James Jeans

How to use this course

James Jeans had two careers. From 1901 to 1928 he was a mathematical physicist of the Cambridge school, applying classical mechanics and kinetic theory to gases, radiation, rotating fluids, star clusters and galaxies. From 1928 until his death he was a writer and broadcaster who brought modern astronomy to a mass audience. Several of his results are permanent tools: the Jeans length and mass of star formation, the Jeans escape of planetary atmospheres, the Rayleigh–Jeans law, and the Jeans theorem and Jeans equations of galactic dynamics. Others, such as his tidal theory of the solar system and his liquid stars, were influential and then abandoned. This course treats both kinds honestly.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The order of Part II roughly follows the chronology of his research: instability (1902), gases (1904), radiation (1905), rotating fluids (1902–1919), stellar dynamics (1913–1922) and cosmogony (1916–1928).

Timeline

Ormskirk to Merchant Taylors’, 1877–1896

James Hopwood Jeans was born on 11 September 1877 in Ormskirk, Lancashire. His father, William Jeans, was a parliamentary journalist of Scottish descent. James was the eldest of three children and the only boy. The family moved to Brighton when he was about eighteen months old and to London when he was three.

In 1890 he entered Merchant Taylors’ School in London, where his mathematical talent was developed for the Cambridge scholarship examinations. In 1896 he won a mathematical scholarship to Trinity College, Cambridge.

Trinity and the Tripos, 1896–1904

Cambridge mathematics was organized around the Tripos, an examination of punishing speed whose ranked list of wranglers was published in the newspapers. Jeans was second wrangler in Part I in 1898, and took a first class in Part II in 1900. In 1901 he shared the Smith’s Prize with G. H. Hardy, who would become the leading English pure mathematician of his generation, and he was elected a Fellow of Trinity. He also held an Isaac Newton Studentship in astronomy and optics.

The Cambridge applied mathematics of the time was the tradition of George Stokes, Lord Kelvin, James Clerk Maxwell and Lord Rayleigh: continuum mechanics, potential theory, waves and the kinetic theory of gases. George Darwin, Plumian professor of astronomy, was studying the figures of rotating fluid masses and the tidal evolution of the Earth–Moon system. These problems became Jeans’s.

In the early 1900s he developed a tuberculous infection of a joint and spent long periods in sanatoria. He kept working. His first major paper, on the stability of a spherical nebula, appeared in the Philosophical Transactions in 1902 (Lesson 04), and his first book, The Dynamical Theory of Gases, in 1904 (Lesson 05).

Princeton and the textbooks, 1904–1912

In 1904 Jeans became a university lecturer in mathematics at Cambridge. The next year he accepted a professorship of applied mathematics at Princeton and stayed until 1909. In 1906, at 28, he was elected a Fellow of the Royal Society. In 1907 he married Charlotte Tiffany Mitchell, an American poet.

The Princeton years produced two textbooks: Theoretical Mechanics (1906) and The Mathematical Theory of Electricity and Magnetism (1908). The second, clear and systematic, was used by generations of students in Britain and America and went through many editions.

He returned to Cambridge as Stokes Lecturer in applied mathematics from 1910 to 1912. In 1911 he was among the physicists at the first Solvay Conference in Brussels, devoted to radiation and the quanta. In 1912 he gave up university employment and worked independently in Surrey for the rest of his life. One of the students he taught in this last Cambridge period was Ronald Fisher, who studied statistical mechanics and quantum theory with him after graduating in 1912.

Gravitational instability

For the general reader. A cloud of gas is pulled together by its own gravity and pushed apart by its pressure. Isaac Newton had already wondered whether matter spread evenly through space could stay that way. In 1902 Jeans gave the quantitative answer: small lumps are stable and simply oscillate as sound waves, but lumps larger than a critical size collapse. This critical size, the Jeans length, and the mass it contains, the Jeans mass, are the starting point of every theory of how stars and galaxies form.

