Active galactic nucleus

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Active_galactic_nucleus

An active galactic nucleus (AGN) is a compact region at the center of a galaxy that emits a significant amount of energy across the electromagnetic spectrum, with characteristics indicating that this luminosity is not produced by the stars. Such excess, non-stellar emissions have been observed in the radio, microwave, infrared, optical, ultra-violet, X-ray and gamma ray wavebands. A galaxy hosting an AGN is called an active galaxy. The non-stellar radiation from an AGN is theorized to result from the accretion of matter by a supermassive black hole at the center of its host galaxy.

Active galactic nuclei are the most luminous persistent sources of electromagnetic radiation in the universe and, as such, can be used as a means of discovering distant objects; their evolution as a function of cosmic time also puts constraints on models of the cosmos.

The observed characteristics of an AGN depend on several properties such as the mass of the central black hole, the rate of gas accretion onto the black hole, the orientation of the accretion disk, the degree of obscuration of the nucleus by dust, and presence or absence of jets.

Numerous subclasses of AGN have been defined on the basis of their observed characteristics; the most powerful AGN are classified as quasars. A blazar is an AGN with a jet pointed toward the Earth, in which radiation from the jet is enhanced by relativistic beaming.

History

Quasar 3C 273 observed by the Hubble Space Telescope. The relativistic jet of 3C 273 appears to the left of the bright quasar, and the four straight lines pointing outward from the central source are diffraction spikes caused by the telescope optics.

During the first half of the 20th century, photographic observations of nearby galaxies detected some characteristic signatures of AGN emission, although there was not yet a physical understanding of the nature of the AGN phenomenon. Some early observations included the first spectroscopic detection of emission lines from the nuclei of NGC 1068 and Messier 81 by Edward Fath (published in 1909), and the discovery of the jet in Messier 87 by Heber Curtis (published in 1918). Further spectroscopic studies by astronomers including Vesto Slipher, Milton Humason, and Nicholas Mayall noted the presence of unusual emission lines in some galaxy nuclei. In 1943, Carl Seyfert published a paper in which he described observations of nearby galaxies having bright nuclei that were sources of unusually broad emission lines. Galaxies observed as part of this study included NGC 1068, NGC 4151, NGC 3516, and NGC 7469. Active galaxies such as these are known as Seyfert galaxies in honor of Seyfert's pioneering work.

The development of radio astronomy was a major catalyst to understanding AGN. Some of the earliest detected radio sources are nearby active elliptical galaxies such as Messier 87 and Centaurus A. Another radio source, Cygnus A, was identified by Walter Baade and Rudolph Minkowski as a tidally distorted galaxy with an unusual emission-line spectrum, having a recessional velocity of 16,700 kilometers per second. The 3C radio survey led to further progress in discovery of new radio sources as well as identifying the visible-light sources associated with the radio emission. In photographic images, some of these objects were nearly point-like or quasi-stellar in appearance, and were classified as quasi-stellar radio sources (later abbreviated as "quasars").

Soviet Armenian astrophysicist Viktor Ambartsumian introduced Active Galactic Nuclei in the early 1950s. At the Solvay Conference on Physics in 1958, Ambartsumian presented a report arguing that "explosions in galactic nuclei cause large amounts of mass to be expelled. For these explosions to occur, galactic nuclei must contain bodies of huge mass and unknown nature. From this point forward Active Galactic Nuclei (AGN) became a key component in theories of galactic evolution." His idea was initially accepted skeptically.

A major breakthrough was the measurement of the redshift of the quasar 3C 273 by Maarten Schmidt, published in 1963. Schmidt noted that if this object was extragalactic (outside the Milky Way, at a cosmological distance) then its large redshift of 0.158 implied that it was the nuclear region of a galaxy about 100 times more powerful than other radio galaxies that had been identified. Shortly afterward, optical spectra were used to measure the redshifts of a growing number of quasars including 3C 48, even more distant at redshift 0.37.

The enormous luminosities of these quasars as well as their unusual spectral properties indicated that their power source could not be ordinary stars. Accretion of gas onto a supermassive black hole was suggested as the source of quasars' power in papers by Edwin Salpeter and Yakov Zeldovich in 1964. In 1969 Donald Lynden-Bell proposed that nearby galaxies contain supermassive black holes at their centers as relics of "dead" quasars, and that black hole accretion was the power source for the non-stellar emission in nearby Seyfert galaxies. In the 1960s and 1970s, early X-ray astronomy observations demonstrated that Seyfert galaxies and quasars are powerful sources of X-ray emission, which originates from the inner regions of black hole accretion disks.

