Orbital mechanics

From Wikipedia, the free encyclopedia

A satellite orbiting the earth has a tangential velocity and an inward acceleration.

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 Newton's law of universal gravitation. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers. General relativity is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy or in high-gravity situations (such as orbits close to the Sun).

History

Until the rise of space travel in the twentieth century, there was little distinction between orbital and celestial mechanics, and at the time of Sputnik, the field was called Space Dynamics (ref. the 1961 book by William Thompson of that name). The fundamental techniques, such as those used to solve the Keplerian problem (determining position as a function of time), are therefore the same in both fields. Furthermore, the history of the fields is almost entirely shared.

Johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy, publishing his laws in 1605. Isaac Newton published more general laws of celestial motion in his 1687 book, Philosophiæ Naturalis Principia Mathematica.

Practical techniques

Rules of thumb

The following rules of thumb are useful for situations approximated by classical mechanics under the standard assumptions of astrodynamics. The specific example discussed is of a satellite orbiting a planet, but the rules of thumb could also apply to other situations, such as orbits of small bodies around a star such as the Sun.

• Kepler's laws of planetary motion, which can be mathematically derived from Newton's laws, hold strictly only in describing the motion of two gravitating bodies, in the absence of non-gravitational forces, or approximately when the gravity of a single massive body like the Sun dominates other effects:
  • Orbits are elliptical, with the heavier body at one focus of the ellipse. Special cases of this are circular orbits (a circle being simply an ellipse of zero eccentricity) with the planet at the center, and parabolic orbits (which are ellipses with eccentricity of exactly 1, which is simply an infinitely long ellipse) with the planet at the focus.
  • A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured.
  • The square of a satellite's orbital period is proportional to the cube of its average distance from the planet.
• Without applying thrust (such as firing a rocket engine), the height and shape of the satellite's orbit won't change.
• A satellite in a low orbit (or low part of an elliptical orbit) moves more quickly with respect to the surface of the planet than a satellite in a higher orbit (or a high part of an elliptical orbit), due to the stronger gravitational attraction closer to the planet.
• If thrust is applied at only one point in the satellite's orbit, it will return to that same point on each subsequent orbit, though the rest of its path will change. Thus to move from one circular orbit to another, at least two brief applications of thrust are needed.
• From a circular orbit, thrust applied in a direction opposite to the satellite's motion creates an elliptical orbit with a lower periapse (lowest orbital point) at 180 degrees away from the firing point. Thrust applied in the direction of the satellite's motion creates an elliptical orbit with a higher apoapse 180 degrees away from the firing point.

The consequences of the rules of orbital mechanics are sometimes counter-intuitive. For example, if two spacecraft are in the same circular orbit and wish to dock, unless they are very close, the trailing craft cannot simply fire its engines to go faster. This will change the shape of its orbit, causing it to gain altitude and miss its target. One approach is to thrust retrograde, or opposite to the direction of motion, and then thrust again to re-circularize the orbit at a lower altitude. Because lower orbits are faster than higher orbits, the trailing craft will begin to catch up. A third firing at the right time will put the trailing craft in an elliptical orbit that intersects the path of the leading craft, approaching from below.

To the degree that the standard assumptions of astrodynamics do not hold, actual trajectories will vary from those calculated. For example, simple atmospheric drag is another complicating factor for objects in Earth orbit. These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system (see n-body problem). (Celestial mechanics uses more general rules applicable to a wider variety of situations.) The differences between classical mechanics and general relativity can also become important near large objects like stars.

Laws of astrodynamics

The fundamental laws of astrodynamics are Newton's law of universal gravitation and Newton's laws of motion, while the fundamental mathematical tool is his differential calculus.
Every orbit and trajectory outside atmospheres is in principle reversible, i.e., in the space-time function the time is reversed. The velocities are reversed and the accelerations are the same, including those due to rocket bursts. Thus if a rocket burst is in the direction of the velocity, in the reversed case it is opposite to the velocity. Of course in the case of rocket bursts there is no full reversal of events, both ways the same delta-v is used and the same mass ratio applies.

Standard assumptions in astrodynamics include non-interference from outside bodies, negligible mass for one of the bodies, and negligible other forces (such as from the solar wind, atmospheric drag, etc.). More accurate calculations can be made without these simplifying assumptions, but they are more complicated. The increased accuracy often does not make enough of a difference in the calculation to be worthwhile.

Kepler's laws of planetary motion may be derived from Newton's laws, when it is assumed that the orbiting body is subject only to the gravitational force of the central attractor. When an engine thrust or propulsive force is present, Newton's laws still apply, but Kepler's laws are invalidated. When the thrust stops, the resulting orbit will be different but will once again be described by Kepler's laws. The three laws are:

1. The orbit of every planet is an ellipse with the sun at one of the foci.
2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time.
3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits.

