Non-keplerian orbit features

Non-keplerian orbit features

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For the first Kepler's law the orbit of a planet is an ellipse, with the Sun in one of the foci, but I've read that in the case of an homogeneous sphere distribution, orbits can be elliptic, but the azimutal period is twice the radial period, meaning that the gravity source is at the center of the ellipse and pericenter and apocenter distances match the semiminor and semimajor axes respectively. How can the latter be found?

This sounds like a homework problem so I'll give you most of it but you'll have to work out the details yourself.

Newton's shell theorem tells us that for any spherical mass distribution, the potential at a distance $r_0$ from the center is the same as if all the mass within the shell of radius $r_0$ were at $r=0$, and all the mass outside of the shell didn't exist at all (it cancels):

$$m(r_0) = int_0^{r_0}4 pi r^2 ho(r)dr$$

and if we let $ ho(r)$ to be constant (the sphere is of uniform density):

$$m(r_0) = 4 pi ho int_0^{r_0}r^2 dr = frac{4}{3} pi ho r_0^3$$

The gravitational potential for a point mass is

$$V(r) = frac{Gm}{r}$$

Put them together and you have

$$V(r) = frac{4 pi G ho r^3}{3r} = frac{4}{3} pi ho G r^2$$

and a parabolic $~r^2$ potential is called a harmonic oscillator potential. Motion in this potential (one or many dimensions) is simple harmonic motion. That's because the potential is proportional to $r^2 = x^2 + y^2 + z^2$ and the restoring force is proportional to $ abla r^2 / 2 = x mathbf{hat{x}} + y mathbf{hat{y}} + z mathbf{hat{z}}$. Oscillations have the same restoring force in all directions, so the periods $omega_x, omega_y, omega_z$ will be the same.

Yes, these will be ellipses and since $r = sqrt{x^2 + y^2 + z^2}$ the value of $r$ will reach a maximum twice during each orbit.

If for example $mathbf{r}(t) = cos(omega t)mathbf{hat{x}} + frac{1}{2}sin(omega t)mathbf{hat{z}}$ then you will see that $r(t)$ has two maxima and two minima per cycle.

import numpy as np import matplotlib.pyplot as plt wt = np.linspace(0, 2*np.pi, 1001) x, z = np.cos(wt), 0.5*np.sin(wt) r = np.sqrt(x**2 + z**2) plt.figure() plt.plot(wt, x) plt.plot(wt, z) plt.plot(wt, r)

The study of perturbations began with the first attempts to predict planetary motions in the sky. In ancient times the causes were a mystery. Newton, at the time he formulated his laws of motion and of gravitation, applied them to the first analysis of perturbations, [2] recognizing the complex difficulties of their calculation. [3] Many of the great mathematicians since then have given attention to the various problems involved throughout the 18th and 19th centuries there was demand for accurate tables of the position of the Moon and planets for marine navigation.

The complex motions of gravitational perturbations can be broken down. The hypothetical motion that the body follows under the gravitational effect of one other body only is typically a conic section, and can be readily described with the methods of geometry. This is called a two-body problem, or an unperturbed Keplerian orbit. The differences between that and the actual motion of the body are perturbations due to the additional gravitational effects of the remaining body or bodies. If there is only one other significant body then the perturbed motion is a three-body problem if there are multiple other bodies it is an n-body problem. A general analytical solution (a mathematical expression to predict the positions and motions at any future time) exists for the two-body problem when more than two bodies are considered analytic solutions exist only for special cases. Even the two-body problem becomes insoluble if one of the bodies is irregular in shape. [4]

Most systems that involve multiple gravitational attractions present one primary body which is dominant in its effects (for example, a star, in the case of the star and its planet, or a planet, in the case of the planet and its satellite). The gravitational effects of the other bodies can be treated as perturbations of the hypothetical unperturbed motion of the planet or satellite around its primary body.

General perturbations Edit

In methods of general perturbations, general differential equations, either of motion or of change in the orbital elements, are solved analytically, usually by series expansions. The result is usually expressed in terms of algebraic and trigonometric functions of the orbital elements of the body in question and the perturbing bodies. This can be applied generally to many different sets of conditions, and is not specific to any particular set of gravitating objects. [5] Historically, general perturbations were investigated first. The classical methods are known as variation of the elements, variation of parameters or variation of the constants of integration. In these methods, it is considered that the body is always moving in a conic section, however the conic section is constantly changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this (now unchanging) conic section indefinitely this conic is known as the osculating orbit and its orbital elements at any particular time are what are sought by the methods of general perturbations. [2]

General perturbations takes advantage of the fact that in many problems of celestial mechanics, the two-body orbit changes rather slowly due to the perturbations the two-body orbit is a good first approximation. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body. [4] In the Solar System, this is usually the case Jupiter, the second largest body, has a mass of about 1/1000 that of the Sun.

General perturbation methods are preferred for some types of problems, as the source of certain observed motions are readily found. This is not necessarily so for special perturbations the motions would be predicted with similar accuracy, but no information on the configurations of the perturbing bodies (for instance, an orbital resonance) which caused them would be available. [4]

Special perturbations Edit

In methods of special perturbations, numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis of numerical integration of the differential equations of motion. [6] In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or the orbital elements. [2]

Special perturbations can be applied to any problem in celestial mechanics, as it is not limited to cases where the perturbing forces are small. [4] Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated planetary ephemerides of the great astronomical almanacs. [2] [7] Special perturbations are also used for modeling an orbit with computers.