Today, AGN are a major topic of astrophysical research, both observational and theoretical. AGN research encompasses observational surveys to find AGN over broad ranges of luminosity and redshift, examination of the cosmic evolution and growth of black holes, studies of the physics of black hole accretion and the emission of electromagnetic radiation from AGN, examination of the properties of jets and outflows of matter from AGN, and the impact of black hole accretion and quasar activity on galaxy evolution.

Models

Since the late 1960s it has been argued that an AGN must be powered by accretion of mass onto massive black holes (10⁶ to 10¹⁰ times the Solar mass). AGN are both compact and persistently extremely luminous. Accretion can potentially give very efficient conversion of potential and kinetic energy to radiation, and a massive black hole has a high Eddington luminosity, and as a result, it can provide the observed high persistent luminosity. Supermassive black holes are now believed to exist in the centres of most if not all massive galaxies since the mass of the black hole correlates well with the velocity dispersion of the galactic bulge (the M–sigma relation) or with bulge luminosity. Thus, AGN-like characteristics are expected whenever a supply of material for accretion comes within the sphere of influence of the central black hole.

Accretion disc

Main article: Accretion disc

In the standard model of AGN, cold material close to a black hole forms an accretion disc. Dissipative processes in the accretion disc transport matter inwards and angular momentum outwards, while causing the accretion disc to heat up. The expected spectrum of an accretion disc peaks in the optical-ultraviolet waveband; in addition, a corona of hot material forms above the accretion disc and can inverse-Compton scatter photons up to X-ray energies. The radiation from the accretion disc excites cold atomic material close to the black hole and this in turn radiates at particular emission lines. A large fraction of the AGN's radiation may be obscured by interstellar gas and dust close to the accretion disc, but (in a steady-state situation) this will be re-radiated at some other waveband, most likely the infrared.

Relativistic jets

Main article: Relativistic jet

Image taken by the Hubble Space Telescope of a 5000-light-year-long jet ejected from the active galaxy M87. The blue synchrotron radiation contrasts with the yellow starlight from the host galaxy.

Some accretion discs produce jets of twin, highly collimated, and fast outflows that emerge in opposite directions from close to the disc. The direction of the jet ejection is determined either by the angular momentum axis of the accretion disc or the spin axis of the black hole. The jet production mechanism and indeed the jet composition on very small scales are not understood at present due to the resolution of astronomical instruments being too low. The jets have their most obvious observational effects in the radio waveband, where very-long-baseline interferometry can be used to study the synchrotron radiation they emit at resolutions of sub-parsec scales. However, they radiate in all wavebands from the radio through to the gamma-ray range via the synchrotron and the inverse-Compton scattering process, and so AGN jets are a second potential source of any observed continuum radiation.

Radiatively inefficient AGN

There exists a class of "radiatively inefficient" solutions to the equations that govern accretion. Several theories exist, but the most widely known of these is the Advection Dominated Accretion Flow (ADAF). In this type of accretion, which is important for accretion rates well below the Eddington limit, the accreting matter does not form a thin disc and consequently does not efficiently radiate away the energy that it acquired as it moved close to the black hole. Radiatively inefficient accretion has been used to explain the lack of strong AGN-type radiation from massive black holes at the centres of elliptical galaxies in clusters, where otherwise we might expect high accretion rates and correspondingly high luminosities. Radiatively inefficient AGN would be expected to lack many of the characteristic features of standard AGN with an accretion disc.

Particle acceleration

AGN are a candidate source of high and ultra-high energy cosmic rays (see also Centrifugal mechanism of acceleration).

Observational characteristics

Among the many interesting characteristics of AGNs:

• very high luminosity, visible out to very high red shifts,
• small emitting regions, milli-parsecs in diameter,
• strong evolution of luminosity functions,
• detectable emission across the entire electromagnetic spectrum.