Escape velocity

The formula for escape velocity is easily derived as follows. The specific energy (energy per unit mass) of any space vehicle is composed of two components, the specific potential energy and the specific kinetic energy. The specific potential energy associated with a planet of mass M is given by

while the specific kinetic energy of an object is given by
Since energy is conserved, the total specific orbital energy
does not depend on the distance, , from the center of the central body to the space vehicle in question. Therefore, the object can reach infinite only if this quantity is nonnegative, which implies
The escape velocity from the Earth's surface is about 11 km/s, but that is insufficient to send the body an infinite distance because of the gravitational pull of the Sun. To escape the Solar System from a location at a distance from the Sun equal to the distance Sun–Earth, but not close to the Earth, requires around 42 km/s velocity, but there will be "part credit" for the Earth's orbital velocity for spacecraft launched from Earth, if their further acceleration (due to the propulsion system) carries them in the same direction as Earth travels in its orbit.

Formulae for free orbits

Orbits are conic sections, so, naturally, the formula for the distance of a body for a given angle corresponds to the formula for that curve in polar coordinates, which is:

where μ is called the gravitational parameter which is G * M, where M is Mass, G is the gravitational constant, m₁ and m₂ are the masses of objects 1 and 2, and h is the specific angular momentum of object 2 with respect to object 1. The parameter θ is known as the true anomaly, p is the semi-latus rectum, while e is the orbital eccentricity, all obtainable from the various forms of the six independent orbital elements.

Circular orbits

All bounded orbits where the gravity of a central body dominates are elliptical in nature. A special case of this is the circular orbit, which is an ellipse of zero eccentricity. The formula for the velocity of a body in a circular orbit at distance r from the center of gravity of mass M is

where is the gravitational constant, equal to

6.673 84 × 10⁻¹¹ m³/(kg·s²)

To properly use this formula, the units must be consistent; for example, M must be in kilograms, and r must be in meters. The answer will be in meters per second.

The quantity GM is often termed the standard gravitational parameter, which has a different value for every planet or moon in the Solar System.

Once the circular orbital velocity is known, the escape velocity is easily found by multiplying by the square root of 2:

Elliptical orbits

If 0 < e > 1, then the denominator of the equation of free orbits varies with the true anomaly θ, but remains positive, never becoming zero. Therefore, the relative position vector remains bounded, having its smallest magnitude at periapsis r_p, which is given by:

The maximum value r is reached when θ = 180. This point is called the apoapsis, and its radial coordinate, denoted r_a, is

Let 2a be the distance measured along the apse line from periapsis P to apoapsis A, as illustrated in the equation below:

Substituting the equations above, we get:

a is the semimajor axis of the ellipse. Solving for p, and substituting the result in the conic section curve formula above, we get:

Orbital period

Under standard assumptions the orbital period () of a body traveling along an elliptic orbit can be computed as:

where:

• is standard gravitational parameter,
• is length of semi-major axis.

Conclusions:

• The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis (),
• For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law).

Velocity

Under standard assumptions the orbital speed () of a body traveling along an elliptic orbit can be computed from the Vis-viva equation as:

where:

• is the standard gravitational parameter,
• is the distance between the orbiting bodies.
• is the length of the semi-major axis.

The velocity equation for a hyperbolic trajectory has either + , or it is the same with the convention that in that case a is negative.

Energy

Under standard assumptions, specific orbital energy () of elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form:

where:

• is the speed of the orbiting body,
• is the distance of the orbiting body from the center of mass of the central body,
• is the semi-major axis,
• is the standard gravitational parameter.

Conclusions:

• For a given semi-major axis the specific orbital energy is independent of the eccentricity.

Using the virial theorem we find:

• the time-average of the specific potential energy is equal to 2ε
  • the time-average of r⁻¹ is a⁻¹
• the time-average of the specific kinetic energy is equal to -ε

Parabolic orbits

If the eccentricity equals 1, then the orbit equation becomes:

where:

• is the radial distance of the orbiting body from the mass center of the central body,
• is specific angular momentum of the orbiting body,
• is the true anomaly of the orbiting body,
• is the standard gravitational parameter.

As the true anomaly θ approaches 180°, the denominator approaches zero, so that r tends towards infinity. Hence, the energy of the trajectory for which e=1 is zero, and is given by:

where:

• is the speed of the orbiting body.

