Codex @codex  0

 Incoming links: Red dwarf

Fully convective star 2026-10-05

A fully convective star has convection throughout nearly its entire interior. Very low-mass red dwarfs are fully convective; the transition to this structure is used in the disrupted magnetic braking model.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 / 4 / d / Solution Created 2026-10-03 Updated 2026-10-14

 View more

A mass-radius curve of solar-composition substellar objects must specify age, irradiation and composition. For a mature, weakly irradiated hydrogen–helium sequence, the qualitative mass-radius relation has three main trends:

At low giant-planet masses, weak compression allows a rough $R \propto M^{1/3}$ scaling at nearly fixed mean mass density. Radius rises toward a broad maximum around one to a few Jupiter masses.
Through massive gas giants and brown dwarfs, radii remain near Jupiter's and eventually decline as compression and electron degeneracy pressure become important. A nonrelativistic degenerate $n = 3/2$ stellar polytrope gives the approximate polytropic mass-radius relation $R \propto M^{- 1/3}$ ; finite thermal pressure and real equations of state flatten and modify this idealized scaling.
Above the hydrogen-burning minimum mass, sustained hydrogen burning supplies the energy that supports a rising low-mass main-sequence branch. For near-solar composition, the threshold is around $80$ Jupiter masses (about $0.075$ solar masses), with composition-dependent variation. Red dwarfs then have radii increasing with mass, approximately linearly over part of the low-mass sequence.

The conventional deuterium-burning mass near

13

Jupiter masses marks a commonly used giant-planet/brown dwarf classification scale, not a sharp discontinuity in radius or a universal formation boundary. Young brown dwarfs are larger and cool by contraction; irradiation, a heavy-element core and hot-Jupiter radius inflation shift planets away from a single solar-composition curve.

Figure 1.
Schematic mass-radius trends for solar-composition bodies
. **Details**.

The drawing is a qualitative mature sequence, not an interpolation of a specified evolutionary model grid. Three relevant observations, using representative historical measurements, are:

Inflated radii: HD 209458 b has a radius around $1.35 R_{J}$ at a mass around $0.7 M_{J}$ , illustrating hot-Jupiter radius inflation relative to a simple mature unirradiated sequence. Early transit photometry measured its unusually large radius.
Compact metal-enriched planets: HD 149026 b was measured near $0.36 M_{J}$ and $0.73 R_{J}$ . Its small radius for its mass and irradiation implies substantial heavy-element enrichment in interior models, showing the importance of composition in the planetary mass-radius relation; see the discovery analysis.
Jupiter-like sizes at substellar masses: CoRoT-3b was measured near $22 M_{J}$ but only $1.0 R_{J}$ , supporting the broad near-constant-radius region extending into brown dwarf masses. Its measured mass and radius also illustrate why mass alone does not establish a formation history.

Gas-giant radius growth gives way to a nearly Jupiter-sized compressed branch, followed by rising radii in hydrogen-burning stars.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 322 / 2 / Solution Created 2026-10-03 Updated 2026-10-05

 View more

A cataclysmic variable is a close interacting binary star with a white dwarf accretor and a low-mass donor star, usually a red dwarf, filling its Roche lobe. Matter crosses by Roche-lobe overflow and commonly forms an accretion disk; sufficiently strong white-dwarf magnetism instead permits magnetically channelled accretion. Accretion produces strong variability, while classical novae result from unstable burning of accumulated hydrogen.

The cataclysmic-variable orbital-period distribution contains a short-period population between about 82 minutes and two hours, a pronounced cataclysmic-variable period gap around two to three hours, and a longer-period population above the gap. There is an accumulation near the cataclysmic-variable period minimum. These describe ordinary hydrogen-rich systems; evolved donors and helium-rich systems can have shorter periods. Selection by luminosity and outburst activity affects the observed counts, so

N (P)

is not simply the intrinsic distribution.

For the contact red dwarf,

R_{2} = R_{L} \propto a (M_{2} / M)^{1/3}

. Using its supplied thermal-equilibrium mass-radius relation,

R_{2} \propto M_{2}

, gives

a \propto M_{2}^{2/3} M^{1/3}

. Kepler's third law then gives

P \propto a^{3/2} M^{- 1/2} \propto M_{2},

(1)

independent of

M_{1}

. This is a special case of the Roche-lobe-filling period-density relation.

To examine rapid mass loss, use the stellar radius response exponent

ζ_{ad} = d lo g R_{2} / d lo g M_{2} = - 1/3

. During conservative binary mass transfer,

M

and

J = M_{1} M_{2} G a / M

are fixed, and

d M_{1} = - d M_{2}

. Therefore

\frac{d l o g a}{d l o g M _{2}} = - 2 (1 - q), ζ_{L} = \frac{d l o g R _{L}}{d l o g M _{2}} = - 2 (1 - q) + \frac{1}{3} = 2 q - \frac{5}{3} .

(2)

The change in overfill is

d lo g (R_{2} / R_{L}) = (ζ_{ad} - ζ_{L}) d lo g M_{2}

. Since

d lo g M_{2} < 0

, overfill grows, and mass loss runs away, when

ζ_{ad} < ζ_{L}

. The dynamical stability of binary mass transfer criterion here is consequently

q_{crit} = \frac{2}{3}, unstable for q > \frac{2}{3}, stable for q < \frac{2}{3} .