Types of active galaxy

It is convenient to divide AGN into two classes, conventionally called radio-quiet and radio-loud. Radio-loud objects have emission contributions from both the jet(s) and the lobes that the jets inflate. These emission contributions dominate the luminosity of the AGN at radio wavelengths and possibly at some or all other wavelengths. Radio-quiet objects are simpler since jet and any jet-related emission can be neglected at all wavelengths.

AGN terminology is often confusing, since the distinctions between different types of AGN sometimes reflect historical differences in how the objects were discovered or initially classified, rather than real physical differences.

Radio-quiet AGN

• Low-ionization nuclear emission-line regions (LINERs). As the name suggests, these systems show only weak nuclear emission-line regions, and no other signatures of AGN emission. It is debatable whether all such systems are true AGN (powered by accretion on to a supermassive black hole). If they are, they constitute the lowest-luminosity class of radio-quiet AGN. Some may be radio-quiet analogues of the low-excitation radio galaxies (see below).
• Seyfert galaxies. Seyferts were the earliest distinct class of AGN to be identified. They show optical range nuclear continuum emission, narrow and occasionally broad emission lines, occasionally strong nuclear X-ray emission and sometimes a weak small-scale radio jet. Originally they were divided into two types known as Seyfert 1 and 2: Seyfert 1s show strong broad emission lines while Seyfert 2s do not, and Seyfert 1s are more likely to show strong low-energy X-ray emission. Various forms of elaboration on this scheme exist: for example, Seyfert 1s with relatively narrow broad lines are sometimes referred to as narrow-line Seyfert 1s. The host galaxies of Seyferts are usually spiral or irregular galaxies.
• Radio-quiet quasars/QSOs. These are essentially more luminous versions of Seyfert 1s: the distinction is arbitrary and is usually expressed in terms of a limiting optical magnitude. Quasars were originally 'quasi-stellar' in optical images as they had optical luminosities that were greater than that of their host galaxy. They always show strong optical continuum emission, X-ray continuum emission, and broad and narrow optical emission lines. Some astronomers use the term QSO (Quasi-Stellar Object) for this class of AGN, reserving 'quasar' for radio-loud objects, while others talk about radio-quiet and radio-loud quasars. The host galaxies of quasars can be spirals, irregulars or ellipticals. There is a correlation between the quasar's luminosity and the mass of its host galaxy, in that the most luminous quasars inhabit the most massive galaxies (ellipticals).
• 'Quasar 2s'. By analogy with Seyfert 2s, these are objects with quasar-like luminosities but without strong optical nuclear continuum emission or broad line emission. They are scarce in surveys, though a number of possible candidate quasar 2s have been identified.

Radio-loud AGN

There are several subtypes of radio-loud active galactic nuclei.

• Radio-loud quasars behave exactly like radio-quiet quasars with the addition of emission from a jet. Thus they show strong optical continuum emission, broad and narrow emission lines, and strong X-ray emission, together with nuclear and often extended radio emission.
• "Blazars" (BL Lac objects and OVV quasars) classes are distinguished by rapidly variable, polarized optical, radio and X-ray emission. BL Lac objects show no optical emission lines, broad or narrow, so that their redshifts can only be determined from features in the spectra of their host galaxies. The emission-line features may be intrinsically absent or simply swamped by the additional variable component. In the latter case, emission lines may become visible when the variable component is at a low level. OVV quasars behave more like standard radio-loud quasars with the addition of a rapidly variable component. In both classes of source, the variable emission is believed to originate in a relativistic jet oriented close to the line of sight. Relativistic effects amplify both the luminosity of the jet and the amplitude of variability.
• Radio galaxies. These objects show nuclear and extended radio emission. Their other AGN properties are heterogeneous. They can broadly be divided into low-excitation and high-excitation classes. Low-excitation objects show no strong narrow or broad emission lines, and the emission lines they do have may be excited by a different mechanism. Their optical and X-ray nuclear emission is consistent with originating purely in a jet. They may be the best current candidates for AGN with radiatively inefficient accretion. By contrast, high-excitation objects (narrow-line radio galaxies) have emission-line spectra similar to those of Seyfert 2s. The small class of broad-line radio galaxies, which show relatively strong nuclear optical continuum emission probably includes some objects that are simply low-luminosity radio-loud quasars. The host galaxies of radio galaxies, whatever their emission-line type, are essentially always ellipticals.