Cowell's formulation Edit

Cowell's formulation (so named for Philip H. Cowell, who, with A.C.D. Cromellin, used a similar method to predict the return of Halley's comet) is perhaps the simplest of the special perturbation methods. [8] In a system of n mutually interacting bodies, this method mathematically solves for the Newtonian forces on body i by summing the individual interactions from the other j bodies:

Encke's method Edit

Encke's method begins with the osculating orbit as a reference and integrates numerically to solve for the variation from the reference as a function of time. [11] Its advantages are that perturbations are generally small in magnitude, so the integration can proceed in larger steps (with resulting lesser errors), and the method is much less affected by extreme perturbations. Its disadvantage is complexity it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as rectification. [9] Encke's method is similar to the general perturbation method of variation of the elements, except the rectification is performed at discrete intervals rather than continuously. [12]

New families of Sun-centered non-Keplerian orbits over cylinders and spheres

This paper introduces new families of Sun-centered non-Keplerian orbits (NKOs) that are constrained to a three-dimensional, cylindrical or spherical surface. As such, they are an extension to the well-known families of displaced NKOs that are confined to a two-dimensional plane. The cylindrical and spherical orbits are found by investigating the geometrically constrained spacecraft dynamics. By imposing further constraints on the orbit’s angular velocity and propulsive acceleration, the set of feasible orbits is defined. Additionally, the phase spaces of the orbits are explored and a numerical analysis is developed to find periodic orbits. The richness of the problem is further enhanced by considering both an inverse square acceleration law (mimicking solar electric propulsion) and a solar sail acceleration law to maintain the spacecraft on the three-dimensional surface. The wealth of orbits that these new families of NKOs generate allows for a range of novel space applications.

This is a preview of subscription content, access via your institution.

Solar-terrestrial Magnetic Activity and Space Environment

H. Koshiishi , . T. Goka , in COSPAR Colloquia Series , 2002

Launch and operation

MDS-1 will be launched aboard H- II A vehicle test flight #2 in 2002 from Tanegashima Space Center, and placed into GTO with a perigee of 500 km, an apogee of 36000 km, and an orbital period of 10 hours and 45 minutes. After an initial checkout of each components for 10 days, the experiments on the five experimental modules, and the measurements of radiation environment by SEDA will be carried out for over a year. The command/telemetry will be planned/processed/stored in the MDS-1 Mission Interface System (MMIS) in Tsukuba Space Center and transmitted/received via the Mission Data Recorder Equipment (MDRE) in Masuda Tracking and Communication Station. MDS-1 has two telemetry rates of 16 kbps and 4 kbps which are operationally selected and control the temporal resolution of SDOM and MAM.

Astronomy Without A Telescope – Galactic Gravity Lab

Many an alternative theory of gravity has been dreamt up in the bath, while waiting for a bus – or maybe over a light beverage or two. These days it’s possible to debunk (or otherwise) your own pet theory by predicting on paper what should happen to an object that is closely orbiting a black hole – and then test those predictions against observations of S2 and perhaps other stars that are closely orbiting our galaxy’s central supermassive black hole – thought to be situated at the radio source Sagittarius A*.

S2, a bright B spectral class star, has been closely observed since 1995 during which time it has completed over one orbit of the black hole, given its orbital period is less than 16 years. S2’s orbital dynamics can be expected to differ from what would be predicted by Kepler’s 3 rd law and Newton’s law of gravity, by an amount that is three orders of magnitude greater than the anomalous amount seen in the orbit of Mercury. In both Mercury’s and S2’s cases, these apparently anomalous effects are predicted by Einstein’s theory of general relativity, as a result of the curvature of spacetime caused by a nearby massive object – the Sun in Mercury’s case and the black hole in S2’s case.

S2 travels at an orbital speed of about 5,000 kilometers per second – which is nearly 2% of the speed of light. At the periapsis (closest-in point) of its orbit, it is thought to come within 5 billion kilometres of the Schwarzschild radius of the supermassive blackhole, being the boundary beyond which light can no longer escape – and a point we might loosely regard as the surface of the black hole. The supermassive black hole’s Schwarzschild radius is roughly the distance from the Sun to the orbit of Mercury – and at periapsis, S2 is roughly the same distance away from the black hole as Pluto is from the Sun.

The supermassive black hole is estimated to have a mass of roughly four million solar masses, meaning it may have dined upon several million stars since its formation in the early universe – and meaning that S2 only manages to cling on to existence by virtue of its stupendous orbital speed – which keeps it falling around, rather than falling into, the black hole. For comparison, Pluto stays in orbit around the Sun by maintaining a leisurely orbital speed of nearly 5 kilometers per second.

Some astrometrics of S2's orbit around the supermassive black hole Sagittarius A* at the center of the Milky Way. Credit: Schödel et al (2002), published in Nature.

The detailed data set of S2’s astrometric position (right ascension and declination) changes over time – and from there, its radial velocity calculated at different points along its orbit – provides an opportunity to test theoretical predictions against observations.

For example, with these data, it’s possible to track various non-Keplerian and non-Newtonian features of S2’s orbit including:

– the effects of general relativity (from a external frame of reference, clocks slow and lengths contract in stronger gravity fields). These are features expected from orbiting a classic Schwarzschild black hole
– the quadrapole mass moment (a way of accounting for the fact that the gravitational field of a celestial body may not be quite spherical due to its rotation). These are additional features expected from orbiting a Kerr black hole – i.e. a black hole with spin and
– dark matter (conventional physics suggests that the galaxy should fly apart given the speed it’s rotating at – leading to the conclusion that there is more mass present than meets the eye).