Equations

An ideal fluid with density \(\rho\), velocity \(\mathbf v\) and pressure \(P\), in its own gravitational potential \(\Phi\), obeys

Take a uniform state \(\rho_0\), \(\mathbf v=0\), and perturb it with \(\rho=\rho_0+\rho_1\), \(P_1=c_s^2\rho_1\), where \(c_s\) is the sound speed. Linearizing, taking the divergence of the momentum equation and eliminating \(\mathbf v_1\) and \(\Phi_1\), one finds

The Jeans mass is the mass inside a sphere of diameter \(\lambda_J\), \(M_J=\tfrac{\pi}{6}\rho_0\lambda_J^3\) (other conventions differ by factors of order unity). For an isothermal gas of mean molecular mass \(\mu m_H\), \(c_s^2=kT/\mu m_H\), so \(M_J\propto T^{3/2}\rho_0^{-1/2}\): cold, dense clouds collapse most easily. The long-wavelength growth rate is \(\sqrt{4\pi G\rho_0}\), the inverse of the free-fall time up to a numerical factor.

The swindle

A uniform infinite static medium cannot satisfy the unperturbed Poisson equation with \(\rho_0\ne0\) unless \(\nabla^2\Phi_0=0\). Jeans implicitly set \(\Phi_0=0\) and applied the Poisson equation only to the perturbations. This step is called the Jeans swindle. A careful treatment in an expanding universe shows that the expansion supplies exactly the missing background, and the Jeans criterion survives, with growth that becomes a power law in time instead of an exponential.

Kinetic theory and escaping atmospheres

For the general reader. Gas molecules move at random speeds, and a few in the high-speed tail of the distribution move fast enough to escape a planet’s gravity. Jeans worked out how fast an atmosphere leaks away this way. The answer explains why the Moon has no air, why Earth has lost almost all its hydrogen, and why the giant planets have kept theirs. The calculation appears in his treatise on gases, the standard English text on kinetic theory for decades.

The Maxwell distribution and equipartition

In equilibrium, molecules of mass \(m\) have speeds distributed as

The most probable speed is \(v_0=\sqrt{2kT/m}\). Each quadratic degree of freedom carries mean energy \(\tfrac12kT\). The Dynamical Theory of Gases (1904) developed these results from Boltzmann’s theory with Cambridge thoroughness, including the transport coefficients of viscosity, conduction and diffusion. Jeans was a strong defender of the H-theorem and of equipartition in classical mechanics, a position that shaped his view of radiation (Lesson 06).

Derivation. Count upward-moving particles with \(v>v_{\text{esc}}\): \(\Phi=\int_{v>v_{\text{esc}}}\int_{\theta<\pi/2}v\cos\theta\,f\,v^2\,dv\,d\Omega\). The angular integral gives \(\pi\). With \(x=v/v_0\), the remaining integral is \(\pi n(\pi v_0^2)^{-3/2}v_0^4\int_{x_e}^\infty x^3e^{-x^2}dx\), and \(\int_{x_e}^\infty x^3e^{-x^2}dx=\tfrac12(1+x_e^2)e^{-x_e^2}\). Since \(x_e^2=\lambda\), this gives the formula.

The factor \(e^{-\lambda}\) makes escape exquisitely sensitive to mass and temperature. A gas is safe over billions of years when \(\lambda\) exceeds about 20 to 25, and leaks quickly when \(\lambda\) is below about 10.

The Rayleigh–Jeans law

For the general reader. A hot cavity is filled with light in many frequencies. Classical physics, applied honestly, says that every possible wave pattern in the box should carry the same average energy. Since there are infinitely many patterns at high frequency, the total energy would be infinite. Rayleigh and Jeans made this argument exact in 1905. Their formula matches experiment at long wavelengths and fails completely at short ones. Its failure was a central piece of evidence for the quantum.

