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.

International Celestial Reference System and its realizations

From Wikipedia, the free encyclopedia

The International Celestial Reference System (ICRS) is the current standard celestial reference system adopted by the International Astronomical Union (IAU). Its origin is at the barycenter of the Solar System, with axes that are intended to "show no global rotation with respect to a set of distant extragalactic objects". This fixed reference system differs from previous reference systems, which had been based on Catalogues of Fundamental Stars that had published the positions of stars based on direct "observations of [their] equatorial coordinates, right ascension and declination" and had adopted as "privileged axes ... the mean equator and the dynamical equinox" at a particular date and time.

The International Celestial Reference Frame (ICRF) is a realization of the International Celestial Reference System using reference celestial sources observed at radio wavelengths. In the context of the ICRS, a reference frame (RF) is the physical realization of a reference system, i.e., the reference frame is the set of numerical coordinates of the reference sources, derived using the procedures spelled out by the ICRS.

More specifically, the ICRF is an inertial barycentric reference frame whose axes are defined by the measured positions of extragalactic sources (mainly quasars) observed using very-long-baseline interferometry while the Gaia-CRF is an inertial barycentric reference frame defined by optically measured positions of extragalactic sources by the Gaia satellite and whose axes are rotated to conform to the ICRF. Although general relativity implies that there are no true inertial frames around gravitating bodies, these reference frames are important because they do not exhibit any measurable angular rotation since the extragalactic sources used to define the ICRF and the Gaia-CRF are so far away. The ICRF and the Gaia-CRF are now the standard reference frames used to define the positions of astronomical objects.

Reference systems and frames

It is useful to distinguish reference systems and reference frames. A reference frame has been defined as "a catalogue of the adopted coordinates of a set of reference objects that serves to define, or realize, a particular coordinate frame". A reference system is a broader concept, encompassing "the totality of procedures, models and constants that are required for the use of one or more reference frames".

Realizations

The ICRF is based on hundreds of extra-galactic radio sources, mostly quasars, distributed around the entire sky. Because they are so distant, they are apparently stationary to our current technology, yet their positions can be measured very accurately by Very Long Baseline Interferometry (VLBI). The positions of most are known to 1 milliarcsecond (mas) or better.

In August 1997, the International Astronomical Union resolved in Resolution B2 of its XXIIIrd General Assembly "that the Hipparcos Catalogue shall be the primary realization of the ICRS at optical wavelengths." The Hipparcos Celestial Reference Frame (HCRF) is based on a subset of about 100,000 stars in the Hipparcos Catalogue. In August 2021 the International Astronomical Union decided in Resolution B3 of its XXXIst General Assembly "that as from 1 January 2022, the fundamental realization of the International Celestial Reference System (ICRS) shall comprise the Third Realization of the International Celestial Reference Frame (ICRF3) for the radio domain and the Gaia-CRF3 for the optical domain."

Radio wavelengths (ICRF)

ICRF1

The ICRF, now called ICRF1, was adopted by the International Astronomical Union (IAU) as of 1 January 1998. ICRF1 was oriented to the axes of the ICRS, which reflected the prior astronomical reference frame The Fifth Fundamental Catalog (FK5). It had an angular noise floor of approximately 250 microarcseconds (μas) and a reference axis stability of approximately 20 μas; this was an order-of-magnitude improvement over the previous reference frame derived from (FK5). The ICRF1 contains 212 defining sources and also contains positions of 396 additional non-defining sources for reference. The positions of these sources have been adjusted in later extensions to the catalogue. ICRF1 agrees with the orientation of the Fifth Fundamental Catalog (FK5) "J2000.0" frame to within the (lower) precision of the latter.