But hey, that’s just one way of interpreting the data. If you want to test out some alternative theories – like, say Oceanic String Space Theory – well, here’s your chance.


From ancient times until the 16th and 17th centuries, the motions of the planets were believed to follow perfectly circular geocentric paths as taught by the ancient Greek philosophers Aristotle and Ptolemy. Variations in the motions of the planets were explained by smaller circular paths overlaid on the larger path (see epicycle). As measurements of the planets became increasingly accurate, revisions to the theory were proposed. In 1543, Nicolaus Copernicus published a heliocentric model of the Solar System, although he still believed that the planets traveled in perfectly circular paths centered on the Sun. [1]

History of Kepler and the telescope Edit

Kepler moved to Prague and started working with Tycho Brahe. Tycho gave him the task of reviewing all the information Tycho had on Mars. Kepler noted that the position of Mars was subject to much error and created problems for many models. This led Kepler to configure 3 Laws of Planetary Motion.

First Law: Planets move in ellipses with the Sun at one focus

The law would change an eccentricity of 0.0. and focus more of an eccentricity of 0.8. which show that Circular and Elliptical orbits have the same period and focus, but different sweeps of area defined by the Sun.

This leads to the Second Law: The radius vector describes equal areas in equal times.

These two laws were published in Kepler's book Astronomia Nova in 1609.

For a circles motion is uniform, however for the elliptical to sweep the area in a uniform rate, the object moves quickly when the radius vector is short and moves slower when the radius vector is long.

Kepler published his Third Law of Planetary Motion in 1619, in his book Harmonices Mundi. Newton used the Third Law to define his laws of gravitation.

The Third Law: The squares of the periodic times are to each other as the cubes of the mean distances. [2]

In 1601, Johannes Kepler acquired the extensive, meticulous observations of the planets made by Tycho Brahe. Kepler would spend the next five years trying to fit the observations of the planet Mars to various curves. In 1609, Kepler published the first two of his three laws of planetary motion. The first law states:

"The orbit of every planet is an ellipse with the sun at a focus."

More generally, the path of an object undergoing Keplerian motion may also follow a parabola or a hyperbola, which, along with ellipses, belong to a group of curves known as conic sections. Mathematically, the distance between a central body and an orbiting body can be expressed as:

  • r is the distance
  • a is the semi-major axis, which defines the size of the orbit
  • e is the eccentricity, which defines the shape of the orbit
  • θ is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis).

Alternately, the equation can be expressed as:

Where p is called the semi-latus rectum of the curve. This form of the equation is particularly useful when dealing with parabolic trajectories, for which the semi-major axis is infinite.

Despite developing these laws from observations, Kepler was never able to develop a theory to explain these motions. [3]

Isaac Newton Edit

Between 1665 and 1666, Isaac Newton developed several concepts related to motion, gravitation and differential calculus. However, these concepts were not published until 1687 in the Principia, in which he outlined his laws of motion and his law of universal gravitation. His second of his three laws of motion states:

Strictly speaking, this form of the equation only applies to an object of constant mass, which holds true based on the simplifying assumptions made below.

Newton's law of gravitation states:

From the laws of motion and the law of universal gravitation, Newton was able to derive Kepler's laws, which are specific to orbital motion in astronomy. Since Kepler's laws were well-supported by observation data, this consistency provided strong support of the validity of Newton's generalized theory, and unified celestial and ordinary mechanics. These laws of motion formed the basis of modern celestial mechanics until Albert Einstein introduced the concepts of special and general relativity in the early 20th century. For most applications, Keplerian motion approximates the motions of planets and satellites to relatively high degrees of accuracy and is used extensively in astronomy and astrodynamics.

Simplified two body problem Edit

To solve for the motion of an object in a two body system, two simplifying assumptions can be made:

1. The bodies are spherically symmetric and can be treated as point masses. 2. There are no external or internal forces acting upon the bodies other than their mutual gravitation.

The shapes of large celestial bodies are close to spheres. By symmetry, the net gravitational force attracting a mass point towards a homogeneous sphere must be directed towards its centre. The shell theorem (also proven by Isaac Newton) states that the magnitude of this force is the same as if all mass was concentrated in the middle of the sphere, even if the density of the sphere varies with depth (as it does for most celestial bodies). From this immediately follows that the attraction between two homogeneous spheres is as if both had its mass concentrated to its center.

Smaller objects, like asteroids or spacecraft often have a shape strongly deviating from a sphere. But the gravitational forces produced by these irregularities are generally small compared to the gravity of the central body. The difference between an irregular shape and a perfect sphere also diminishes with distances, and most orbital distances are very large when compared with the diameter of a small orbiting body. Thus for some applications, shape irregularity can be neglected without significant impact on accuracy. This effect is quite noticeable for artificial Earth satellites, especially those in low orbits.

Planets rotate at varying rates and thus may take a slightly oblate shape because of the centrifugal force. With such an oblate shape, the gravitational attraction will deviate somewhat from that of a homogeneous sphere. At larger distances the effect of this oblateness becomes negligible. Planetary motions in the Solar System can be computed with sufficient precision if they are treated as point masses.

