Astronomy · Evidence for planets around other stars

Inferring a planet's size, mass and orbit

Connect each inferred property to the measurements and model parameters it actually needs, with original radius–density examples and an optional exact radial-velocity mass function.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Distinguish relative radius, physical radius, mass and mean density.
Apply the relative-orbit period law with its mass and length definitions.
Explain inclination, host-property and interior-model limits in combined parameter inference.

A measured signal does not determine every property at once

A transit light curve, a stellar velocity curve and an image constrain different combinations of physical parameters. Combining suitable evidence can narrow the possibilities, but each inferred property must retain the inputs and assumptions connecting it to the data.

In the ideal uniform-star full-overlap model, depth δ gives radius ratio k = √δ. Physical radius then follows from R_p = R_star√δ. The host-star radius supplies the absolute scale; the transit is not a ruler independent of the star.

A mass constraint instead needs dynamics or another physical model. A radial-velocity amplitude depends on the companion and host masses, orbital period, eccentricity and inclination. Transits can help constrain geometry, while astrometry or other dynamical evidence can add different information.

Do not combine estimates as though every input were independent. A common stellar model can affect inferred radius, host mass and other quantities together. The uncertainty structure belongs in a joint inference rather than being lost when values are copied from separate tables.

A revised radius changes volume cubically

For an approximately spherical planet, mean density is mass divided by volume: ρ = 3M_p/(4πR_p³). It is a global average in units such as kilograms per cubic metre, rather than the local density at every depth.

Use an original conditional example with ideal depth δ = 0.01 and independently stipulated planet mass 2m₀. Consider two alternative host-radius assumptions, 10R₀ and 12R₀. These are alternative parameter choices, not measurements of a star physically changing during a transit.

The corresponding planet radii are 1R₀ and 1.2R₀. Define density unit ρ₀ = 3m₀/(4πR₀³). The mean densities are then 2ρ₀ and approximately 1.157407ρ₀. Increasing the assumed radius by twenty percent at fixed mass multiplies density by 1/1.2³ ≈ 0.578704.

This sensitivity is arithmetic within the declared model, not a probability distribution for a real measurement. Actual radius and mass estimates can be correlated. A physical uncertainty calculation should propagate their joint constraints and the assumptions behind the radius definition.

Original alternative host-scale assumptions with the same ideal depth 0.01 and independently stipulated planet mass 2m₀.
R_star / R₀	R_p / R₀	M_p / m₀	ρ / ρ₀
10	1	2	2
12	1.2	2	1.157407

Use the full relative semimajor axis

For an isolated bound Newtonian two-body orbit, P² = 4π²a_rel³/[G(M_star + M_p)]. Here P is orbital period, G is the gravitational constant, and a_rel is the physical semimajor axis of the separation vector. It is the sum of the two barycentric semimajor axes.

Thus a_rel = [G(M_star + M_p)P²/(4π²)]¹ᐟ³. Period alone does not fix a physical orbital size without the total gravitational parameter. In many planet systems the host dominates the sum, but that simplification must be justified rather than built into the definition.

As a separate original units example, set P = 2πT₀. Gravitational parameters μg = 1 and 8 L₀³/T₀² then give a_rel = 1 and 2 L₀. Both have the same period because the changed gravitational parameter balances the cubic change of orbital size.

An instantaneous projected separation from an image is a different quantity. Orbital phase, orientation and line-of-sight position intervene between that measurement and a_rel. Likewise, an angular orbit requires distance before its scale can enter this physical period law.

Optional: retain the total mass in the velocity equation

For a single companion in the isolated bound Keplerian model, the host’s orbital radial-velocity semi-amplitude is K = (2πG/P)¹ᐟ³ M_p sin i / [(M_star + M_p)²ᐟ³√(1 − e²)]. Eccentricity obeys 0 ≤ e < 1, period and masses are positive, and inclination i is measured from the sky plane.

