Orbital decay

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Altitude of Tiangong-1 during its final year of uncontrolled reentry. Altitude of Tiangong-1.svg
Altitude of Tiangong-1 during its final year of uncontrolled reentry.

In orbital mechanics, orbitaldecay is a gradual decrease of the distance between two orbiting bodies at their closest approach (the periapsis) over many orbital periods. These orbiting bodies can be a planet and its satellite, a star and any object orbiting it, or components of any binary system. Orbits do not decay without some friction-like mechanism which transfers energy from the orbital motion. This can be any of a number of mechanical, gravitational, or electromagnetic effects. For bodies in low Earth orbit, the most significant effect is atmospheric drag.

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If left unchecked, the decay eventually results in termination of the orbit when the smaller object strikes the surface of the primary; or for objects where the primary has an atmosphere, the smaller object burns, explodes, or otherwise breaks up in the larger object's atmosphere; or for objects where the primary is a star, ends with incineration by the star's radiation (such as for comets), and so on.

Collisions of stellar-mass objects usually produce cataclysmic effects, such as gamma-ray bursts and detectable gravitational waves.

Due to atmospheric drag, the lowest altitude above the Earth at which an object in a circular orbit can complete at least one full revolution without propulsion is approximately 150 km (93 mi) while the lowest perigee of an elliptical revolution is approximately 90 km (56 mi).

Modelling Orbit Decay

A Simplified Orbit Decay Model

A simplified decay model for a near-circular two-body orbit about a central body (or planet) with an atmosphere, in terms of the rate of change of the orbital altitude, is given below. [2]

Where R is the distance of the spacecraft from the planet's origin, αo is the sum of all accelerations projected on the along-track direction of the spacecraft (or parallel to the spacecraft velocity vector), and T is the Keplerian period. Note that αo is often a function of R due to variations in atmospheric density in the altitude, and T is a function of R by virtue of Kepler's laws of planetary motion.

If only atmospheric drag is considered, one can approximate drag deceleration αo as a function of orbit radius R using the Drag equation below:

is the mass density of the atmosphere which is a function of the radius R from the origin,
is the orbital velocity,
is the drag reference area,
is the mass of the satellite, and
is the dimensionless drag coefficient related to the satellite geometry, and accounting for skin friction and form drag (~2.2 for cube satellites).

The orbit decay model has been tested against ~1 year of actual GPS measurements of VELOX-C1, where the mean decay measured via GPS was 2.566km across Dec 2015 to Nov 2016, and the orbit decay model predicted a decay of 2.444km, which amounted to a 5% deviation.

An open-source Python based software, ORBITM (ORBIT Maintenance the Propulsion Sizing), is available freely on GitHub for Python users using the above model.

Proof of Simplified Orbit Decay Model

By the conservation of mechanical energy, the energy of the orbit is simply the sum of kinetic and gravitational potential energies, in an unperturbed two-body orbit. By substituting the Vis-viva equation into the kinetic energy component, the orbital energy of a circular orbit is given by:

Where G is the gravitational constant, ME is the mass of the central body and m is the mass of the orbiting satellite. We take the derivative of the orbital energy with respect to the radius.

The total decelerating force, which is usually atmospheric drag for low Earth orbits, exerted on a satellite of constant mass m is given by some force F. The rate of loss of orbital energy is simply the rate at the external force does negative work on the satellite as the satellite traverses an infinitesimal circular arc-length ds, spanned by some infinitesimal angle and angular rate ω.

The angular rate ω is also known as the Mean motion, where for a two-body circular orbit of radius R, it is expressed as:

and...

Substituting ω into the rate of change of orbital energy above, and expressing the external drag or decay force in terms of the deceleration αo, the orbital energy rate of change with respect to time can be expressed as:

Having an equation for the rate of change of orbital energy with respect to both radial distance and time allows us to find the rate of change of the radial distance with respect to time as per below.

The assumptions used in this derivation above are that the orbit stays very nearly circular throughout the decay process, so that the equations for orbital energy are more or less that of a circular orbit's case. This is often true for orbits that begin as circular, as drag forces are considered "re-circularizing", since drag magnitudes at the periapsis (lower altitude) is expectedly greater than that of the apoapsis, which has the effect of reducing the mean eccentricity.