Dividing by their respective masses and subtracting the second equation from the first yields the equation of motion for the acceleration of the first object with respect to the second:

The word osculate is Latin for "kiss". In mathematics, two curves osculate when they just touch, without (necessarily) crossing, at a point, where both have the same position and slope, i.e. the two curves "kiss".

An osculating orbit and the object's position upon it can be fully described by the six standard Kepler orbital elements (osculating elements), which are easy to calculate as long as one knows the object's position and velocity relative to the central body. The osculating elements would remain constant in the absence of perturbations. Real astronomical orbits experience perturbations that cause the osculating elements to evolve, sometimes very quickly. In cases where general celestial mechanical analyses of the motion have been carried out (as they have been for the major planets, the Moon, and other planetary satellites), the orbit can be described by a set of mean elements with secular and periodic terms. In the case of minor planets, a system of proper orbital elements has been devised to enable representation of the most important aspects of their orbits.

Perturbations that cause an object's osculating orbit to change can arise from:

  • A non-spherical component to the central body (when the central body can be modeled neither with a point mass nor with a spherically symmetrical mass distribution, e.g. when it is an oblate spheroid).
  • A third body or multiple other bodies whose gravity perturbs the object's orbit, for example the effect of the Moon's gravity on objects orbiting Earth.
  • A relativistic correction.
  • A non-gravitational force acting on the body, for example force arising from:
    • Thrust from a rocket engine
    • Releasing, leaking, venting or ablation of a material
    • Collisions with other objects pressure
    • Switch to a non-inertial reference frame (e.g. when a satellite's orbit is described in a reference frame associated with the precessing equator of the planet).

    An object's orbital parameters will be different if they are expressed with respect to a non-inertial reference frame (for example, a frame co-precessing with the primary's equator), than if it is expressed with respect to a (non-rotating) inertial reference frame.

    Put in more general terms, a perturbed trajectory can be analysed as if assembled of points, each of which is contributed by a curve out of a sequence of curves. Variables parameterising the curves within this family can be called orbital elements. Typically (though not necessarily), these curves are chosen as Keplerian conics, all of which share one focus. In most situations, it is convenient to set each of these curves tangent to the trajectory at the point of intersection. Curves that obey this condition (and also the further condition that they have the same curvature at the point of tangency as would be produced by the object's gravity towards the central body in the absence of perturbing forces) are called osculating, while the variables parameterising these curves are called osculating elements. In some situations, description of orbital motion can be simplified and approximated by choosing orbital elements that are not osculating. Also, in some situations, the standard (Lagrange-type or Delaunay-type) equations furnish orbital elements that turn out to be non-osculating. [2]


    This paper discusses the stability, transition and control of displaced non-Keplerian orbits by the spacecraft using low-thrust propulsion. The two-body dynamical model developed in the polar coordinates is parameterized by the thrust pitch angle, and then two of the hyperbolic and elliptic equilibria are solved from it. The bounded motions near two equilibria are investigated by dynamical system techniques to find out all the stable and unstable periodic trajectories, and two scenarios of the resonant periodic trajectory are presented. Regardless of the thrust pitch angle, all the transit orbits are numerically demonstrated to be restricted inside the invariant manifolds of Lyapunov orbit near the hyperbolic equilibrium. Then the transit orbits can be distinguished from non-transit ones by the restriction of three-dimensional invariant manifolds projected onto the Poincaré section or position space. Based on the influence of thrust direction on the system topology, operating the thrust pitch angle is an effective tool to achieve the transfer within different types of KAM tori, or even transfer beyond the KAM tori.

    Non-keplerian orbit features - Astronomy

    NOTE: This document has substantially outpaced implementation. For the time being, it is better read as a design document than as documentation at this point. I'm working hard to bring the code in line with this.

    A Simple Orbital Mechanics package for Keplerian Orbits.

    To produce a pythonic package suitable for use in hobby applications, but with sufficient accuracy to serve as a good first approximation for heavier use.

    Envisioned applications built on Keplerian include:

    Simple, understandable hobby astronomy aids to track the position of planets, satellites and asteroids in the night sky.

    Kerbal Space Program Calculators such as Olex's Interplanetary transfer calculator or Peppe's Battery Capacity spreadsheet created with uniform code and interface style and minimal developer effort.

    Integrated Kerbal Space Program Mission Planning, when combined with other packages.

    A quick and dirty way to generate candidate trajectories for detailed multi-body analysis.

    Complete features for representing trajectories, simulating impulsive maneuvers, and reverse solving maneuvers between orbits with a common point.

    Intelligent constructors. Describe orbits in the terms an units that are easiest for you.

    Excellent for first approximation trajectory and maneuver planning.

    Stretch / Planned Features

    Produce cubic spline approximations of trajectories over specified time periods.

    SGP4 integration, either through the python sgp4 package or other means.

    • Requires the units package to ensure consistent units and avoid unit-based errors. The indexed units package is somewhat incomplete a custom updated version is available here.

    Keplerian defines orbits very loosely. Hyperbolic and Parabolic Trajectories are considered orbits.

    The simplest type of orbit. Only defines the shape and period (if any) of the orbit. Mostly a utility class.

    Oriented within a plane. For documentation convention the plane is assumed to be equatorial with normal in the z direction.

    The most complex orbit. It is oriented in 3D space and has timing / position information.