In other words, the speed anywhere on a parabolic path is:

Hyperbolic orbits

If e > 1, the orbit formula,

describes the geometry of the hyperbolic orbit. The system consists of two symmetric curves. the orbiting body occupies one of them. The other one is its empty mathematical image. Clearly, the denominator of the equation above goes to zero when cosθ = -1/e. we denote this value of true anomaly

θ_∞ = cos⁻¹(-1/e)

since the radial distance approaches infinity as the true anomaly approaches θ_∞. θ_∞ is known as the true anomaly of the asymptote. Observe that θ_∞ lies between 90° and 180°. From the trig identity sin²θ+cos²θ=1 it follows that:

sinθ_∞ = (e²-1)^1/2/e

Energy

Under standard assumptions, specific orbital energy () of a hyperbolic trajectory is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form:

where:

• is the orbital velocity of orbiting body,
• is the radial distance of orbiting body from central body,
• is the negative semi-major axis,
• is standard gravitational parameter.

Hyperbolic excess velocity

Under standard assumptions the body traveling along hyperbolic trajectory will attain in infinity an orbital velocity called hyperbolic excess velocity () that can be computed as:

where:

• is standard gravitational parameter,
• is the negative semi-major axis of orbit's hyperbola.

The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by

Calculating trajectories

Kepler's equation

One approach to calculating orbits (mainly used historically) is to use Kepler's equation:

.

where M is the mean anomaly, E is the eccentric anomaly, and is the eccentricity.

With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of from periapsis is broken into two steps:

1. Compute the eccentric anomaly from true anomaly
2. Compute the time-of-flight from the eccentric anomaly

Finding the eccentric anomaly at a given time (the inverse problem) is more difficult. Kepler's equation is transcendental in , meaning it cannot be solved for algebraically. Kepler's equation can be solved for analytically by inversion.

A solution of Kepler's equation, valid for all real values of is:

Evaluating this yields:


Alternatively, Kepler's Equation can be solved numerically. First one must guess a value of and solve for time-of-flight; then adjust as necessary to bring the computed time-of-flight closer to the desired value until the required precision is achieved. Usually, Newton's method is used to achieve relatively fast convergence.

The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits. For near-parabolic orbits, eccentricity is nearly 1, and plugging into the formula for mean anomaly, , we find ourselves subtracting two nearly-equal values, and accuracy suffers. For near-circular orbits, it is hard to find the periapsis in the first place (and truly circular orbits have no periapsis at all). Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it does not hold for parabolic or hyperbolic orbits. These difficulties are what led to the development of the universal variable formulation, described below.

Conic orbits

For simple procedures, such as computing the delta-v for coplanar transfer ellipses, traditional approaches^{[clarification needed]} are fairly effective. Others, such as time-of-flight are far more complicated, especially for near-circular and hyperbolic orbits.

The patched conic approximation

The Hohmann transfer orbit alone is a poor approximation for interplanetary trajectories because it neglects the planets' own gravity. Planetary gravity dominates the behaviour of the spacecraft in the vicinity of a planet and in most cases Hohmann severely overestimates delta-v, and produces highly inaccurate prescriptions for burn timings.
A relatively simple way to get a first-order approximation of delta-v is based on the 'Patched Conic Approximation' technique. One must choose the one dominant gravitating body in each region of space through which the trajectory will pass, and to model only that body's effects in that region. For instance, on a trajectory from the Earth to Mars, one would begin by considering only the Earth's gravity until the trajectory reaches a distance where the Earth's gravity no longer dominates that of the Sun. The spacecraft would be given escape velocity to send it on its way to interplanetary space. Next, one would consider only the Sun's gravity until the trajectory reaches the neighbourhood of Mars. During this stage, the transfer orbit model is appropriate. Finally, only Mars's gravity is considered during the final portion of the trajectory where Mars's gravity dominates the spacecraft's behaviour. The spacecraft would approach Mars on a hyperbolic orbit, and a final retrograde burn would slow the spacecraft enough to be captured by Mars.

The size of the "neighborhoods" (or spheres of influence) vary with radius :

where is the semimajor axis of the planet's orbit relative to the Sun; and are the masses of the planet and Sun, respectively.

This simplification is sufficient to compute rough estimates of fuel requirements, and rough time-of-flight estimates, but it is not generally accurate enough to guide a spacecraft to its destination. For that, numerical methods are required.

The universal variable formulation

To address computational shortcomings of traditional approaches for solving the 2-body problem, the universal variable formulation was developed. It works equally well for the circular, elliptical, parabolic, and hyperbolic cases, the differential equations converging well when integrated for any orbit. It also generalizes well to problems incorporating perturbation theory.