Counting modes

In a cube of side \(L\) with conducting walls, the standing waves have wave vectors \(\mathbf k=\tfrac{\pi}{L}(n_1,n_2,n_3)\) with non-negative integers \(n_i\), and two polarizations each. The number of modes with \(|\mathbf k|<k\) is two times one octant of a sphere of radius \(kL/\pi\) in \(n\)-space, \(2\cdot\tfrac18\cdot\tfrac43\pi(kL/\pi)^3\). Using \(k=2\pi\nu/c\) and differentiating, the number of modes per unit volume between \(\nu\) and \(\nu+d\nu\) is \(8\pi\nu^2d\nu/c^3\).

In his June 1900 note Rayleigh had the dependence on \(\nu\) and \(T\) but no constant. In 1905 he published a full derivation in Nature, and Jeans pointed out that it counted modes in the whole sphere instead of one octant, overestimating the result by a factor of 8. Rayleigh accepted the correction.

Max Planck’s law, \(u=\dfrac{8\pi h\nu^3}{c^3}\dfrac{1}{e^{h\nu/kT}-1}\), reduces to the Rayleigh–Jeans law when \(h\nu\ll kT\) and falls off exponentially when \(h\nu\gg kT\). The divergence of the classical law was later named the ultraviolet catastrophe by Paul Ehrenfest.

Jeans’s position and its reversal

Jeans was reluctant to give up equipartition. He argued that radiation and matter in a real cavity never reach full equilibrium: the energy of the high-frequency modes grows so slowly that, on any laboratory timescale, they remain nearly empty. The observed spectrum would then be a non-equilibrium state. The argument kept classical mechanics intact at the cost of making the observed spectrum depend on time.

Henri Poincaré’s 1912 analysis, showing that any law with finite total energy requires discontinuities in the energy of oscillators, persuaded him. His Report on Radiation and the Quantum Theory (1914), written for the Physical Society of London, was one of the first surveys of quantum theory in English, and it introduced many British physicists to the work of Planck, Einstein and Bohr.

Rotating fluid masses

For the general reader. A spinning ball of liquid, held together by its own gravity, flattens at the poles. Spin it faster and it flattens more. Beyond a certain speed the flattened shape becomes unstable and the body becomes a lopsided ellipsoid, then perhaps a pear, and perhaps it splits in two. Nineteenth-century mathematicians hoped this sequence would explain double stars and the origin of the Moon. Jeans spent nearly two decades on the problem, and his Adams Prize essay of 1917 was largely about it.

The proof uses the gravitational potential inside a homogeneous ellipsoid, a quadratic form \(\Phi=\pi G\rho(A_1x^2+A_2y^2+A_3z^2)+\text{const}\) with coefficients given by elliptic integrals. On the surface, \(\Phi-\tfrac12\Omega^2(x^2+y^2)\) must be constant, which yields one equation linking \(\Omega\) and \(e\). For small \(e\), the right side is approximately \(\tfrac{8}{15}e^2\).

The sequence of figures

• The right side increases from 0 to a maximum of about 0.4493 at \(e\approx0.930\), then falls. A Maclaurin spheroid cannot rotate faster than this.
• At \(e\approx0.8127\), where \(\Omega^2/\pi G\rho\approx0.374\), Carl Jacobi’s (1834) triaxial ellipsoids branch off. With viscosity, a Maclaurin spheroid past this point slowly turns into a Jacobi ellipsoid at fixed angular momentum.
• Along the Jacobi sequence, Poincaré (1885) found a further bifurcation into pear-shaped figures. George Darwin hoped the pear would deepen its waist and split, giving a double star. This was the fission hypothesis.

Whether the pear-shaped figures are stable was disputed. Aleksandr Lyapunov argued in 1905 that they are unstable from the start. Jeans, after long calculations in 1914–1916, including models of compressible gaseous masses closer to real stars, reached the same conclusion for the incompressible case. A rotating mass that passes the Jacobi stage does not divide gently into two. His conclusions were summarized in Problems of Cosmogony and Stellar Dynamics (1919), which became the reference for the subject until Subrahmanyan Chandrasekhar’s Ellipsoidal Figures of Equilibrium (1969).

