Specific orbital energy

Last updated

In the gravitational two-body problem, the specific orbital energy (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy () and their total kinetic energy (), divided by the reduced mass. [1] According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time:

Contents

where

It is typically expressed in (megajoule per kilogram) or (squared kilometer per squared second). For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as characteristic energy.

Equation forms for different orbits

For an elliptic orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to: [2]

where

Proof

For an elliptic orbit with specific angular momentum h given by

we use the general form of the specific orbital energy equation,

with the relation that the relative velocity at periapsis is

Thus our specific orbital energy equation becomes

and finally with the last simplification we obtain:

For a parabolic orbit this equation simplifies to

For a hyperbolic trajectory this specific orbital energy is either given by

or the same as for an ellipse, depending on the convention for the sign of a.

In this case the specific orbital energy is also referred to as characteristic energy (or ) and is equal to the excess specific energy compared to that for a parabolic orbit.

It is related to the hyperbolic excess velocity (the orbital velocity at infinity) by

It is relevant for interplanetary missions.

Thus, if orbital position vector () and orbital velocity vector () are known at one position, and is known, then the energy can be computed and from that, for any other position, the orbital speed.

Rate of change

For an elliptic orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is

where

In the case of circular orbits, this rate is one half of the gravitation at the orbit. This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy.

Additional energy

If the central body has radius R, then the additional specific energy of an elliptic orbit compared to being stationary at the surface is

The quantity is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth). For the Earth and just little more than the additional specific energy is ; which is the kinetic energy of the horizontal component of the velocity, i.e. , .

Examples

ISS

The International Space Station has an orbital period of 91.74 minutes (5504 s), hence by Kepler's Third Law the semi-major axis of its orbit is 6,738 km.[ citation needed ]

The specific orbital energy associated with this orbit is −29.6 MJ/kg: the potential energy is −59.2 MJ/kg, and the kinetic energy 29.6 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 3.4 MJ/kg, the total extra energy is 33.0 MJ/kg. The average speed is 7.7 km/s, the net delta-v to reach this orbit is 8.1 km/s (the actual delta-v is typically 1.5–2.0 km/s more for atmospheric drag and gravity drag).

The increase per meter would be 4.4 J/kg; this rate corresponds to one half of the local gravity of 8.8 m/s2.

For an altitude of 100 km (radius is 6471 km):

The energy is −30.8 MJ/kg: the potential energy is −61.6 MJ/kg, and the kinetic energy 30.8 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 1.0 MJ/kg, the total extra energy is 31.8 MJ/kg.

The increase per meter would be 4.8 J/kg; this rate corresponds to one half of the local gravity of 9.5 m/s2. The speed is 7.8 km/s, the net delta-v to reach this orbit is 8.0 km/s.

Taking into account the rotation of the Earth, the delta-v is up to 0.46 km/s less (starting at the equator and going east) or more (if going west).

Voyager 1

For Voyager 1 , with respect to the Sun:

Hence:

Thus the hyperbolic excess velocity (the theoretical orbital velocity at infinity) is given by

However, Voyager 1 does not have enough velocity to leave the Milky Way. The computed speed applies far away from the Sun, but at such a position that the potential energy with respect to the Milky Way as a whole has changed negligibly, and only if there is no strong interaction with celestial bodies other than the Sun.

Applying thrust

Assume:

Then the time-rate of change of the specific energy of the rocket is : an amount for the kinetic energy and an amount for the potential energy.

The change of the specific energy of the rocket per unit change of delta-v is

which is |v| times the cosine of the angle between v and a.

Thus, when applying delta-v to increase specific orbital energy, this is done most efficiently if a is applied in the direction of v, and when |v| is large. If the angle between v and g is obtuse, for example in a launch and in a transfer to a higher orbit, this means applying the delta-v as early as possible and at full capacity. See also gravity drag. When passing by a celestial body it means applying thrust when nearest to the body. When gradually making an elliptic orbit larger, it means applying thrust each time when near the periapsis.

When applying delta-v to decrease specific orbital energy, this is done most efficiently if a is applied in the direction opposite to that of v, and again when |v| is large. If the angle between v and g is acute, for example in a landing (on a celestial body without atmosphere) and in a transfer to a circular orbit around a celestial body when arriving from outside, this means applying the delta-v as late as possible. When passing by a planet it means applying thrust when nearest to the planet. When gradually making an elliptic orbit smaller, it means applying thrust each time when near the periapsis.