    Internally, Keplerian represents orbital trajectories by periapsis radius (not height!), the body being orbited, and eccentricity.

    This nonstandard choice of basis means that all trajectories (excepting those that intersect the center of the planet -- unlikely to occur in practice see section on edge cases) can be represented. In addition, it is unambiguous and easy to differentiate between circular, elliptical, parabolic, and hyperbolic orbits.

    This is obviously not convenient. As a result, Keplerian includes means to create orbits from almost any sufficient set of parameters.

    Give the following a try after importing the appropriate units!

    >>> UnorientedTimelessOrbit( r_periapsis = km(1034), r_apoapsis = km(1201) )

    >>> UnorientedTimelessOrbit( eccentricity = 0.0, v_periapsis = km_per_s(2) )

    >>> UnorientedTimelessOrbit( eccentricity = 0.5, v_apoapsis = km_per_s(1) )

    >>> UnorientedTimelessOrbit( eccentricity = 2.0, v_periapsis = km_per_s(2) )

    If you find a constructor combination that should work but doesn't (i.e. sufficient relevant information to totally fix an orbit is given, but an InsufficientInformationError is raised regardless), please submit an bug report with a detailed write-up of how to convert the given quantities into equivalent quantities that do work. I'll do my best to integrate it into the system when I have time.

    No effort is made to detect or correct inconsistencies in provided data.

    Insufficient Information for Construction

    If Keplerian is unable to create an orbit from the provided information, then it will raise InsufficientInformationError("Unable to produce XXX") where XXX is the quantity(s) that couldn't be computed from the provided information.

    If the orbital parameters provided produce an periapsis radius of zero, then Keplerian will raise OrbitRepresentationError("Zero periapsis radius") . In these cases, either the orbit will be falling directly toward the center of the planet, or speeding directly away.

    Both cases can be handled separately by the RadialOrbit class, but this can be complex and is not guaranteed to interoperate.


    Discovery Edit

    Two teams claim credit for the discovery of Haumea. A team consisting of Mike Brown of Caltech, David Rabinowitz of Yale University and Chad Trujillo of Gemini Observatory in Hawaii discovered Haumea on December 28, 2004 on images they had taken on May 6, 2004. On July 20, 2005, they published an online abstract of a report intended to announce the discovery at a conference in September 2005. [24] At around this time, José Luis Ortiz Moreno and his team at the Instituto de Astrofísica de Andalucía at Sierra Nevada Observatory in Spain found Haumea on images taken on March 7–10, 2003. [25] Ortiz emailed the Minor Planet Center with their discovery on the night of July 27, 2005. [25]

    Brown initially conceded discovery credit to Ortiz, [26] but came to suspect the Spanish team of fraud upon learning that the Spanish observatory had accessed Brown's observation logs the day before the discovery announcement.

    These logs included enough information to allow the Ortiz team to precover Haumea in their 2003 images, and they were accessed again just before Ortiz scheduled telescope time to obtain confirmation images for a second announcement to the MPC on July 29. Ortiz later admitted he had accessed the Caltech observation logs but denied any wrongdoing, stating he was merely verifying whether they had discovered a new object. [27] Precovery images of Haumea have been identified back to March 22, 1955. [8]

    IAU protocol is that discovery credit for a minor planet goes to whoever first submits a report to the MPC (Minor Planet Center) with enough positional data for a decent determination of its orbit, and that the credited discoverer has priority in choosing a name. However, the IAU announcement on September 17, 2008, that Haumea had been named by dual committee established for bodies expected to be dwarf planets, did not mention a discoverer. The location of discovery was listed as the Sierra Nevada Observatory of the Spanish team, [28] [29] but the chosen name, Haumea, was the Caltech proposal Ortiz's team had proposed "Ataecina", the ancient Iberian goddess of spring, [25] which as a chthonic deity would have been appropriate for a plutino.

    Name Edit

    Until it was given a permanent name, the Caltech discovery team used the nickname "Santa" among themselves, because they had discovered Haumea on December 28, 2004, just after Christmas. [30] The Spanish team were the first to file a claim for discovery to the Minor Planet Center, in July 2005. On July 29, 2005, Haumea was given the provisional designation 2003 EL61, based on the date of the Spanish discovery image. On September 7, 2006, it was numbered and admitted into the official minor planet catalog as (136108) 2003 EL61.

    Following guidelines established at the time by the IAU that classical Kuiper belt objects be given names of mythological beings associated with creation, [31] in September 2006 the Caltech team submitted formal names from Hawaiian mythology to the IAU for both (136108) 2003 EL61 and its moons, in order "to pay homage to the place where the satellites were discovered". [32] The names were proposed by David Rabinowitz of the Caltech team. [22] Haumea is the matron goddess of the island of Hawaiʻi, where the Mauna Kea Observatory is located. In addition, she is identified with Papa, the goddess of the earth and wife of Wākea (space), [33] which, at the time, seemed appropriate because Haumea was thought to be composed almost entirely of solid rock, without the thick ice mantle over a small rocky core typical of other known Kuiper belt objects. [34] [35] Lastly, Haumea is the goddess of fertility and childbirth, with many children who sprang from different parts of her body [33] this corresponds to the swarm of icy bodies thought to have broken off the main body during an ancient collision. [35] The two known moons, also believed to have formed in this manner, [35] are thus named after two of Haumea's daughters, Hiʻiaka and Nāmaka. [34]

    The proposal by the Ortiz team, Ataecina, did not meet IAU naming requirements, because the names of chthonic deities were reserved for stably resonant trans-Neptunian objects such as plutinos that resonate 3:2 with Neptune, whereas Haumea was in an intermittent 7:12 resonance and so by some definitions was not a resonant body. The naming criteria would be clarified in late 2019, when the IAU decided that chthonic figures were to be used specifically for plutinos. (See Ataecina § Dwarf planet.)