ICRF2

An updated reference frame ICRF2 was created in 2009. The update was a joint collaboration of the International Astronomical Union, the International Earth Rotation and Reference Systems Service, and the International VLBI Service for Geodesy and Astrometry. ICRF2 is defined by the position of 295 compact radio sources (97 of which also define ICRF1). Alignment of ICRF2 with ICRF1-Ext2, the second extension of ICRF1, was made with 138 sources common to both reference frames. Including non-defining sources, it comprises 3414 sources measured using very-long-baseline interferometry. The ICRF2 has a noise floor of approximately 40 μas and an axis stability of approximately 10 μas. Maintenance of the ICRF2 will be accomplished by a set of 295 sources that have especially good positional stability and unambiguous spatial structure.

The data used to derive the reference frame come from approximately 30 years of VLBI observations, from 1979 to 2009. Radio observations in both the S-band (2.3 GHz) and X-band (8.4 GHz) were recorded simultaneously to allow correction for ionospheric effects. The observations resulted in about 6.5 million group-delay measurements among pairs of telescopes. The group delays were processed with software that takes into account atmospheric and geophysical processes. The positions of the reference sources were treated as unknowns to be solved for by minimizing the mean squared error across group-delay measurements. The solution was constrained to be consistent with the International Terrestrial Reference Frame (ITRF2008) and earth orientation parameters (EOP) systems.

ICRF3

Sky distribution of the 303 "defining sources" in the ICRF3

ICRF3 is the third major revision of the ICRF, and was adopted by the IAU in August 2018 and became effective 1 January 2019. The modeling incorporates the effect of the galactocentric acceleration of the solar system, a new feature over and above ICRF2. ICRF3 also includes measurements at three frequency bands, providing three independent, and slightly different, realizations of the ICRS: dual frequency measurements at 8.4 GHz (X band) and 2.3 GHz (S band) for 4536 sources; measurements of 824 sources at 24 GHz (K band), and dual frequency measurements at 32 GHz (Ka band) and 8.4 GHz (X band) for 678 sources. Of these, 303 sources, uniformly distributed on the sky, are identified as "defining sources" which fix the axes of the frame. ICRF3 also increases the number of defining sources in the southern sky.

Optical wavelengths

Hipparcos Celestial Reference Frame (HCRF)

In 1991 the International Astronomical Union recommended "that observing programmes be undertaken or continued in order to ... determine the relationship between catalogues of extragalactic source positions and ... the [stars of the] FK5 and Hipparcos catalogues." Using a variety of linking techniques, the coordinate axes defined by the Hipparcos catalogue were aligned with the extragalactic radio frame. In August 1997, the International Astronomical Union recognized in Resolution B2 of its XXIIIrd General Assembly "That the Hipparcos Catalogue was finalized in 1996 and that its coordinate frame is aligned to that of the frame of the extragalactic sources [ICRF1] with one sigma uncertainties of ±0.6 milliarcseconds (mas)" and resolved "that the Hipparcos Catalogue shall be the primary realization of the ICRS at optical wavelengths."

Second Gaia celestial reference frame (Gaia–CRF2)

The second Gaia celestial reference frame (Gaia–CRF2), based on 22 months of observations of over half a million extragalactic sources by the Gaia spacecraft, appeared in 2018 and has been described as "the first full-fledged optical realisation of the ICRS, that is to say, an optical reference frame built only on extragalactic sources." The axes of Gaia-CRF2 were aligned to a prototype version of the forthcoming ICRF3 using 2820 objects common to Gaia-CRF2 and to the ICRF3 prototype.

Third Gaia celestial reference frame (Gaia–CRF3)

The third Gaia celestial reference frame (Gaia–CRF3) is based on 33 months of observations of 1,614,173 extragalactic sources. As with the earlier Hipparcos and Gaia reference frames, the axes of Gaia-CRF3 were aligned to 3142 optical counterparts of ICRF-3 in the S/X frequency bands. In August 2021 the International Astronomical Union noted that the Gaia-CRF3 had "largely superseded the Hipparcos Catalogue" and was "de facto the optical realization of the Celestial Reference Frame within the astronomical community." Consequently, the IAU decided that Gaia-CRF3 shall be "the fundamental realization of the International Celestial Reference System (ICRS) ... for the optical domain."