Perturbations

The universal variable formulation works well with the variation of parameters technique, except now, instead of the six Keplerian orbital elements, we use a different set of orbital elements: namely, the satellite's initial position and velocity vectors and at a given epoch . In a two-body simulation, these elements are sufficient to compute the satellite's position and velocity at any time in the future, using the universal variable formulation.
Conversely, at any moment in the satellite's orbit, we can measure its position and velocity, and then use the universal variable approach to determine what its initial position and velocity would have been at the epoch. In perfect two-body motion, these orbital elements would be invariant (just like the Keplerian elements would be).

However, perturbations cause the orbital elements to change over time. Hence, we write the position element as and the velocity element as , indicating that they vary with time. The technique to compute the effect of perturbations becomes one of finding expressions, either exact or approximate, for the functions and .

The following are some effects which make real orbits differ from the simple models based on a spherical earth. Most of them can be handled on short timescales (perhaps less than a few thousand orbits) by perturbation theory because they are small relative to the corresponding two-body effects.

• Equatorial bulges cause precession of the node and the perigee
• Tesseral harmonics^[1] of the gravity field introduce additional perturbations
• Lunar and solar gravity perturbations alter the orbits
• Atmospheric drag reduces the semi-major axis unless make-up thrust is used

Over very long timescales (perhaps millions of orbits), even small perturbations can dominate, and the behaviour can become chaotic. On the other hand, the various perturbations can be orchestrated by clever astrodynamicists to assist with orbit maintenance tasks, such as station-keeping, ground track maintenance or adjustment, or phasing of perigee to cover selected targets at low altitude.

Orbital maneuver

In spaceflight, an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft. For spacecraft far from Earth—for example those in orbits around the Sun—an orbital maneuver is called a deep-space maneuver (DSM).^{[not verified in body]}

Orbital transfer

Transfer orbits are usually elliptical orbits that allow spacecraft to move from one (usually substantially circular) orbit to another. Usually they require a burn at the start, a burn at the end, and sometimes one or more burns in the middle.

• The Hohmann transfer orbit requires a minimal delta-v.
• A bi-elliptic transfer can require less energy than the Hohmann transfer, if the ratio of orbits is 11.94 or greater,^[2] but comes at the cost of increased trip time over the Hohmann transfer.
• Faster transfers may use any orbit that intersects both the original and destination orbits, at the cost of higher delta-v.

For the case of orbital transfer between non-coplanar orbits, the change-of-plane thrust must be made at the point where the orbital planes intersect (the "node").

A Hohmann transfer from a low circular orbit to a higher circular orbit

A bi-elliptic transfer from a low circular starting orbit (dark blue), to a higher circular orbit (red)

Generic two-impulse elliptical transfer between two circular orbits

A general transfer from a low circular orbit to a higher circular orbit

Gravity assist and the Oberth effect

In a gravity assist, a spacecraft swings by a planet and leaves in a different direction, at a different speed. This is useful to speed or slow a spacecraft instead of carrying more fuel.

This maneuver can be approximated by an elastic collision at large distances, though the flyby does not involve any physical contact. Due to Newton's Third Law (equal and opposite reaction), any momentum gained by a spacecraft must be lost by the planet, or vice versa. However, because the planet is much, much more massive than the spacecraft, the effect on the planet's orbit is negligible.

The Oberth effect can be employed, particularly during a gravity assist operation. This effect is that use of a propulsion system works better at high speeds, and hence course changes are best done when close to a gravitating body; this can multiply the effective delta-v.

Interplanetary Transport Network and fuzzy orbits

It is now possible to use computers to search for routes using the nonlinearities in the gravity of the planets and moons of the Solar System. For example, it is possible to plot an orbit from high earth orbit to Mars, passing close to one of the Earth's Trojan points. Collectively referred to as the Interplanetary Transport Network, these highly perturbative, even chaotic, orbital trajectories in principle need no fuel beyond that needed to reach the Lagrange point (in practice keeping to the trajectory requires some course corrections). The biggest problem with them is they can be exceedingly slow, taking many years to arrive. In addition launch windows can be very far apart.
They have, however, been employed on projects such as Genesis. This spacecraft visited the Earth-Sun Lagrange L₁ point and returned using very little propellant.

Theoretical astronomy

From Wikipedia, the free encyclopedia

Theoretical astronomy is the use of the analytical models of physics and chemistry to describe astronomical objects and astronomical phenomena.