Features of different types of galaxies

Galaxy type	Active nuclei	Emission lines		X-rays	Excess of		Strong radio	Jets	Variable	Radio loud
		Narrow	Broad		UV	Far-IR
Normal (non-AGN)	no	weak	no	weak	no	no	no	no	no	no
LINER	unknown	weak	weak	weak	no	no	no	no	no	no
Seyfert I	yes	yes	yes	some	some	yes	few	no	yes	no
Seyfert II	yes	yes	no	some	some	yes	few	no	yes	no
Quasar	yes	yes	yes	some	yes	yes	some	some	yes	some
Blazar	yes	no	some	yes	yes	no	yes	yes	yes	yes
BL Lac	yes	no	no/faint	yes	yes	no	yes	yes	yes	yes
OVV	yes	no	stronger than BL Lac	yes	yes	no	yes	yes	yes	yes
Radio galaxy	yes	some	some	some	some	yes	yes	yes	yes	yes

Unification of AGN species

Unified AGN models

Unified models propose that different observational classes of AGN are a single type of physical object observed under different conditions. The currently favoured unified models are 'orientation-based unified models' meaning that they propose that the apparent differences between different types of objects arise simply because of their different orientations to the observer. However, they are debated (see below).

Radio-quiet unification

At low luminosities, the objects to be unified are Seyfert galaxies. The unification models propose that in Seyfert 1s the observer has a direct view of the active nucleus. In Seyfert 2s the nucleus is observed through an obscuring structure which prevents a direct view of the optical continuum, broad-line region or (soft) X-ray emission. The key insight of orientation-dependent accretion models is that the two types of object can be the same if only certain angles to the line of sight are observed. The standard picture is of a torus of obscuring material surrounding the accretion disc. It must be large enough to obscure the broad-line region but not large enough to obscure the narrow-line region, which is seen in both classes of object. Seyfert 2s are seen through the torus. Outside the torus there is material that can scatter some of the nuclear emission into our line of sight, allowing us to see some optical and X-ray continuum and, in some cases, broad emission lines—which are strongly polarized, showing that they have been scattered and proving that some Seyfert 2s really do contain hidden Seyfert 1s. Infrared observations of the nuclei of Seyfert 2s also support this picture.

At higher luminosities, quasars take the place of Seyfert 1s, but, as already mentioned, the corresponding 'quasar 2s' are elusive at present. If they do not have the scattering component of Seyfert 2s they would be hard to detect except through their luminous narrow-line and hard X-ray emission.

Radio-loud unification

Historically, work on radio-loud unification has concentrated on high-luminosity radio-loud quasars. These can be unified with narrow-line radio galaxies in a manner directly analogous to the Seyfert 1/2 unification (but without the complication of much in the way of a reflection component: narrow-line radio galaxies show no nuclear optical continuum or reflected X-ray component, although they do occasionally show polarized broad-line emission). The large-scale radio structures of these objects provide compelling evidence that the orientation-based unified models really are true. X-ray evidence, where available, supports the unified picture: radio galaxies show evidence of obscuration from a torus, while quasars do not, although care must be taken since radio-loud objects also have a soft unabsorbed jet-related component, and high resolution is necessary to separate out thermal emission from the sources' large-scale hot-gas environment. At very small angles to the line of sight, relativistic beaming dominates, and we see a blazar of some variety.

However, the population of radio galaxies is completely dominated by low-luminosity, low-excitation objects. These do not show strong nuclear emission lines—broad or narrow—they have optical continua which appear to be entirely jet-related, and their X-ray emission is also consistent with coming purely from a jet, with no heavily absorbed nuclear component in general. These objects cannot be unified with quasars, even though they include some high-luminosity objects when looking at radio emission, since the torus can never hide the narrow-line region to the required extent, and since infrared studies show that they have no hidden nuclear component: in fact there is no evidence for a torus in these objects at all. Most likely, they form a separate class in which only jet-related emission is important. At small angles to the line of sight, they will appear as BL Lac objects.

Criticism of the radio-quiet unification

In the recent literature on AGN, being subject to an intense debate, an increasing set of observations appear to be in conflict with some of the key predictions of the Unified Model, e.g. that each Seyfert 2 has an obscured Seyfert 1 nucleus (a hidden broad-line region).

