have two tails: Dust Tail: Composed of dusty particles swept back in a curved path by solar radiation.
[23]
Catalog of Earth Satellite Orbits - NASA Earth Observatory
Sep 4, 2009 · The Molniya orbit combines high inclination (63.4°) with high eccentricity (0.722) to maximize viewing time over high latitudes. Each orbit ...
[24]
Kepler's equation and some of its pearls - AIP Publishing
Nov 1, 2018 · Kepler's equation (11) was obtained for the first time in chapter 60 of Kepler's famous book Astronomia Nova published in 1609.7,8 After ...Missing: digital source
[25]
[PDF] Optimized solution of kepler's equation
Kepler's equation, E = M + e sin E, is solved for E by use of a second-order Newton-Raphson iterative algorithm. The derivation of this algorithm is discussed ...Missing: authoritative | Show results with:authoritative
[26]
None
Summary of each segment:
[27]
[PDF] Orbital Mechanics
In analytical geometry, the general equation of an ellipse in polar coordinates, r and θ, with one of the ellipse's foci as the origin of the coordinate frame ( ...
[28]
[PDF] The Two-Body Problem - UCSB Physics
This of course corresponds to a circular orbit. Thus, we see that so long as the effective potential admits a minimum, there will exist a circular orbit.Missing: centripetal | Show results with:centripetal
[29]
5.5 Motion in Polar Coordinates - Physics Bootcamp
Polar coordinates are more natural for circular and elliptical trajectories. In this section, we will introduce polar coordinates and define new unit vectors ...
[30]
Orbits and the Ecliptic Plane - HyperPhysics
Synodic and Sidereal Periods The period of a planet's orbital period around the Sun with respect to the distant stars is called its sidereal period. The ...
[31]
Orbits and Kepler's Laws - NASA Science
May 21, 2024 · Half of the major axis is termed a semi-major axis. Knowing then that the orbits of the planets are elliptical, johannes Kepler formulated three ...
[32]
SLR Orbit Predictions | NASA Earthdata
Predicted orbit files typically contain orbit information for multiple days. Predictions are primarily available in daily files sorted by year and satellite; ...Missing: period | Show results with:period
[33]
[PDF] Introduction to Orbital Mechanics and Spacecraft Attitudes ... - NASA
Mar 20, 2020 · Kepler formulated his three laws of planetary motion: Kepler's 1st Law: The orbit of each planet is an ellipse, with the sun as a focus.
[34]
[PDF] Lecture D29 - Central Force Motion: Orbits - DSpace@MIT
This expression shows that the energy (and the period) of an elliptical orbit depends only on the major semi-axis. We also see that for a fixed a, the value of ...
[35]
[PDF] Using the Two-Burn Escape Maneuver for Fast Transfers in the ...
The right hand side of the equation expresses specific orbital energy as a function of the orbits semi-major axis. The equations above define the major ...
[36]
[PDF] Astrodynamics Summary - Aerostudents
The flight path angle can be derived from the above equations, yielding: tan γ = ˙r r ˙θ. = e sin θ. 1 + e cos θ. (5.18). The velocity components Vn and Vl can ...
[37]
Chapter 2 – Orbit Geometry – Introduction to Orbital Mechanics
In Chapter 1, the Two Body Equation of Motion was developed and we discussed how the elliptical orbit was one possible solution. In general, however, the ...
[38]
Planet Compare - Solar System Exploration - NASA
Orbit Eccentricity, 0.20563593, 0.00677672, 0.01671123, 0.0933941, 0.04838624, 0.05386179, 0.04725744, 0.00859048. Orbit Inclination, 7.0 degrees, 3.39 degrees ...
[39]
Small-Body Element Tables - JPL Solar System Dynamics
Sep 13, 2021 · Orbital elements below are available as fixed-format ASCII tables. Asteroid elements are available as Keplerian elements while comet elements use perihelion ...<|control11|><|separator|>
[40]
[PDF] 6 The Two-Body Problem and Kepler's Laws
motion (not to be confused with Asimov's Three Laws of Robotics):. 1. Kepler's First Law: objects orbit along elliptical trajectories, as shown in. Fig. 4 ...Missing: definition | Show results with:definition
[41]
Kepler Orbits - Galileo and Einstein
The Runge Lenz vector gives a very quick derivation of the elliptic orbit, without Bernoulli's unobvious tricks in the standard derivation presented above.
[42]
Binary Black Hole Mergers: Formation and Populations - Frontiers
We review the main physical processes that lead to the formation of stellar binary black holes (BBHs) and to their merger.
[43]
[PDF] Binary black hole merger dynamics and waveforms
Feb 8, 2006 · Simulations of excised black holes allowed to move through the grid on one or more orbits and then through merger have been carried out, with ...
[44]
Johannes Kepler - Stanford Encyclopedia of Philosophy
May 2, 2011 · Johannes Kepler (1571–1630) is one of the most significant representatives of the so-called Scientific Revolution of the 16 th and 17 th centuries.4. Epistemology And... · 4.2 Causality · B. Secondary Literature
[45]
https://plato.stanford.edu/entries/kepler/#CopReaThrPlaLaw
[46]
https://plato.stanford.edu/entries/kepler/#HarSou
[47]
Newton’s Philosophiae Naturalis Principia Mathematica (Stanford Encyclopedia of Philosophy)
Summary of each segment:
[48]
[PDF] NEWTON'S Principia - UTK Math
11-12-13: let a body revolve in an ellipse; the law of centripetal force toward a focus of the ellipse is: inversely as the square of the distance from the body ...
[49]
Elliptical Planetary Orbits Derived from Newtonian Mechanics
The notion that planets revolve about the sun in elliptical orbits was found empirically by Johannes Kepler during his little “war with Mars.”
[50]
Perihelion Precession of Mercury - Richard Fitzpatrick
It turns out that the cause of this discrepancy is the general relativistic correction to Newtonian gravity. General relativity gives rise to a small correction ...