Sources of Orbital Decay

Atmospheric drag

Atmospheric drag at orbital altitude is caused by frequent collisions of gas molecules with the satellite. It is the major cause of orbital decay for satellites in low Earth orbit. It results in the reduction in the altitude of a satellite's orbit. For the case of Earth, atmospheric drag resulting in satellite re-entry can be described by the following sequence:

lower altitude → denser atmosphere → increased drag → increased heat → usually burns on re-entry

Orbital decay thus involves a positive feedback effect, where the more the orbit decays, the lower its altitude drops, and the lower the altitude, the faster the decay. Decay is also particularly sensitive to external factors of the space environment such as solar activity, which are not very predictable. During solar maxima the Earth's atmosphere causes significant drag up to altitudes much higher than during solar minima. [3]

Atmospheric drag exerts a significant effect at the altitudes of space stations, Space Shuttles and other manned Earth-orbit spacecraft, and satellites with relatively high "low Earth orbits" such as the Hubble Space Telescope. Space stations typically require a regular altitude boost to counteract orbital decay (see also orbital station-keeping). Uncontrolled orbital decay brought down the Skylab space station, and (relatively) controlled orbital decay was used to de-orbit the Mir space station.[ citation needed ]

Reboosts for the Hubble Space Telescope are less frequent due to its much higher altitude. However, orbital decay is also a limiting factor to the length of time the Hubble can go without a maintenance rendezvous, the most recent having been performed successfully by STS-125, with Space Shuttle Atlantis in 2009. Newer space telescopes are in much higher orbits, or in some cases in solar orbit, so orbital boosting may not be needed. [4]

Tidal effects

An orbit can also decay by negative tidal acceleration when the orbiting body is large enough to raise a significant tidal bulge on the body it is orbiting and is either in a retrograde orbit or is below the synchronous orbit. This saps momentum from the orbiting body and transfers it to the primary's rotation, lowering the orbit's altitude.

Examples of satellites undergoing tidal orbital decay are Mars' moon Phobos, Neptune's moon Triton, and the extrasolar planet TrES-3b.

Light and thermal radiation

Small objects in the Solar System also experience an orbital decay due to the forces applied by asymmetric radiation pressure. Ideally, energy absorbed would equal blackbody energy emitted at any given point, resulting in no net force. However, the Yarkovsky effect is the phenomenon that, because absorption and radiation of heat are not instantaneous, objects which are not terminally locked absorb sunlight energy on surfaces exposed to the Sun, but those surfaces do not re-emit much of that energy until after the object has rotated, so that the emission is parallel to the object's orbit. This results in a very small acceleration parallel to the orbital path, yet one which can be significant for small objects over millions of years. The Poynting-Robertson effect is a force opposing the object's velocity caused by asymmetric incidence of light, i.e., aberration of light. For an object with prograde rotation, these two effects will apply opposing, but generally unequal, forces.

Gravitational radiation

Gravitational radiation is another mechanism of orbital decay. It is negligible for orbits of planets and planetary satellites (when considering their orbital motion on time scales of centuries, decades, and less), but is noticeable for systems of compact objects, as seen in observations of neutron star orbits. All orbiting bodies radiate gravitational energy, hence no orbit is infinitely stable.

Electromagnetic drag

Satellites using an electrodynamic tether, moving through the Earth's magnetic field, creates drag force that could eventually deorbit the satellite.

Stellar collision

A stellar collision is the coming together of two binary stars when they lose energy and approach each other. Several things can cause the loss of energy including tidal forces, mass transfer, and gravitational radiation. The stars describe the path of a spiral as they approach each other. This sometimes results in a merger of the two stars or the creation of a black hole. In the latter case, the last several revolutions of the stars around each other take only a few seconds. [5]

Mass concentration

While not a direct cause of orbital decay, uneven mass distributions (known as mascons) of the body being orbited can perturb orbits over time, and extreme distributions can cause orbits to be highly unstable. The resulting unstable orbit can mutate into an orbit where one of the direct causes of orbital decay can take place.