Therefore, one cannot know whether the gas in all Seyfert 2 galaxies is ionized due to photoionization from a single, non-stellar continuum source in the center or due to shock-ionization from e.g. intense, nuclear starbursts. Spectropolarimetric studies reveal that only 50% of Seyfert 2s show a hidden broad-line region and thus split Seyfert 2 galaxies into two populations. The two classes of populations appear to differ by their luminosity, where the Seyfert 2s without a hidden broad-line region are generally less luminous. This suggests absence of broad-line region is connected to low Eddington ratio, and not to obscuration.

The covering factor of the torus might play an important role. Some torus models predict how Seyfert 1s and Seyfert 2s can obtain different covering factors from a luminosity and accretion rate dependence of the torus covering factor, something supported by studies in the x-ray of AGN. The models also suggest an accretion-rate dependence of the broad-line region and provide a natural evolution from more active engines in Seyfert 1s to more "dead" Seyfert 2s and can explain the observed break-down of the unified model at low luminosities and the evolution of the broad-line region.

While studies of single AGN show important deviations from the expectations of the unified model, results from statistical tests have been contradictory. The most important short-coming of statistical tests by direct comparisons of statistical samples of Seyfert 1s and Seyfert 2s is the introduction of selection biases due to anisotropic selection criteria.

Studying neighbour galaxies rather than the AGN themselves first suggested the numbers of neighbours were larger for Seyfert 2s than for Seyfert 1s, in contradiction with the Unified Model. Today, having overcome the previous limitations of small sample sizes and anisotropic selection, studies of neighbours of hundreds to thousands of AGN have shown that the neighbours of Seyfert 2s are intrinsically dustier and more star-forming than Seyfert 1s and a connection between AGN type, host galaxy morphology and collision history. Moreover, angular clustering studies of the two AGN types confirm that they reside in different environments and show that they reside within dark matter halos of different masses. The AGN environment studies are in line with evolution-based unification models where Seyfert 2s transform into Seyfert 1s during merger, supporting earlier models of merger-driven activation of Seyfert 1 nuclei.

While controversy about the soundness of each individual study still prevails, they all agree on that the simplest viewing-angle based models of AGN Unification are incomplete. Seyfert-1 and Seyfert-2 seem to differ in star formation and AGN engine power.

While it still might be valid that an obscured Seyfert 1 can appear as a Seyfert 2, not all Seyfert 2s must host an obscured Seyfert 1. Understanding whether it is the same engine driving all Seyfert 2s, the connection to radio-loud AGN, the mechanisms of the variability of some AGN that vary between the two types at very short time scales, and the connection of the AGN type to small and large-scale environment remain important issues to incorporate into any unified model of active galactic nuclei.

A study of Swift/BAT AGN published in July 2022 adds support to the "radiation-regulated unification model" outlined in 2017. In this model, the relative accretion rate (termed the "Eddington ratio") of the black hole has a significant impact on the observed features of the AGN. Black Holes with higher Eddington ratios appear to be more likely to be unobscured, having cleared away locally obscuring material in a very short timescale.

Cosmological uses and evolution

For a long time, active galaxies held all the records for the highest-redshift objects known either in the optical or the radio spectrum, because of their high luminosity. They still have a role to play in studies of the early universe, but it is now recognised that an AGN gives a highly biased picture of the "typical" high-redshift galaxy.

Most luminous classes of AGN (radio-loud and radio-quiet) seem to have been much more numerous in the early universe. This suggests that massive black holes formed early on and that the conditions for the formation of luminous AGN were more common in the early universe, such as a much higher availability of cold gas near the centre of galaxies than at present. It also implies that many objects that were once luminous quasars are now much less luminous, or entirely quiescent. The evolution of the low-luminosity AGN population is much less well understood due to the difficulty of observing these objects at high redshifts.

Three-body problem

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Three-body_problem

Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities. The center of mass, in accordance with the law of conservation of momentum, remains in place.

In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses that orbit each other in space and calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.

Unlike the two-body problem, the three-body problem has no general closed-form solution, meaning there is no equation that always solves it. When three bodies orbit each other, the resulting dynamical system is chaotic for most initial conditions. Because there are no solvable equations for most three-body systems, the only way to predict the motions of the bodies is to estimate them using numerical methods.

