Evidence for stellar radii and masses

A stellar surface is a modelled emitting region

A star’s photosphere is an atmospheric region from which much of its observed visible light escapes. It is not a solid surface with one sharply marked edge at every wavelength. A quoted stellar radius therefore needs a definition and a measurement or model linking it to the data.

For a spherical emitting surface, effective temperature T_eff is defined through its bolometric outward flux per area, σT_eff⁴. Here σ is the Stefan–Boltzmann constant. Combining that definition with surface area gives L = 4πR²σT_eff⁴, so R = √[L/(4πσT_eff⁴)].

Using effective temperature this way does not assert that every part of a real stellar spectrum is an exact blackbody spectrum. The luminosity, temperature calibration and adopted spherical radius must refer to a compatible physical description. Rotation, surface structure or distortion can complicate that description.

With common reference quantities satisfying the same relation, R/R_ref = √(L/L_ref)/(T_eff/T_ref)². In an original stipulated case with luminosity ratio 9 and temperature ratio 1.5, the radius ratio is 3/2.25 = 4/3. A more luminous star is not necessarily larger by the same factor as its luminosity.

An angular diameter provides a different route to radius

Interferometric measurements can constrain a stellar angular size by comparing the received wave signals across separated telescopes. The fitted visibility pattern depends on the source brightness distribution. Converting a fitted disk diameter into a photospheric radius involves an atmosphere or limb-darkening model.

For a small angular diameter θ measured in radians and a source distance d, the geometric approximation is R ≈ dθ/2. The factor of two matters: θ is the full angular diameter, while R is the physical radius. An angular radius would use a different convention.

As an original calculation, take a stipulated model angular diameter of 0.4 milliarcseconds and distance of 10 parsecs. The corresponding small-angle physical diameter is 0.004 astronomical units, so the radius is 0.002 au, about 2.992 × 10⁸ metres.

This is a conditional size estimate, not an actual observed star. It assumes the stated angular diameter and distance apply to the same object and compatible photospheric definition. An unresolved companion or an unsuitable brightness-profile model can affect the angular inference; distance uncertainty affects the physical scale.

Use the relative orbit rather than one component’s path

For an isolated Newtonian two-body orbit, the relative orbital semimajor axis is a = a₁ + a₂, where a₁ and a₂ are the component axes around their common centre of mass. The orbital period P obeys M₁ + M₂ = 4π²a³/(GP²). Both masses contribute to the gravitational dynamics.

This equation needs the reconstructed physical relative semimajor axis. An instantaneous projected separation on an image is generally different from a. A measured angular orbit also needs a distance to supply its physical length scale.

Construct an original circular binary using arbitrary length unit ℓ₀, mass unit m₀ and time unit τ₀. Set G = 1 ℓ₀³/(m₀τ₀²), relative separation a = 3 ℓ₀ and period P = 3πτ₀. Substitution gives the full mass sum 12 m₀.

Let its component masses be M₁ = 8 m₀ and M₂ = 4 m₀. Centre-of-mass balance M₁a₁ = M₂a₂ gives a₁ = 1 ℓ₀ and a₂ = 2 ℓ₀. These are orbital dimensions, separate from either star’s photospheric radius. No physical stellar radii have been supplied for this binary.

Projected velocities can supply a ratio without the full mass scale

In that circular binary, component orbital speeds are 2πa₁/P = 2/3 ℓ₀/τ₀ and 2πa₂/P = 4/3 ℓ₀/τ₀. A line-of-sight velocity semiamplitude is K_j = (2πa_j/P) sin i, where i = 90 degrees is edge-on and i = 0 degrees is face-on.

When both nonzero component amplitudes are measured, their ratio gives M₁/M₂ = K₂/K₁ in this two-body model. The common projection factor cancels. But the circular spectroscopic mass combination is (M₁ + M₂) sin³i = P(K₁ + K₂)³/(2πG), so an unknown inclination still prevents a unique total mass.

Changing the viewing inclination of the supplied binary from 90 to 30 degrees halves both amplitudes without changing either physical mass. Cubing their sum changes the directly constrained projected mass sum by a factor of eight, from 12 m₀ to 1.5 m₀.

The table describes the same physical binary viewed from two directions. It is not a pair of observed systems or a general formula for eccentric velocity curves. Eccentric orbits require the appropriate eccentricity terms and a fit to their orbital phase dependence.

Complementary evidence can separate otherwise coupled quantities

An eclipsing binary light curve constrains how the apparent emitting disks overlap through time. Its durations, depths and shapes carry information about relative radii, inclination and surface brightness within an orbital model. The light curve alone does not directly provide each star’s absolute dynamical mass.

Combining suitable eclipse modelling with both component velocity curves can supply the orbital scale, inclination and mass ratio. Relative radii then become physical radii through that scale. Limb darkening, third light, surface activity, eccentricity and data quality can affect the joint inference.

A luminosity-and-temperature radius, an angular-diameter radius and a binary radius can be compared, provided their definitions are compatible. Agreement tests parts of the measurement and physical description. Shared distance or atmospheric assumptions can make those tests partly correlated.

Report which quantities the data constrain directly within the adopted model and which depend on an added calibration, distance or evolutionary relation. A stellar mass–luminosity relation can be useful in a stated regime, but it should not replace a dynamical measurement with an unexplained universal power law.

Worked examples

Try the reasoning

1. A small-angle model gives a physical stellar diameter of 0.004 au. What is its radius?

   Show the answer and reasoning

   0.002 au.

   Radius is half the diameter. Do not apply a full angular diameter as though it were an angular radius.

2. The component barycentric orbital axes are 1 and 2 length units. Which semimajor axis enters the full two-body mass-sum equation?

   Show the answer and reasoning

   The relative axis a = 3 length units.

   Use a₁ + a₂, together with the common period. Neither component axis alone represents the relative orbit.

3. For the same circular binary, both radial velocity amplitudes halve because sin i changes from 1 to 1/2. By what factor does the constrained projected mass sum change?

   Show the answer and reasoning

   It becomes one eighth as large.

   P and G stay fixed, while (K₁ + K₂)³ is multiplied by (1/2)³. The true masses do not change.

4. Does detecting an eclipse by itself directly determine both stars’ absolute masses?

   Show the answer and reasoning

   No.

   An eclipse supplies geometric and surface-brightness constraints within a model; velocities, scale and inclination information must be combined appropriately.

What to carry forward

• A stellar radius depends on a stated emitting-surface definition and geometric or atmospheric evidence.
• Binary dynamics uses the full relative orbit and total mass.
• Two velocity amplitudes can give a mass ratio while an unknown inclination leaves the absolute mass scale unresolved.