    Haumea has an orbital period of 284 Earth years, a perihelion of 35 AU, and an orbital inclination of 28°. [8] It passed aphelion in early 1992, [36] and is currently more than 50 AU from the Sun. [20] It will come to perihelion in 2133. [36] Haumea's orbit has a slightly greater eccentricity than that of the other members of its collisional family. This is thought to be due to Haumea's weak 7:12 orbital resonance with Neptune gradually modifying its initial orbit over the course of a billion years, [35] [37] through the Kozai effect, which allows the exchange of an orbit's inclination for increased eccentricity. [35] [38] [39]

    With a visual magnitude of 17.3, [20] Haumea is the third-brightest object in the Kuiper belt after Pluto and Makemake, and easily observable with a large amateur telescope. [40] However, because the planets and most small Solar System bodies share a common orbital alignment from their formation in the primordial disk of the Solar System, most early surveys for distant objects focused on the projection on the sky of this common plane, called the ecliptic. [41] As the region of sky close to the ecliptic became well explored, later sky surveys began looking for objects that had been dynamically excited into orbits with higher inclinations, as well as more distant objects, with slower mean motions across the sky. [42] [43] These surveys eventually covered the location of Haumea, with its high orbital inclination and current position far from the ecliptic.

    Possible resonance with Neptune Edit

    Rotation Edit

    Haumea displays large fluctuations in brightness over a period of 3.9 hours, which can only be explained by a rotational period of this length. [45] This is faster than any other known equilibrium body in the Solar System, and indeed faster than any other known body larger than 100 km in diameter. [40] While most rotating bodies in equilibrium are flattened into oblate spheroids, Haumea rotates so quickly that it is distorted into a triaxial ellipsoid. If Haumea were to rotate much more rapidly, it would distort itself into a dumbbell shape and split in two. [22] This rapid rotation is thought to have been caused by the impact that created its satellites and collisional family. [35]

    The plane of Haumea's equator is oriented nearly edge-on from Earth at present and is also slightly offset to the orbital planes of its ring and its outermost moon Hiʻiaka. Although initially assumed to be coplanar to Hiʻiaka's orbital plane by Ragozzine and Brown in 2009, their models of the collisional formation of Haumea's satellites consistently suggested Haumea's equatorial plane to be at least aligned with Hiʻiaka's orbital plane by approximately 1°. [14] This was supported with observations of a stellar occultation by Haumea in 2017, which revealed the presence of a ring approximately coincident with the plane of Hiʻiaka's orbit and Haumea's equator. [11] A mathematical analysis of the occultation data by Kondratyev and Kornoukhov in 2018 was able to constrain the relative inclination angles of Haumea's equator to the orbital planes of its ring and Hiʻiaka, which were found to be inclined 3.2° ± 1.4° and 2.0° ± 1.0° relative to Haumea's equator, respectively. They also derived two solutions for Haumea's north pole direction, pointing at the equatorial coordinates ( α , δ ) = (282.6°, –13.0°) or (282.6°, –11.8°). [16]

    Size, shape, and composition Edit

    The size of a Solar System object can be deduced from its optical magnitude, its distance, and its albedo. Objects appear bright to Earth observers either because they are large or because they are highly reflective. If their reflectivity (albedo) can be ascertained, then a rough estimate can be made of their size. For most distant objects, the albedo is unknown, but Haumea is large and bright enough for its thermal emission to be measured, which has given an approximate value for its albedo and thus its size. [46] However, the calculation of its dimensions is complicated by its rapid rotation. The rotational physics of deformable bodies predicts that over as little as a hundred days, [40] a body rotating as rapidly as Haumea will have been distorted into the equilibrium form of a triaxial ellipsoid. It is thought that most of the fluctuation in Haumea's brightness is caused not by local differences in albedo but by the alternation of the side view and ends view as seen from Earth. [40]

    The rotation and amplitude of Haumea's light curve were argued to place strong constraints on its composition. If Haumea were in hydrostatic equilibrium and had a low density like Pluto, with a thick mantle of ice over a small rocky core, its rapid rotation would have elongated it to a greater extent than the fluctuations in its brightness allow. Such considerations constrained its density to a range of 2.6–3.3 g/cm 3 . [47] [40] By comparison, the Moon, which is rocky, has a density of 3.3 g/cm 3 , whereas Pluto, which is typical of icy objects in the Kuiper belt, has a density of 1.86 g/cm 3 . Haumea's possible high density covered the values for silicate minerals such as olivine and pyroxene, which make up many of the rocky objects in the Solar System. This also suggested that the bulk of Haumea was rock covered with a relatively thin layer of ice. A thick ice mantle more typical of Kuiper belt objects may have been blasted off during the impact that formed the Haumean collisional family. [35]