If a is in the direction of v:

Tangential velocities at altitude

Orbit Center-to-center
distance
Altitude above
the Earth's surface
Speed Orbital period Specific orbital energy
Earth's own rotation at surface (for comparison— not an orbit)6,378 km0 km 465.1 m/s (1,674 km/h or 1,040 mph)23 h 56 min 4.09 sec−62.6 MJ/kg
Orbiting at Earth's surface (equator) theoretical6,378 km0 km7.9 km/s (28,440 km/h or 17,672 mph)1 h 24 min 18 sec−31.2 MJ/kg
Low Earth orbit 6,600–8,400 km200–2,000 km
  • Circular orbit: 7.7–6.9 km/s (27,772–24,840 km/h or 17,224–15,435 mph) respectively
  • Elliptic orbit: 10.07–8.7 km/s respectively
1 h 29 min – 2 h 8 min−29.8 MJ/kg
Molniya orbit 6,900–46,300 km500–39,900 km1.5–10.0 km/s (5,400–36,000 km/h or 3,335–22,370 mph) respectively11 h 58 min−4.7 MJ/kg
Geostationary 42,000 km35,786 km3.1 km/s (11,600 km/h or 6,935 mph)23 h 56 min 4.09 sec−4.6 MJ/kg
Orbit of the Moon 363,000–406,000 km357,000–399,000 km0.97–1.08 km/s (3,492–3,888 km/h or 2,170–2,416 mph) respectively27.27 days−0.5 MJ/kg
The lower axis gives orbital speeds of some orbits Comparison satellite navigation orbits.svg
The lower axis gives orbital speeds of some orbits

See also

Related Research Articles

<span class="mw-page-title-main">Electric field</span> Physical field surrounding an electric charge

An electric field is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental interactions of nature.

<span class="mw-page-title-main">Stress–energy tensor</span> Tensor describing energy momentum density in spacetime

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

<span class="mw-page-title-main">Noether's theorem</span> Statement relating differentiable symmetries to conserved quantities

Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

<span class="mw-page-title-main">Orbital mechanics</span> 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, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

<span class="mw-page-title-main">Hyperbolic trajectory</span>

In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In astrodynamics, the characteristic energy is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared, or energy per mass.

<span class="mw-page-title-main">Elliptic orbit</span> Kepler orbit with an eccentricity of less than one

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

<span class="mw-page-title-main">Circular orbit</span> Orbit with a fixed distance from the barycenter

A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle. In this case, not only the distance, but also the speed, angular speed, potential and kinetic energy are constant. There is no periapsis or apoapsis. This orbit has no radial version.

In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations.

<span class="mw-page-title-main">Vis-viva equation</span> Equation to model the motion of orbiting bodies

In astrodynamics, the vis-viva equation, also referred to as orbital-energy-invariance law or Burgas formula, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which applies when the only force acting on an object is its own weight which is the gravitational force determined by the product of the mass of the object and the strength of the surrounding gravitational field.

A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature.

<span class="mw-page-title-main">Spacecraft flight dynamics</span> 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.

<span class="mw-page-title-main">Maxwell stress tensor</span>

The Maxwell stress tensor is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.

<span class="mw-page-title-main">Covariant formulation of classical electromagnetism</span> Ways of writing certain laws of physics

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

In electrical engineering, dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy. It can be parameterized in terms of either the loss angleδ or the corresponding loss tangenttan(δ). Both refer to the phasor in the complex plane whose real and imaginary parts are the resistive (lossy) component of an electromagnetic field and its reactive (lossless) counterpart.

<span class="mw-page-title-main">Semi-major and semi-minor axes</span> Term in geometry; longest and shortest semidiameters of an ellipse

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

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.

<span class="mw-page-title-main">Relativistic Lagrangian mechanics</span> Mathematical formulation of special and general relativity

In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity.

<span class="mw-page-title-main">Dual graviton</span> Hypothetical particle found in supergravity

In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality, as an S-duality, predicted by some formulations of supergravity in eleven dimensions.

References

  1. "Specific energy". Marspedia. Retrieved 2022-08-12.
  2. Wie, Bong (1998). "Orbital Dynamics" . Space Vehicle Dynamics and Control. AIAA Education Series. Reston, Virginia: American Institute of Aeronautics and Astronautics. p.  220. ISBN   1-56347-261-9.