The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Earth, the Moon, and the Sun. In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.

Mathematical description

The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions ${\displaystyle {\mathbf{r}}_{\mathbf{i}}=(x_i,y_i,z_i)}$ of three gravitationally interacting bodies with masses ${\displaystyle m_i}$:

$${\displaystyle \begin{array}{rl}{\ddot{\mathbf{r}}}_{\mathbf{1}}& =-G{m}_2\frac{{\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_{\mathbf{2}}}{|{\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_{\mathbf{2}}{|}^3}-G{m}_3\frac{{\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_{\mathbf{3}}}{|{\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_{\mathbf{3}}{|}^3},\\ {\ddot{\mathbf{r}}}_{\mathbf{2}}& =-G{m}_3\frac{{\mathbf{r}}_{\mathbf{2}}-{\mathbf{r}}_{\mathbf{3}}}{|{\mathbf{r}}_{\mathbf{2}}-{\mathbf{r}}_{\mathbf{3}}{|}^3}-G{m}_1\frac{{\mathbf{r}}_{\mathbf{2}}-{\mathbf{r}}_{\mathbf{1}}}{|{\mathbf{r}}_{\mathbf{2}}-{\mathbf{r}}_{\mathbf{1}}{|}^3},\\ {\ddot{\mathbf{r}}}_{\mathbf{3}}& =-G{m}_1\frac{{\mathbf{r}}_{\mathbf{3}}-{\mathbf{r}}_{\mathbf{1}}}{|{\mathbf{r}}_{\mathbf{3}}-{\mathbf{r}}_{\mathbf{1}}{|}^3}-G{m}_2\frac{{\mathbf{r}}_{\mathbf{3}}-{\mathbf{r}}_{\mathbf{2}}}{|{\mathbf{r}}_{\mathbf{3}}-{\mathbf{r}}_{\mathbf{2}}{|}^3}.\end{array}}$$where ${\displaystyle G}$ is the gravitational constant. As astronomer Juhan Frank describes, "These three second-order vector differential equations are equivalent to 18 first order scalar differential equations." As June Barrow-Green notes with regard to an alternative presentation, if

${\displaystyle P_i}$ represent three particles with masses ${\displaystyle m_i}$, distances ${\displaystyle P_i}$${\displaystyle P_j}$ = ${\displaystyle r_{ij}}$, and coordinates ${\displaystyle q_{ij}}$ (i,j = 1,2,3) in an inertial coordinate system ... the problem is described by nine second-order differential equations.

The problem can also be stated equivalently in the Hamiltonian formalism, in which case it is described by a set of 18 first-order differential equations, one for each component of the positions ${\displaystyle {\mathbf{r}}_{\mathbf{i}}}$ and momenta ${\displaystyle {\mathbf{p}}_{\mathbf{i}}}$:

$${\displaystyle \frac{d{\mathbf{r}}_{\mathbf{i}}}{dt}=\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{\mathbf{p}}_{\mathbf{i}}},\frac{d{\mathbf{p}}_{\mathbf{i}}}{dt}=-\frac{\mathrm{\partial }\mathcal{H}}{\mathrm{\partial }{\mathbf{r}}_{\mathbf{i}}},}$$

where ${\displaystyle \mathcal{H}}$ is the Hamiltonian:

$${\displaystyle \mathcal{H}=-\frac{G{m}_1{m}_2}{|{\mathbf{r}}_{\mathbf{1}}-{\mathbf{r}}_{\mathbf{2}}|}-\frac{G{m}_2{m}_3}{|{\mathbf{r}}_{\mathbf{3}}-{\mathbf{r}}_{\mathbf{2}}|}-\frac{G{m}_3{m}_1}{|{\mathbf{r}}_{\mathbf{3}}-{\mathbf{r}}_{\mathbf{1}}|}+\frac{{{\mathbf{p}}_{\mathbf{1}}}^2}{2m_1}+\frac{{{\mathbf{p}}_{\mathbf{2}}}^2}{2m_2}+\frac{{{\mathbf{p}}_{\mathbf{3}}}^2}{2m_3}.}$$

In this case, ${\displaystyle \mathcal{H}}$ is simply the total energy of the system, gravitational plus kinetic.