    Because Haumea has moons, the mass of the system can be calculated from their orbits using Kepler's third law. The result is 4.2 × 10 21 kg , 28% the mass of the Plutonian system and 6% that of the Moon. Nearly all of this mass is in Haumea. [14] [48] Several ellipsoid-model calculations of Haumea's dimensions have been made. The first model produced after Haumea's discovery was calculated from ground-based observations of Haumea's light curve at optical wavelengths: it provided a total length of 1,960 to 2,500 km and a visual albedo (pv) greater than 0.6. [40] The most likely shape is a triaxial ellipsoid with approximate dimensions of 2,000 × 1,500 × 1,000 km, with an albedo of 0.71. [40] Observations by the Spitzer Space Telescope give a diameter of 1,150 +250
    −100 km and an albedo of 0.84 +0.1
    −0.2 , from photometry at infrared wavelengths of 70 μm. [46] Subsequent light-curve analyses have suggested an equivalent circular diameter of 1,450 km. [49] In 2010 an analysis of measurements taken by Herschel Space Telescope together with the older Spitzer Telescope measurements yielded a new estimate of the equivalent diameter of Haumea—about 1300 km. [50] These independent size estimates overlap at an average geometric mean diameter of roughly 1,400 km. In 2013 the Herschel Space Telescope measured Haumea's equivalent circular diameter to be roughly 1,240 +69
    −58 km . [51]

    However the observations of a stellar occultation in January 2017 cast a doubt on all those conclusions. The measured shape of Haumea, while elongated as presumed before, appeared to have significantly larger dimensions – according to the data obtained from the occultation Haumea is approximately the diameter of Pluto along its longest axis and about half that at its poles. [11] The resulting density calculated from the observed shape of Haumea was about 1.8 g/cm 3 – more in line with densities of other large TNOs. This resulting shape appeared to be inconsistent with a homogenous body in hydrostatic equilibrium, [11] though Haumea appears to be one of the largest trans-Neptunian objects discovered nonetheless, [46] smaller than Eris, Pluto, similar to Makemake, and possibly Gonggong, and larger than Sedna, Quaoar, and Orcus.

    A 2019 study attempted to resolve the conflicting measurements of Haumea's shape and density using numerical modeling of Haumea as a differentiated body. It found that dimensions of ≈ 2,100 × 1,680 × 1,074 km (modeling the long axis at intervals of 25 km) were a best-fit match to the observed shape of Haumea during the 2017 occultation, while also being consistent with both surface and core scalene ellipsoid shapes in hydrostatic equilibrium. [10] The revised solution for Haumea's shape implies that it has a core of approximately 1,626 × 1,446 × 940 km, with a relatively high density of ≈ 2.68 g/cm 3 , indicative of a composition largely of hydrated silicates such as kaolinite. The core is surrounded by an icy mantle that ranges in thickness from about 70 at the poles to 170 km along its longest axis, comprising up to 17% of Haumea's mass. Haumea's mean density is estimated at ≈ 2.018 g/cm 3 , with an albedo of ≈ 0.66. [10]

    Surface Edit

    In 2005, the Gemini and Keck telescopes obtained spectra of Haumea which showed strong crystalline water ice features similar to the surface of Pluto's moon Charon. [17] This is peculiar, because crystalline ice forms at temperatures above 110 K, whereas Haumea's surface temperature is below 50 K, a temperature at which amorphous ice is formed. [17] In addition, the structure of crystalline ice is unstable under the constant rain of cosmic rays and energetic particles from the Sun that strike trans-Neptunian objects. [17] The timescale for the crystalline ice to revert to amorphous ice under this bombardment is on the order of ten million years, [52] yet trans-Neptunian objects have been in their present cold-temperature locations for timescales of billions of years. [37] Radiation damage should also redden and darken the surface of trans-Neptunian objects where the common surface materials of organic ices and tholin-like compounds are present, as is the case with Pluto. Therefore, the spectra and colour suggest Haumea and its family members have undergone recent resurfacing that produced fresh ice. However, no plausible resurfacing mechanism has been suggested. [19]

    Haumea is as bright as snow, with an albedo in the range of 0.6–0.8, consistent with crystalline ice. [40] Other large TNOs such as Eris appear to have albedos as high or higher. [53] Best-fit modeling of the surface spectra suggested that 66% to 80% of the Haumean surface appears to be pure crystalline water ice, with one contributor to the high albedo possibly hydrogen cyanide or phyllosilicate clays. [17] Inorganic cyanide salts such as copper potassium cyanide may also be present. [17]

    However, further studies of the visible and near infrared spectra suggest a homogeneous surface covered by an intimate 1:1 mixture of amorphous and crystalline ice, together with no more than 8% organics. The absence of ammonia hydrate excludes cryovolcanism and the observations confirm that the collisional event must have happened more than 100 million years ago, in agreement with the dynamic studies. [54] The absence of measurable methane in the spectra of Haumea is consistent with a warm collisional history that would have removed such volatiles, [17] in contrast to Makemake. [55]

    In addition to the large fluctuations in Haumea's light curve due to the body's shape, which affect all colours equally, smaller independent colour variations seen in both visible and near-infrared wavelengths show a region on the surface that differs both in colour and in albedo. [56] [57] More specifically, a large dark red area on Haumea's bright white surface was seen in September 2009, possibly an impact feature, which indicates an area rich in minerals and organic (carbon-rich) compounds, or possibly a higher proportion of crystalline ice. [45] [58] Thus Haumea may have a mottled surface reminiscent of Pluto, if not as extreme.