Stellar dynamics and Jeans’s theorem

For the general reader. A galaxy contains so many stars, so far apart, that they almost never collide or even pass close to each other. Each star moves in the smooth gravitational field of all the others. Jeans saw that such a system is a gas without collisions, and he adapted the kinetic theory of gases to it. His 1915 paper on star-streaming founded the mathematical study of galaxies.

The collisionless Boltzmann equation

Describe the stars by a distribution function \(f(\mathbf x,\mathbf v,t)\) on phase space, the number of stars per unit volume of \((\mathbf x,\mathbf v)\). If encounters between stars are negligible, each star follows the equations of motion in the smooth potential \(\Phi\), and by Liouville’s theorem \(f\) is constant along these trajectories:

Jeans introduced this equation into astronomy in 1915. Plasma physicists know the same system, with electric in place of gravitational forces, as the Vlasov equation.

Proof. The equation says \(df/dt=0\) along orbits. If \(f\) is steady, it is constant along each orbit, so it is an integral of motion. Conversely, if \(I_1,\dots,I_n\) are integrals, \(dI_k/dt=0\) along orbits, so \(f(I_1,\dots,I_n)\) satisfies \(df/dt=\sum_k(\partial f/\partial I_k)\,dI_k/dt=0\).

In a spherical potential the energy \(E=\tfrac12v^2+\Phi\) and the angular momentum vector \(\mathbf L\) are integrals, so \(f(E)\) gives isotropic spherical models and \(f(E,L)\) anisotropic ones. The strong Jeans theorem (later sharpened by Donald Lynden-Bell) says that for regular orbits a function of the isolating integrals suffices. Jeans’s 1915 paper used this to argue that the star streams found by Jacobus Kapteyn could not be explained by an axisymmetric distribution depending only on \(E\) and \(L_z\), because that would force equal velocity dispersions in the radial and vertical directions. Observations show they are unequal, which points to a third integral, a puzzle that lasted into the 1960s.

The Jeans equations and relaxation

For the general reader. We cannot observe the full distribution of stellar velocities, only averages: densities, mean motions and the spread of velocities. Jeans derived the equations that link these averages to the gravitational field. Astronomers still use them to weigh galaxies, find dark matter and detect black holes at galactic centres. He also estimated how long it takes for encounters between stars to matter, and found it is far longer than the age of a galaxy.

Moments

Integrate the collisionless Boltzmann equation over velocities, after multiplying by 1 and by \(v_j\). With \(\nu=\int f\,d^3v\), mean velocity \(\bar v_j\) and dispersion tensor \(\sigma^2_{ij}=\overline{v_iv_j}-\bar v_i\bar v_j\):

These are the Jeans equations. They resemble the Euler equations, with the stress tensor \(\nu\sigma^2_{ij}\) in place of the pressure, and they are not closed: each moment equation involves the next moment.

Given an observed tracer density \(\nu(r)\) and dispersion \(\sigma_r(r)\), the equation yields the enclosed mass \(M(r)\), including any unseen matter. The uncertainty in \(\beta\) is the main limitation, the mass–anisotropy degeneracy.

Relaxation

Encounters do change stellar orbits, slowly. Summing small deflections from many distant passes, one finds that a star’s velocity is changed by order of itself after about

crossing times, for a system of \(N\) equal stars. Jeans’s estimates of such timescales in the 1910s and 1920s showed that galaxies are effectively collisionless, which justifies everything above. Globular clusters, with about \(10^5\) stars, are partly relaxed; galaxies are not.