Cubing and rearranging defines the mass function f_M = PK³(1 − e²)³ᐟ²/(2πG) = M_p³sin³i/(M_star + M_p)². This function has mass units. A fitted K, P and e therefore constrain one combination of companion mass, host mass and inclination.

In an original arbitrary-unit case, take G = 1 L₀³/(m₀T₀²), M_star = 1000m₀, M_p = 1m₀, P = 2πT₀, e = 0.6 and i = 30 degrees. Substitution gives K ≈ 0.00624584 L₀/T₀ and f_M ≈ 1.24750375 × 10⁻⁷ m₀. This is separate from the circular projection examples in the preceding motion lesson.

For the same K, P, e and assumed host mass, the smallest compatible companion mass occurs at sin i = 1. Solving the exact mass function gives approximately 0.49983343m₀ in this example. The true system’s M_p sin i is 0.5m₀: that familiar quantity is close, but not identical, to the exact minimum after retaining the companion in the mass sum.

A constrained property can still permit several physical interiors

The approximation commonly written M_p sin i is useful when the companion is much lighter than the host. Unknown inclination still leaves a range of true masses; stellar activity, calibration and additional companions must also be assessed before treating a velocity fit as a planetary dynamical measurement.

Mass and radius constrain mean density and can rule out some interior models. They do not uniquely specify the proportions, layering or state of all materials. Different combinations of core, mantle, volatiles and atmosphere can produce similar integrated quantities within model and measurement uncertainty.

Additional chemical and atmospheric information can narrow those families, subject to its own calibration and modelling limits. An atmosphere-sensitive transit radius may correspond to a chosen opacity or pressure level rather than a directly observed solid boundary. The inferred volume must retain that interpretation.

A useful parameter summary distinguishes measured quantities, fitted parameters and derived properties, alongside shared assumptions and uncertainties. The next lesson considers wavelength-dependent atmospheric signals and the further steps required before making habitability or biological claims.

Worked examples

Supply the stellar scale explicitly

For ideal depth 0.01, how does changing assumed R_star from 10R₀ to 12R₀ affect R_p?

The radius ratio is √0.01 = 0.1 in both cases.
Multiplying by each assumed stellar radius gives 1R₀ and 1.2R₀.
At fixed stipulated mass, the density ratio is 1/1.2³ ≈ 0.578704.

Radius increases by twenty percent; the inferred mean density becomes about 57.87 percent of its former value under these assumptions.

Retain the total gravitational parameter

With P = 2πT₀, why can μg values 1 and 8 L₀³/T₀² give different orbital sizes?

The period law gives a_rel³ = μgP²/(4π²).
In the stated units P²/(4π²) = 1 T₀².
Cube roots give a_rel = 1 L₀ and 2 L₀ respectively.

The period constrains orbital size together with the total gravitational parameter, not independently of mass.

Try the reasoning

Does a transit radius ratio supply an absolute radius without a host scale?
Show the answer and reasoning
No.
The physical planet radius is the ratio multiplied by the stellar radius.
Is one image’s projected separation automatically the relative semimajor axis?
Show the answer and reasoning
No.
Projection, phase and the three-dimensional orbital model connect those different quantities.
Is the shorthand M_p sin i an exact mass-function inversion for every companion-to-host mass ratio?
Show the answer and reasoning
No.
The exact relation retains M_star + M_p; the shorthand uses a light-companion approximation.
Does one mass-and-radius pair uniquely determine a planet’s internal layering?
Show the answer and reasoning
No.
Different material and structure models can share similar mean density; further evidence and assumptions are needed.

What to carry forward

Radius ratio, physical size, mass, density and orbital scale need different inputs.
The total-mass relative-orbit law and full radial-velocity mass function preserve the model’s definitions.
Joint uncertainties and model degeneracies limit how far integrated properties determine composition or structure.

References

Teaching resources and research supporting this lesson’s explanations and examples.