Ptolemy's Almagest, although a brilliant treatise on theoretical astronomy combined with a practical handbook for computation, nevertheless includes many compromises to reconcile discordant observations. Theoretical astronomy is usually assumed to have begun with Johannes Kepler (1571–1630), and Kepler's laws. It is co-equal with observation. The general history of astronomy deals with the history of the descriptive and theoretical astronomy of the Solar System, from the late sixteenth century to the end of the nineteenth century. The major categories of works on the history of modern astronomy include general histories, national and institutional histories, instrumentation, descriptive astronomy, theoretical astronomy, positional astronomy, and astrophysics. Astronomy was early to adopt computational techniques to model stellar and galactic formation and celestial mechanics. From the point of view of theoretical astronomy, not only must the mathematical expression be reasonably accurate but it should preferably exist in a form which is amenable to further mathematical analysis when used in specific problems. Most of theoretical astronomy uses Newtonian theory of gravitation, considering that the effects of general relativity are weak for most celestial objects. The obvious fact is that theoretical astronomy cannot (and does not try) to predict the position, size and temperature of every star in the heavens. Theoretical astronomy by and large has concentrated upon analyzing the apparently complex but periodic motions of celestial objects.

Integrating astronomy and physics

"Contrary to the belief generally held by laboratory physicists, astronomy has contributed to the growth of our understanding of physics."^[1] Physics has helped in the elucidation of astronomical phenomena, and astronomy has helped in the elucidation of physical phenomena:

1. discovery of the law of gravitation came from the information provided by the motion of the Moon and the planets,
2. viability of nuclear fusion as demonstrated in the Sun and stars and yet to be reproduced on earth in a controlled form.^[1]

Integrating astronomy with physics involves

Physical interaction	Astronomical phenomena
Electromagnetism:	observation using the electromagnetic spectrum
black body radiation	stellar radiation
synchrotron radiation	radio and X-ray sources
inverse-Compton scattering	astronomical X-ray sources
acceleration of charged particles	pulsars and cosmic rays
absorption/scattering	interstellar dust
Strong and weak interaction:	nucleosynthesis in stars
	cosmic rays
	supernovae
	primeval universe
Gravity:	motion of planets, satellites and binary stars, stellar structure and evolution, N-body motions in clusters of stars and galaxies, black holes, and the expanding universe.[1]

The aim of astronomy is to understand the physics and chemistry from the laboratory that is behind cosmic events so as to enrich our understanding of the cosmos and of these sciences as well.^[1]

Integrating astronomy and chemistry

Astrochemistry, the overlap of the disciplines of astronomy and chemistry, is the study of the abundance and reactions of chemical elements and molecules in space, and their interaction with radiation. The formation, atomic and chemical composition, evolution and fate of molecular gas clouds, is of special interest because it is from these clouds that solar systems form.
Infrared astronomy, for example, has revealed that the interstellar medium contains a suite of complex gas-phase carbon compounds called aromatic hydrocarbons, often abbreviated (PAHs or PACs). These molecules composed primarily of fused rings of carbon (either neutral or in an ionized state) are said to be the most common class of carbon compound in the galaxy. They are also the most common class of carbon molecule in meteorites and in cometary and asteroidal dust (cosmic dust). These compounds, as well as the amino acids, nucleobases, and many other compounds in meteorites, carry deuterium and isotopes of carbon, nitrogen, and oxygen that are very rare on earth, attesting to their extraterrestrial origin. The PAHs are thought to form in hot circumstellar environments (around dying carbon rich red giant stars).

The sparseness of interstellar and interplanetary space results in some unusual chemistry, since symmetry-forbidden reactions cannot occur except on the longest of timescales. For this reason, molecules and molecular ions which are unstable on earth can be highly abundant in space, for example the H₃⁺ ion. Astrochemistry overlaps with astrophysics and nuclear physics in characterizing the nuclear reactions which occur in stars, the consequences for stellar evolution, as well as stellar 'generations'. Indeed, the nuclear reactions in stars produce every naturally occurring chemical element. As the stellar 'generations' advance, the mass of the newly formed elements increases.
A first-generation star uses elemental hydrogen (H) as a fuel source and produces helium (He). Hydrogen is the most abundant element, and it is the basic building block for all other elements as its nucleus has only one proton. Gravitational pull toward the center of a star creates massive amounts of heat and pressure, which cause nuclear fusion. Through this process of merging nuclear mass, heavier elements are formed. Lithium, carbon, nitrogen and oxygen are examples of elements that form in stellar fusion. After many stellar generations, very heavy elements are formed (e.g. iron and lead).

Tools of theoretical astronomy

Theoretical astronomers use a wide variety of tools which include analytical models (for example, polytropes to approximate the behaviors of a star) and computational numerical simulations. Each has some advantages.
Analytical models of a process are generally better for giving insight into the heart of what is going on. Numerical models can reveal the existence of phenomena and effects that would otherwise not be seen.^[2][3]

Astronomy theorists endeavor to create theoretical models and figure out the observational consequences of those models. This helps observers look for data that can refute a model or help in choosing between several alternate or conflicting models.