Cosmogony and the stars

For the general reader. Where did the planets come from, what makes stars shine, and how old are the stars? In the 1910s and 1920s Jeans proposed bold answers to all three. The planets, he suggested, were torn out of the Sun by a passing star. Stars, he thought, shine by slowly annihilating their own matter, and live for millions of millions of years. Almost all of this turned out to be wrong, yet the arguments he used, and the debates they caused, shaped modern astrophysics.

The tidal hypothesis

The nebular hypothesis of Pierre-Simon Laplace derived the planets from a contracting, rotating cloud. It faced a serious objection: the Sun carries less than 1% of the solar system’s angular momentum, though it holds more than 99.8% of its mass. Jeans proposed instead that a star passing close to the Sun raised a tidal filament of gas, which broke up into planets by gravitational instability, the mechanism of Lesson 04. The passing star would supply angular momentum to the planets. Harold Jeffreys developed related versions. The theory fell in the late 1930s, when Lyman Spitzer showed that hot gas pulled from the Sun would disperse before condensing, and when the rarity of close stellar encounters made planetary systems seem implausibly rare. Modern theory returns to a disk, with angular momentum moved outward by magnetic and viscous processes.

Stellar energy and the long timescale

Gravitational contraction, the source proposed by Hermann Helmholtz and Kelvin, gives the Sun a lifetime of tens of millions of years, too short for geology. Already in 1904 Jeans had suggested that the mutual annihilation of positive and negative charges might power the stars. Converting the Sun’s entire mass to energy would last trillions of years, and Jeans adopted a correspondingly long timescale for stellar evolution. Arthur Eddington, from 1917, built the theory of radiative stellar interiors and favoured a less drastic subatomic source. The two clashed repeatedly in the 1920s, notably over Jeans’s 1928 proposal that stars are partly liquid. Hydrogen fusion, worked out by Hans Bethe in 1938–39, settled the question: it releases about 0.7% of the rest mass, giving the Sun a main-sequence life of about ten billion years.

Continuous creation

In Astronomy and Cosmogony (1928) Jeans speculated that the centres of spiral nebulae might be places where matter enters our universe. The suggestion anticipated, in loose form, the continuous creation of the steady-state cosmology of Hermann Bondi, Thomas Gold and Fred Hoyle (1948).

Secretary, popularizer, philosopher, 1919–1935

In 1917 Jeans won the Adams Prize at Cambridge and gave the Bakerian Lecture of the Royal Society. The prize essay became Problems of Cosmogony and Stellar Dynamics (1919). He received the Royal Medal in 1919 and the Gold Medal of the Royal Astronomical Society in 1922, and served as President of the Royal Astronomical Society from 1925 to 1927. As one of the two Secretaries of the Royal Society from 1919 to 1929 he carried a heavy load of editorial and administrative work. From 1923 he was a research associate of the Mount Wilson Observatory in California, which kept him close to the observations of the largest telescopes of the day. He was knighted in 1928.

Astronomy and Cosmogony (1928) closed his research career. That year a series of radio talks led to The Universe Around Us (1929), and his 1930 Rede Lecture at Cambridge became The Mysterious Universe, which sold in enormous numbers. In it he argued that modern physics pointed toward a universe best described in mathematical terms, so that it looked more like a great thought than like a great machine. Philosophers, including Susan Stebbing, criticized this idealism sharply. He followed with The Stars in Their Courses (1931), The New Background of Science (1933) and Through Space and Time (1934), from his Royal Institution Christmas Lectures of 1933.

He received the Franklin Medal in 1931 and presided over the British Association at Aberdeen in 1934.

Last years and legacy, 1934–1946

Charlotte Jeans died in 1934. In 1935 Jeans married Susi Hock, a Viennese organist and harpsichordist who became a noted performer and authority on early keyboard music as Susi Jeans. They had three children. Music had always been his recreation; he owned a large organ at his home, Cleveland Lodge in Dorking, and in 1937 he published Science and Music, an account of the physics of sound and of musical scales. From 1935 he held the professorship of astronomy at the Royal Institution. He received the Order of Merit in 1939, and wrote Physics and Philosophy (1942) during the war.