Restricted three-body problem

The circular restricted three-body problem is a valid approximation of elliptical orbits found in the Solar System, and this can be visualized as a combination of the potentials due to the gravity of the two primary bodies along with the centrifugal effect from their rotation (Coriolis effects are dynamic and not shown). The Lagrange points can then be seen as the five places where the gradient on the resultant surface is zero, indicating that the forces are in balance there.

In the restricted three-body problem formulation, in the description of Barrow-Green,

two... bodies revolve around their centre of mass in circular orbits under the influence of their mutual gravitational attraction, and... form a two body system... [whose] motion is known. A third body (generally known as a planetoid), assumed massless with respect to the other two, moves in the plane defined by the two revolving bodies and, while being gravitationally influenced by them, exerts no influence of its own.

Per Barrow-Green, "[t]he problem is then to ascertain the motion of the third body."

That is to say, this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits. (That is, it is useful to consider the effective potential.) With respect to a rotating reference frame, the two co-orbiting bodies are stationary, and the third can be stationary as well at the Lagrangian points, or move around them, for instance on a horseshoe orbit.

The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.

Mathematically, the problem is stated as follows. Let ${\displaystyle m_{1,2}}$ be the masses of the two massive bodies, with (planar) coordinates ${\displaystyle (x_1,y_1)}$ and ${\displaystyle (x_2,y_2)}$, and let ${\displaystyle (x,y)}$ be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to ${\displaystyle 1}$. Then, the motion of the planetoid is given by:

$${\displaystyle \begin{array}{r}\frac{d^{2}x}{d{t}^2}=-m_1\frac{x-x_1}{r_1^3}-m_2\frac{x-x_2}{r_2^3},\\ \frac{d^{2}y}{d{t}^2}=-m_1\frac{y-y_1}{r_1^3}-m_2\frac{y-y_2}{r_2^3},\end{array}}$$

where ${\displaystyle r_i=\sqrt{(x-x_i{)}^2+(y-y_i{)}^2}}$. In this form the equations of motion carry an explicit time dependence through the coordinates ${\displaystyle x_i(t),y_i(t)}$; however, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis.

Solutions

General solution

While a system of 3 bodies interacting gravitationally is chaotic, a system of 3 bodies interacting elastically is not.

There is no general closed-form solution to the three-body problem. In other words, it does not have a general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.

However, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists an analytic solution to the three-body problem in the form of a Puiseux series, specifically a power series in terms of powers of t^1/3. This series converges for all real t, except for initial conditions corresponding to zero angular momentum. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having Lebesgue measure zero.

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the three-body problems. As is briefly discussed below, the only singularities in the three-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).

Collisions of any number are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. But there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:

1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.
2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the three-body problem.
3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (related to the Cauchy–Kovalevskaya theorem).
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t^1/3 (the new variable after the regularization) and if |ln s| ≤ β, then this map is given by $${\displaystyle \sigma =\frac{e^{\frac{\pi s}{2\beta }}-1}{e^{\frac{\pi s}{2\beta }}+1}.}$$

This finishes the proof of Sundman's theorem.

The corresponding series converges extremely slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^8000000 terms.

Special-case solutions

In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant.

In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant. Together with Euler's collinear solutions, these solutions form the central configurations for the three-body problem. These solutions are valid for any mass ratios, and the masses move on Keplerian ellipses. These four families are the only known solutions for which there are explicit analytic formulae. In the special case of the circular restricted three-body problem, these solutions, viewed in a frame rotating with the primaries, become points called Lagrangian points and labeled L₁, L₂, L₃, L₄, and L₅, with L₄ and L₅ being symmetric instances of Lagrange's solution.

In work summarized in 1892–1899, Henri Poincaré established the existence of an infinite number of periodic solutions to the restricted three-body problem, together with techniques for continuing these solutions into the general three-body problem.

In 1893, Meissel stated what is now called the Pythagorean three-body problem: three masses in the ratio 3:4:5 are placed at rest at the vertices of a 3:4:5 right triangle, with the heaviest body at the right angle and the lightest at the smaller acute angle. Burrau further investigated this problem in 1913. In 1967 Victor Szebehely and C. Frederick Peters established eventual escape of the lightest body for this problem using numerical integration, while at the same time finding a nearby periodic solution.