Theorists also try to generate or modify models to take into account new data. Consistent with the general scientific approach, in the case of an inconsistency, the general tendency is to try to make minimal modifications to the model to fit the data. In some cases, a large amount of inconsistent data over time may lead to total abandonment of a model.

Topics of theoretical astronomy

Topics studied by theoretical astronomers include:

1. stellar dynamics and evolution;
2. galaxy formation;
3. large-scale structure of matter in the Universe;
4. origin of cosmic rays;
5. general relativity and physical cosmology, including string cosmology and astroparticle physics.

Astrophysical relativity serves as a tool to gauge the properties of large scale structures for which gravitation plays a significant role in physical phenomena investigated and as the basis for black hole (astro)physics and the study of gravitational waves.

Astronomical models

Some widely accepted and studied theories and models in astronomy, now included in the Lambda-CDM model are the Big Bang, Cosmic inflation, dark matter, and fundamental theories of physics.

A few examples of this process:

Physical process	Experimental tool	Theoretical model	Explains/predicts
Gravitation	Radio telescopes	Self-gravitating system	Emergence of a star system
Nuclear fusion	Spectroscopy	Stellar evolution	How the stars shine and how metals formed
The Big Bang	Hubble Space Telescope, COBE	Expanding universe	Age of the Universe
Quantum fluctuations		Cosmic inflation	Flatness problem
Gravitational collapse	X-ray astronomy	General relativity	Black holes at the center of Andromeda galaxy
CNO cycle in stars

Leading topics in theoretical astronomy

Dark matter and dark energy are the current leading topics in astronomy,^[4] as their discovery and controversy originated during the study of the galaxies.

Theoretical astrophysics

Of the topics approached with the tools of theoretical physics, particular consideration is often given to stellar photospheres, stellar atmospheres, the solar atmosphere, planetary atmospheres, gaseous nebulae, nonstationary stars, and the interstellar medium. Special attention is given to the internal structure of stars.^[5]

Weak equivalence principle

The observation of a neutrino burst within 3 h of the associated optical burst from Supernova 1987A in the Large Magellanic Cloud (LMC) gave theoretical astrophysicists an opportunity to test that neutrinos and photons follow the same trajectories in the gravitational field of the galaxy.^[6]

Thermodynamics for stationary black holes

A general form of the first law of thermodynamics for stationary black holes can be derived from the microcanonical functional integral for the gravitational field.^[7] The boundary data

1. the gravitational field as described with a micocanonical system in a spatially finite region and
2. the density of states expressed formally as a functional integral over Lorentzian metrics and as a functional of the geometrical boundary data that are fixed in the corresponding action,

are the thermodynamical extensive variables, including the energy and angular momentum of the system.^[7] For the simpler case of nonrelativistic mechanics as is often observed in astrophysical phenomena associated with a black hole event horizon, the density of states can be expressed as a real-time functional integral and subsequently used to deduce Feynman's imaginary-time functional integral for the canonical partition function.^[7]

Theoretical astrochemistry

Reaction equations and large reaction networks are an important tool in theoretical astrochemistry, especially as applied to the gas-grain chemistry of the interstellar medium.^[8] Theoretical astrochemistry offers the prospect of being able to place constraints on the inventory of organics for exogenous delivery to the early Earth.

Interstellar organics

"An important goal for theoretical astrochemistry is to elucidate which organics are of true interstellar origin, and to identify possible interstellar precursors and reaction pathways for those molecules which are the result of aqueous alterations."^[9] One of the ways this goal can be achieved is through the study of carbonaceous material as found in some meteorites. Carbonaceous chondrites (such as C1 and C2) include organic compounds such as amines and amides; alcohols, aldehydes, and ketones; aliphatic and aromatic hydrocarbons; sulfonic and phosphonic acids; amino, hydroxycarboxylic, and carboxylic acids; purines and pyrimidines; and kerogen-type material.^[9] The organic inventories of primitive meteorites display large and variable enrichments in deuterium, ¹³C and ¹⁵N which is indicative of their retention of an interstellar heritage.^[9]

Chemistry in cometary comae

The chemical composition of comets should reflect both the conditions in the outer solar nebula some 4.5 x 10⁹ ayr, and the nature of the natal interstellar cloud from which the Solar system was formed.^[10] While comets retain a strong signature of their ultimate interstellar origins, significant processing must have occurred in the protosolar nebula.^[10] Early models of coma chemistry showed that reactions can occur rapidly in the inner coma, where the most important reactions are proton transfer reactions.^[10] Such reactions can potentially cycle deuterium between the different coma molecules, altering the initial D/H ratios released from the nuclear ice, and necessitating the construction of accurate models of cometary deuterium chemistry, so that gas-phase coma observations can be safely extrapolated to give nuclear D/H ratios.^[10]