Ill health forced him to give up the Royal Institution chair in 1946. He died at Cleveland Lodge on 16 September 1946, aged 69. The Growth of Physical Science appeared posthumously in 1947. Craters on the Moon and on Mars bear his name.

Legacy

• Star formation: the Jeans length and mass remain the first estimates in every theory of gravitational collapse, from molecular cloud cores to the first structures in cosmology.
• Galactic dynamics: the collisionless Boltzmann equation, Jeans’s theorem and the Jeans equations are the foundation of the subject, as in James Binney and Scott Tremaine’s Galactic Dynamics.
• Planetary science: Jeans escape is the baseline thermal loss rate for every planetary atmosphere.
• Quantum history: the Rayleigh–Jeans law defined the classical limit that Planck’s law must reproduce, and his 1914 report brought the quantum to British physics.
• Ideas that failed: the tidal hypothesis, liquid stars and the long timescale show how a powerful mathematical method can be applied with confidence to the wrong physical model.

Connections

People linked to James Jeans across this collection. A name without a link has no page here yet. See James Jeans in the network of all 337 lives →

Students

• Ronald Fisher 1912-1913

Influenced by

• Aleksandr Lyapunov
• Henri Poincaré
• Ludwig Boltzmann
• Max Planck
• George Darwin

Colleagues and collaborators

• G. H. Hardy 1901
• John William Strutt, Lord Rayleigh 1905
• Edward Arthur Milne
• Harold Jeffreys

Rivals

• Aleksandr Lyapunov
• Arthur Eddington 1917-1930

Family

• Susi Jeans

Also mentioned in this life

• Hans Bethe
• Niels Bohr
• Subrahmanyan Chandrasekhar
• Alexis Clairaut
• Paul Ehrenfest
• Albert Einstein
• Leonhard Euler
• Benjamin Franklin
• Hermann von Helmholtz
• Carl Gustav Jacob Jacobi
• William Thomson, Lord Kelvin
• Pierre-Simon Laplace
• Joseph Liouville
• Colin Maclaurin
• James Clerk Maxwell
• Isaac Newton
• Siméon Denis Poisson
• George Gabriel Stokes

Fields

• mathematical physics 45 lives
• astrophysics 6 lives
• kinetic theory 4 lives

Sources and further reading

Biography

1. James Hopwood Jeans, MacTutor History of Mathematics.
2. Obituary of Sir James Jeans, Monthly Notices of the Royal Astronomical Society, via MacTutor.
3. James Jeans, Wikipedia.
4. Jeans, James Hopwood, Encyclopedia.com (Dictionary of Scientific Biography and other entries).
5. E. A. Milne, Sir James Jeans: A Biography, Cambridge University Press, 1952.
6. E. A. Milne, James Hopwood Jeans, Obituary Notices of Fellows of the Royal Society 5 (1947).

Physics

1. J. H. Jeans, The stability of a spherical nebula, Phil. Trans. R. Soc. A 199 (1902) 1–53.
2. J. H. Jeans, The Dynamical Theory of Gases, Cambridge University Press, 1904.
3. J. H. Jeans, On the theory of star-streaming and the structure of the universe, MNRAS 76 (1915) 70–84.
4. J. H. Jeans, Report on Radiation and the Quantum Theory, Physical Society of London, 1914.
5. J. H. Jeans, Problems of Cosmogony and Stellar Dynamics, Cambridge University Press, 1919.
6. J. H. Jeans, Astronomy and Cosmogony, Cambridge University Press, 1928.
7. Jeans instability and Rayleigh–Jeans law, Wikipedia.
8. J. Binney and S. Tremaine, Galactic Dynamics, 2nd ed., Princeton University Press, 2008.
9. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Yale University Press, 1969.
10. T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894–1912, Oxford University Press, 1978.