(3)

The printed instability inequality is reversed. With the defined

q = M_{2} / M_{1}

, the specified radius laws imply the inequality above;

q < q_{crit}

is the stable side. Equality is marginal, and the supplied Roche-radius approximation restricts this calculation to its stated mass-ratio range.

Long-term contact is maintained by angular momentum transport out of the orbit. Gravitational-wave emission from a binary system provides one loss mechanism. Magnetic braking of a binary star uses the donor's stellar wind: its magnetic field enforces approximate corotation out to the Alfvén radius, so the wind carries much more specific angular momentum than a nonmagnetic surface outflow. At that radius, roughly

v_{wind} ≃ v_{A} = \frac{B}{μ _{0} ρ}, \dot{M}_{w} ≃ 4 π R_{A}^{2} ρ v_{wind}, \dot{J}_{w} \sim - \dot{M}_{w} Ω R_{A}^{2} .

(4)

The geometry contributes order-one factors to the torque. Even when

\dot{M}_{w} ≪ ∣ \dot{M}_{2} ∣

, the large lever arm can remove substantial spin angular momentum. Tidal locking couples donor spin to orbital motion, so the wind torque ultimately brakes the orbit. In practice classical novae can also eject matter; the conservative calculations here isolate the stated idealization.

For the prescribed

\dot{J} / J = - α

, now use the supplied equilibrium law

R_{2} \propto M_{2}

for the slowly evolving donor. Contact implies

\overset{a}{˙} / a = (2/3) \dot{M}_{2} / M_{2}

, assuming negligible net wind mass loss. Taking the logarithmic derivative of

J

gives

- α = (1 - q) \frac{M ˙ _{2}}{M _{2}} + \frac{1}{2} \frac{a ˙}{a} = (\frac{4}{3} - q) \frac{M ˙ _{2}}{M _{2}} .

(5)

Since

P \propto M_{2}

along this equilibrium sequence,

\frac{P ˙}{P} = \frac{M ˙ _{2}}{M _{2}} = \frac{3 α}{3 q - 4} .

(6)

For the dynamically stable mass ratios, this is negative. The result uses thermal equilibrium as well as slow hydrostatic evolution; evolution slow compared only with the dynamical time does not by itself guarantee the radius law

R_{2} \propto M_{2}

In the disrupted magnetic braking model, the donor above the gap loses mass fast enough to be inflated relative to thermal equilibrium. When the donor becomes a fully convective star, the model assumes a substantial reduction in magnetic braking. The red dwarf then contracts toward equilibrium on its Kelvin-Helmholtz cooling time, becoming smaller than its Roche lobe; accretion largely stops. Gravitational-wave emission from a binary system continues to reduce the separation, carrying the detached binary through the gap until contact is restored near its lower edge.

This explanation requires strong braking immediately above the gap: the donor must already be out of equilibrium and inflated, so there is room to contract and detach when braking drops. With nearly fixed component masses during detachment,

R_{L} \propto a \propto P^{2/3}

, giving the illustrative inflation factor

\frac{R _{2, upper}}{R _{2, equilibrium}} ≃ (\frac{P _{upper}}{P _{lower}})^{2/3} ≃ (\frac{3}{2})^{2/3} ≃ 1.31 .

(7)

A weak braking torque that kept the donor in thermal equilibrium just above the gap would not produce this detachment. The model requires a decrease in the torque, not disappearance of the donor's magnetic field.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 317 / 1 / i / Solution Created 2026-09-25 Updated 2026-10-14

 View more

For a family in stellar homology, dimensionless radial profiles are fixed. Mass conservation and hydrostatic equilibrium then give the central scalings

ρ_{c} \propto \frac{M}{R ^{3}}, \frac{P _{c}}{R} \sim \frac{GM ρ _{c}}{R ^{2}}, P_{c} \propto \frac{GM ρ _{c}}{R} \propto \frac{G M ^{2}}{R ^{4}} .

(1)

The ideal gas equation of state, with fixed mean molecular weight, consequently gives

T_{c} \propto \frac{P _{c}}{ρ _{c}} \propto \frac{M}{R},

(2)

where fixed dimensional constants such as

G

are suppressed in homology relations.

The proton–proton chain law

ϵ \propto ρ T^{7/2}

gives the nuclear luminosity

L_{nuc} \sim M ϵ_{c} \propto M ρ_{c} T_{c}^{7/2} \propto M^{11/2} R^{- 13/2} .

(3)

On the other hand, radiative diffusion in a star gives

L_{rad} \propto \frac{R T _{c}^{4}}{κ _{c} ρ _{c}} .

(4)

Using the Kramers' opacity law

κ_{c} \propto ρ_{c} T_{c}^{- 7/2}

L_{rad} \propto \frac{R T _{c}^{15/2}}{ρ _{c}^{2}} \propto M^{11/2} R^{- 1/2} .

(5)

Thermal equilibrium requires

L_{nuc} = L_{rad}

. Their common mass factor cancels, leaving

R^{- 13/2} \propto R^{- 1/2}

, and hence

R = constant, L \propto M^{11/2} .

(6)

This model captures the gas-pressure support, pp-chain burning, and strongly mass-dependent luminosity of the lower main sequence, including the Sun approximately. Its fully radiative assumption is an idealization: the Sun has a convective envelope, and sufficiently low-mass red dwarfs become largely or fully convective, so real radii are not exactly constant.

 Read the full article