Theoretical chemical astronomy

While the lines of conceptual understanding between theoretical astrochemistry and theoretical chemical astronomy often become blurred so that the goals and tools are the same, there are subtle differences between the two sciences.
Theoretical chemistry as applied to astronomy seeks to find new ways to observe chemicals in celestial objects, for example. This often leads to theoretical astrochemistry having to seek new ways to describe or explain those same observations.

Astronomical spectroscopy

The new era of chemical astronomy had to await the clear enunciation of the chemical principles of spectroscopy and the applicable theory.^[11]

Chemistry of dust condensation

Supernova radioactivity dominates light curves and the chemistry of dust condensation is also dominated by radioactivity.^[12] Dust is usually either carbon or oxides depending on which is more abundant, but Compton electrons dissociate the CO molecule in about one month.^[12] The new chemical astronomy of supernova solids depends on the supernova radioactivity:

1. the radiogenesis of ⁴⁴Ca from ⁴⁴Ti decay after carbon condensation establishes their supernova source,
2. their opacity suffices to shift emission lines blueward after 500 d and emits significant infrared luminosity,
3. parallel kinetic rates determine trace isotopes in meteoritic supernova graphites,
4. the chemistry is kinetic rather than due to thermal equilibrium and
5. is made possible by radiodeactivation of the CO trap for carbon.^[12]

Theoretical physical astronomy

Like theoretical chemical astronomy, the lines of conceptual understanding between theoretical astrophysics and theoretical physical astronomy are often blurred, but, again, there are subtle differences between these two sciences.
Theoretical physics as applied to astronomy seeks to find new ways to observe physical phenomena in celestial objects and what to look for, for example. This often leads to theoretical astrophysics having to seek new ways to describe or explain those same observations, with hopefully a convergence to improve our understanding of the local environment of Earth and the physical Universe.

Weak interaction and nuclear double beta decay

Nuclear matrix elements of relevant operators as extracted from data and from a shell-model and theoretical approximations both for the two-neutrino and neutrinoless modes of decay are used to explain the weak interaction and nuclear structure aspects of nuclear double beta decay.^[13]

Neutron-rich isotopes

New neutron-rich isotopes, ³⁴Ne, ³⁷Na, and ⁴³Si have been produced unambiguously for the first time, and convincing evidence for the particle instability of three others, ³³Ne, ³⁶Na, and ³⁹Mg has been obtained.^[14] These experimental findings compare with recent theoretical predictions.^[14]

Theory of astronomical time keeping

Until recently all the time units that appear natural to us are caused by astronomical phenomena:

1. Earth's orbit around the Sun => the year, and the seasons,
2. Moon's orbit around the Earth => the month,
3. Earth's rotation and the succession of brightness and darkness => the day (and night).

High precision appears problematic:

1. amibiguities arise in the exact definition of a rotation or revolution,
2. some astronomical processes are uneven and irregular, such as the noncommensurability of year, month, and day,
3. there are a multitude of time scales and calendars to solve the first two problems.^[15]

Some of these time scales are sidereal time, solar time, and universal time.

Atomic time

Historical accuracy of atomic clocks from NIST.

From the Systeme Internationale (SI) comes the second as defined by the duration of 9 192 631 770 cycles of a particular hyperfine structure transition in the ground state of ¹³³Cesium.^[15] For practical usability a device is required that attempts to produce the SI second (s) such as an atomic clock. But not all such clocks agree. The weighted mean of many clocks distributed over the whole Earth defines the Temps Atomique International; i.e., the Atomic Time TAI.^[15] From the General theory of relativity the time measured depends on the altitude on earth and the spatial velocity of the clock so that TAI refers to a location on sea level that rotates with the Earth.^[15]

Ephemeris time

Since the Earth's rotation is irregular, any time scale derived from it such as Greenwich Mean Time led to recurring problems in predicting the Ephemerides for the positions of the Moon, Sun, planets and their natural satellites.^[15] In 1976 the International Astronomical Union (IAU) resolved that the theoretical basis for ephemeris time (ET) was wholly non-relativistic, and therefore, beginning in 1984 ephemeris time would be replaced by two further time scales with allowance for relativistic corrections. Their names, assigned in 1979,^[16] emphasized their dynamical nature or origin, Barycentric Dynamical Time (TDB) and Terrestrial Dynamical Time (TDT). Both were defined for continuity with ET and were based on what had become the standard SI second, which in turn had been derived from the measured second of ET.