Related Research Articles

Acceleration Rate of change of velocity

In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities. The orientation of an object's acceleration is given by the orientation of the net force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law, is the combined effect of two causes:

Angular momentum Physical quantity

In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Motorcycles, frisbees and rifled bullets all owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes have spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system but does not uniquely determine it.

Orbit Curved path of an object around a point

In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

Potential energy Energy held by an object because of its position relative to other objects

In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.

Tidal acceleration Natural phenomenon due to which tidal locking occurs

Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller body first, and later the larger body. The Earth–Moon system is the best-studied case.

Escape velocity Concept in celestial mechanics

In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.

Roche limit Orbital radius that will break up a satellite

In celestial mechanics, the Roche limit, also called Roche radius, is the distance from a celestial body within which a second celestial body, held together only by its own force of gravity, will disintegrate because the first body's tidal forces exceed the second body's gravitational self-attraction. Inside the Roche limit, orbiting material disperses and forms rings, whereas outside the limit, material tends to coalesce. The Roche radius depends on the radius of the first body and on the ratio of the bodies' densities.

Free fall Motion of a body subject only to gravity

In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.

Work (physics) Process of energy transfer to an object via force application through displacement

In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, it is often represented as the product of force and displacement. A force is said to do positive work if it has a component in the direction of the displacement of the point of application. A force does negative work if it has a component opposite to the direction of the displacement at the point of application of the force.

Orbital mechanics Field of classical mechanics concerned with the motion of spacecraft

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body. In circular motion, the distance between the body and a fixed point on the surface remains the same.

Flight dynamics (spacecraft) Application of mechanical dynamics to model the flight of space vehicles

Spacecraft flight dynamics is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

Rotation around a fixed axis Type of motion

Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will appear.

Geodetic effect

The geodetic effect represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the Gravity Probe B experiment. The geodetic effect was first predicted by Willem de Sitter in 1916, who provided relativistic corrections to the Earth–Moon system's motion. De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker. It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace–Runge–Lenz vector.

In general relativity, Lense–Thirring precession or the Lense–Thirring effect is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a gravitomagnetic frame-dragging effect. It is a prediction of general relativity consisting of secular precessions of the longitude of the ascending node and the argument of pericenter of a test particle freely orbiting a central spinning mass endowed with angular momentum .

Kepler orbit

In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

Frame-dragging Effect of general relativity

Frame-dragging is an effect on spacetime, predicted by Albert Einstein's general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, but the masses causing that field may be non-static ⁠— rotating, for instance. More generally, the subject that deals with the effects caused by mass–energy currents is known as gravitoelectromagnetism, which is analogous to the magnetism of classical electromagnetism.

Nodal precession is the precession of the orbital plane of a satellite around the rotational axis of an astronomical body such as Earth. This precession is due to the non-spherical nature of a rotating body, which creates a non-uniform gravitational field. The following discussion relates to low Earth orbit of artificial satellites, which have no measurable effect on the motion of Earth. The nodal precession of more massive, natural satellites like the Moon is more complex.

Orbit modeling is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects. Directly modeling an orbit can push the limits of machine precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit in order to achieve better accuracy.

References

  1. "Tiangong-1 Orbital Status". Official Website of China Manned Space. China Manned Space Engineering Office. 1 April 2018. Retrieved 1 April 2018.
  2. Low, Samuel Y. W. (August 2018). "Assessment of Orbit Maintenance Strategies for Small Satellites". AIAA/USU Conference on Small Satellites. 32. doi:10.26077/bffw-p652.
  3. Nwankwo, Victor U. J.; Chakrabarti, Sandip K. (1 May 2013). "Effects of Plasma Drag on Low Earth Orbiting Satellites due to Heating of Earth's Atmosphere by Coronal Mass Ejections". arXiv: 1305.0233 [physics.space-phn].
  4. The Hubble Program – Servicing Missions – SM4
  5. "INSPIRAL GRAVITATIONAL WAVES". LIGO. Retrieved 1 May 2015.