An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259

20 examples of periodic solutions to the three-body problem

In the 1970s, Michel Hénon and Roger A. Broucke each found a set of solutions that form part of the same family of solutions: the Broucke–Hénon–Hadjidemetriou family. In this family, the three objects all have the same mass and can exhibit both retrograde and direct forms. In some of Broucke's solutions, two of the bodies follow the same path.

In 1993, physicist Cris Moore at the Santa Fe Institute found a zero angular momentum solution with three equal masses moving around a figure-eight shape. In 2000, mathematicians Alain Chenciner and Richard Montgomery proved its formal existence. The solution has been shown numerically to be stable for small perturbations of the mass and orbital parameters, which makes it possible for such orbits to be observed in the physical universe. But it has been argued that this is unlikely since the domain of stability is small. For instance, the probability of a binary–binary scattering eventresulting in a figure-8 orbit has been estimated to be a small fraction of a percent.

In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.

In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.

In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem. This was followed in 2018 by an additional 1,223 new solutions for a zero-angular-momentum system of unequal masses.

In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem. The free-fall formulation starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forward and backward along an open "track".

In 2023, Ivan Hristov, Radoslava Hristova, Dmitrašinović and Kiyotaka Tanikawa published a search for "periodic free-fall orbits" three-body problem, limited to the equal-mass case, and found 12,409 distinct solutions.

Numerical approaches

Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. There have been attempts of creating computer programs that numerically solve the three-body problem (and by extension, the n-body problem) involving both electromagnetic and gravitational interactions, and incorporating modern theories of physics such as special relativity. In addition, using the theory of random walks, an approximate probability of different outcomes may be computed.

History

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Philosophiæ Naturalis Principia Mathematica, in which Newton attempted to figure out if any long term stability is possible especially for such a system like that of our Earth, the Moon, and the Sun. Guided by major Renaissance astronomers Nicolaus Copernicus, Tycho Brahe and Johannes Kepler he introduced later generations to the beginning of the gravitational three-body problem. In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of Earth and the Sun. Later, this problem was also applied to other planets' interactions with the Earth and the Sun.

The physical problem was first addressed by Amerigo Vespucci and subsequently by Galileo Galilei, as well as Simon Stevin, but they did not realize what they contributed. Though Galileo determined that the speed of fall of all bodies changes uniformly and in the same way, he did not apply it to planetary motions. Whereas in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around Earth.

Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747. It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.

From the end of the 19th century to early 20th century, the approach to solve the three-body problem with the usage of short-range attractive two-body forces was developed by scientists, which offered P.F. Bedaque, H.-W. Hammer and U. van Kolck an idea to renormalize the short-range three-body problem, providing scientists a rare example of a renormalization group limit cycle at the beginning of the 21st century. George William Hill worked on the restricted problem in the late 19th century with an application of motion of Venus and Mercury.

At the beginning of the 20th century, Karl Sundman approached the problem mathematically and systematically by providing a functional theoretical proof to the problem valid for all values of time. It was the first time scientists theoretically solved the three-body problem. However, because there was not a qualitative enough solution of this system, and it was too slow for scientists to practically apply it, this solution still left some issues unresolved. In the 1970s, implication to three-body from two-body forces had been discovered by V. Efimov, which was named the Efimov effect.

In 2017, Shijun Liao and Xiaoming Li applied a new strategy of numerical simulation for chaotic systems called the clean numerical simulation (CNS), with the use of a national supercomputer, to successfully gain 695 families of periodic solutions of the three-body system with equal mass.

In 2019, Breen et al. announced a fast neural network solver for the three-body problem, trained using a numerical integrator.

In September 2023, several possible solutions have been found to the problem according to reports.

Other problems involving three bodies

The term "three-body problem" is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.

A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.

In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force that do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force. This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.

Within the point vortex model, the motion of vortices in a two-dimensional ideal fluid is described by equations of motion that contain only first-order time derivatives. I.e. in contrast to Newtonian mechanics, it is the velocity and not the acceleration that is determined by their relative positions. As a consequence, the three-vortex problem is still integrable, while at least four vortices are required to obtain chaotic behavior. One can draw parallels between the motion of a passive tracer particle in the velocity field of three vortices and the restricted three-body problem of Newtonian mechanics.

The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.

n-body problem

The three-body problem is a special case of the n-body problem, which describes how n objects move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes; therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.