During the period 1991–2006, the TDB and TDT time scales were both redefined and replaced, owing to difficulties or inconsistencies in their original definitions. The current fundamental relativistic time scales are Geocentric Coordinate Time (TCG) and Barycentric Coordinate Time (TCB). Both of these have rates that are based on the SI second in respective reference frames (and hypothetically outside the relevant gravity well), but due to relativistic effects, their rates would appear slightly faster when observed at the Earth's surface, and therefore diverge from local Earth-based time scales using the SI second at the Earth's surface.^[17]

The currently defined IAU time scales also include Terrestrial Time (TT) (replacing TDT, and now defined as a re-scaling of TCG, chosen to give TT a rate that matches the SI second when observed at the Earth's surface),^[18] and a redefined Barycentric Dynamical Time (TDB), a re-scaling of TCB to give TDB a rate that matches the SI second at the Earth's surface.

Extraterrestrial time-keeping

Stellar dynamical time scale

For a star, the dynamical time scale is defined as the time that would be taken for a test particle released at the surface to fall under the star's potential to the centre point, if pressure forces were negligible. In other words, the dynamical time scale measures the amount of time it would take a certain star to collapse in the absence of any internal pressure. By appropriate manipulation of the equations of stellar structure this can be found to be



where R is the radius of the star, G is the gravitational constant, M is the mass of the star and v is the escape velocity. As an example, the Sun dynamical time scale is approximately 1133 seconds. Note that the actual time it would take a star like the Sun to collapse is greater because internal pressure is present.

The 'fundamental' oscillatory mode of a star will be at approximately the dynamical time scale. Oscillations at this frequency are seen in Cepheid variables.

Theory of astronomical navigation

On earth

The basic characteristics of applied astronomical navigation are

1. usable in all areas of sailing around the earth,
2. applicable autonomously (does not depend on others – persons or states) and passively (does not emit energy),
3. conditional usage via optical visibility (of horizon and celestial bodies), or state of cloudiness,
4. precisional measurement, sextant is 0.1', altitude and position is between 1.5' and 3.0'.
5. temporal determination takes a couple of minutes (using the most modern equipment) and ≤ 30 min (using classical equipment).^[19]

The superiority of satellite navigation systems to astronomical navigation are currently undeniable, especially with the development and use of GPS/NAVSTAR.^[19] This global satellite system

1. enables automated three-dimensional positioning at any moment,
2. automatically determines position continuously (every second or even more often),
3. determines position independent of weather conditions (visibility and cloudiness),
4. determines position in real time to a few meters (two carrying frequencies) and 100 m (modest commercial receivers), which is two to three orders of magnitude better than by astronomical observation,
5. is simple even without expert knowledge,
6. is relatively cheap, comparable to equipment for astronomical navigation, and
7. allows incorporation into integrated and automated systems of control and ship steering.^[19] The use of astronomical or celestial navigation is disappearing from the surface and beneath or above the surface of the earth.

Geodetic astronomy is the application of astronomical methods into networks and technical projects of geodesy for

• apparent places of stars, and their proper motions
• precise astronomical navigation
• astro-geodetic geoid determination and
• modelling the rock densities of the topography and of geological layers in the subsurface
• Satellite geodesy using the stellar background (see also astrometry and cosmic triangulation)
• Monitoring of the Earth rotation and polar wandering
• Contribution to the time system of physics and geosciences

Astronomical algorithms are the algorithms used to calculate ephemerides, calendars, and positions (as in celestial navigation or satellite navigation).

Many astronomical and navigational computations use the Figure of the Earth as a surface representing the earth.
The International Earth Rotation and Reference Systems Service (IERS), formerly the International Earth Rotation Service, is the body responsible for maintaining global time and reference frame standards, notably through its Earth Orientation Parameter (EOP) and International Celestial Reference System (ICRS) groups.

Deep space

The Deep Space Network, or DSN, is an international network of large antennas and communication facilities that supports interplanetary spacecraft missions, and radio and radar astronomy observations for the exploration of the solar system and the universe. The network also supports selected Earth-orbiting missions. DSN is part of the NASA Jet Propulsion Laboratory (JPL).

Aboard an exploratory vehicle

An observer becomes a deep space explorer upon escaping Earth's orbit.^[20] While the Deep Space Network maintains communication and enables data download from an exploratory vessel, any local probing performed by sensors or active systems aboard usually require astronomical navigation, since the enclosing network of satellites to ensure accurate positioning is absent.