    A stellar occultation observed on January 21, 2017 and described in an October 2017 Nature article indicated the presence of a ring around Haumea. This represents the first ring system discovered for a TNO. [11] [59] The ring has a radius of about 2,287 km, a width of

    70 km and an opacity of 0.5. It is well within Haumea's Roche limit, which would be at a radius of about 4,400 km if it were spherical (being nonspherical pushes the limit out farther). [11] The ring plane is inclined 3.2° ± 1.4° with respect to Haumea's equatorial plane and approximately coincides with the orbital plane of its larger, outer moon Hiʻiaka. [11] [60] The ring is also close to the 1:3 orbit-spin resonance with Haumea's rotation (which is at a radius of 2,285 ± 8 km from Haumea's center). The ring is estimated to contribute 5% to the total brightness of Haumea. [11]

    In a study about the dynamics of ring particles published in 2019, Othon Cabo Winter and colleagues have shown that the 1:3 resonance with Haumea's rotation is dynamically unstable, but that there is a stable region in the phase space consistent with the location of Haumea's ring. This indicates that the ring particles originate on circular, periodic orbits that are close to, but not inside, the resonance. [61]

    Two small satellites have been discovered orbiting Haumea, (136108) Haumea I Hiʻiaka and (136108) Haumea II Namaka. [28] Darin Ragozzine and Michael Brown discovered both in 2005, through observations of Haumea using the W. M. Keck Observatory.

    Hiʻiaka, at first nicknamed "Rudolph" by the Caltech team, [62] was discovered January 26, 2005. [48] It is the outer and, at roughly 310 km in diameter, the larger and brighter of the two, and orbits Haumea in a nearly circular path every 49 days. [63] Strong absorption features at 1.5 and 2 micrometres in the infrared spectrum are consistent with nearly pure crystalline water ice covering much of the surface. [64] The unusual spectrum, along with similar absorption lines on Haumea, led Brown and colleagues to conclude that capture was an unlikely model for the system's formation, and that the Haumean moons must be fragments of Haumea itself. [37]

    Namaka, the smaller, inner satellite of Haumea, was discovered on June 30, 2005, [65] and nicknamed "Blitzen". It is a tenth the mass of Hiʻiaka, orbits Haumea in 18 days in a highly elliptical, non-Keplerian orbit, and as of 2008 [update] is inclined 13° from the larger moon, which perturbs its orbit. [66] The relatively large eccentricities together with the mutual inclination of the orbits of the satellites are unexpected as they should have been damped by the tidal effects. A relatively recent passage by a 3:1 resonance with Hiʻiaka might explain the current excited orbits of the Haumean moons. [67]

    At present, the orbits of the Haumean moons appear almost exactly edge-on from Earth, with Namaka periodically occulting Haumea. [68] Observation of such transits would provide precise information on the size and shape of Haumea and its moons, [69] as happened in the late 1980s with Pluto and Charon. [70] The tiny change in brightness of the system during these occultations will require at least a medium-aperture professional telescope for detection. [69] [71] Hiʻiaka last occulted Haumea in 1999, a few years before discovery, and will not do so again for some 130 years. [72] However, in a situation unique among regular satellites, Namaka's orbit is being greatly torqued by Hiʻiaka, which preserved the viewing angle of Namaka–Haumea transits for several more years. [66] [69] [71]

    Haumea is the largest member of its collisional family, a group of astronomical objects with similar physical and orbital characteristics thought to have formed when a larger progenitor was shattered by an impact. [35] This family is the first to be identified among TNOs and includes—beside Haumea and its moons— (55636) 2002 TX 300 (≈364 km), (24835) 1995 SM 55 (≈174 km), (19308) 1996 TO 66 (≈200 km), (120178) 2003 OP 32 (≈230 km), and (145453) 2005 RR 43 (≈252 km). [3] Brown and colleagues proposed that the family were a direct product of the impact that removed Haumea's ice mantle, [35] but a second proposal suggests a more complicated origin: that the material ejected in the initial collision instead coalesced into a large moon of Haumea, which was later shattered in a second collision, dispersing its shards outwards. [74] This second scenario appears to produce a dispersion of velocities for the fragments that is more closely matched to the measured velocity dispersion of the family members. [74]

    The presence of the collisional family could imply that Haumea and its "offspring" might have originated in the scattered disc. In today's sparsely populated Kuiper belt, the chance of such a collision occurring over the age of the Solar System is less than 0.1 percent. [75] The family could not have formed in the denser primordial Kuiper belt because such a close-knit group would have been disrupted by Neptune's migration into the belt—the believed cause of the belt's current low density. [75] Therefore, it appears likely that the dynamic scattered disc region, in which the possibility of such a collision is far higher, is the place of origin for the object that generated Haumea and its kin. [75]

    Because it would have taken at least a billion years for the group to have diffused as far as it has, the collision which created the Haumea family is believed to have occurred very early in the Solar System's history. [3]

    Joel Poncy and colleagues calculated that a flyby mission to Haumea could take 14.25 years using a gravity assist at Jupiter, based on a launch date of 25 September 2025. Haumea would be 48.18 AU from the Sun when the spacecraft arrives. A flight time of 16.45 years can be achieved with launch dates on 1 November 2026, 23 September 2037 and 29 October 2038. [76] Haumea could become a target for an exploration mission, [77] and an example of this work is a preliminary study on a probe to Haumea and its moons (at 35–51 AU). [78] Probe mass, power source, and propulsion systems are key technology areas for this type of mission. [77]