Boson Star Mass–Radius Calculator

Physical background

Boson stars are hypothetical compact objects composed of bosons occupying a single macroscopic quantum state, similar in spirit to a Bose–Einstein condensate. They arise in many models of dark matter, including ultralight axion-like particles and scalar-field dark matter. Gravity tries to make the configuration collapse, while quantum or interaction-induced pressure provides support.

In relativistic field theory, the system is described by the Einstein–Klein–Gordon equations. In the dilute, weak-field regime relevant here, it can be approximated by the coupled Gross–Pitaevskii–Poisson system. Simple scaling relations derived from numerical solutions and dimensional analysis allow quick estimates of the characteristic maximum mass and size as functions of the boson mass and interaction strength.

Key formulas used in the calculator

Noninteracting boson stars

For bosons without self-interactions, the maximum stable mass and its corresponding radius scale with the Planck mass M_Pl and the particle mass m as

and

Here the expressions are written in natural units (c = ℏ = 1), and the calculator converts them to physical units (solar masses and kilometers) internally. The key feature is that lighter bosons lead to larger, more massive stars, with M_max proportional to 1 / m² and R proportional to 1 / m once proper unit factors are restored.

Repulsively self-interacting boson stars

If the bosons have a repulsive contact interaction, for example a quartic potential with coupling constant λ, an additional pressure term appears. In the Thomas–Fermi limit where this interaction pressure dominates over quantum gradients, one finds approximately

and

The interaction strength λ is related to the scattering length by

λ = 8 π a_s m

after converting a_s from femtometers to natural units. Even very small positive scattering lengths can dramatically increase both the mass and the radius compared to the noninteracting case.

Interpreting the calculator outputs

When you enter a boson mass and scattering length, the calculator first converts them into natural units, then chooses the appropriate regime:

• a_s = 0: uses the noninteracting formulas to return the maximum mass and its radius.
• a_s > 0: computes λ from the scattering length and applies the self-interacting formulas.

The mass output is presented in units of M_☉ (solar masses) and the radius in km. Broadly:

• Masses of order 1–10 M_☉ and radii of order 10 km resemble neutron-star scales.
• Much larger radii (up to parsec or kiloparsec scales in extreme self-interacting cases) indicate halo-like or galactic-core configurations.
• Very small radii at a given mass may suggest that the configuration is beyond the validity of the simple scaling formulas and closer to black-hole-like compactness.

Worked example

Suppose you are exploring an ultralight scalar dark matter candidate with mass

• Boson mass: m = 1 × 10⁻¹⁰ eV
• Scattering length: a_s = 0 fm (noninteracting case)

Enter 1e-10 into the boson mass field and 0 into the scattering length field, then run the calculation. Using the noninteracting relations above, the tool will return a maximum mass on the order of a few solar masses and a radius of a few tens of kilometers. This is a compact object roughly comparable in scale to a neutron star, though supported by bosonic quantum pressure instead of nucleon degeneracy.

Now repeat the calculation with a tiny positive scattering length, for example

• Scattering length: a_s = 1 × 10⁻⁶ fm

After converting to λ and applying the self-interacting scaling, the calculator will report a significantly larger maximum mass and a radius that can be orders of magnitude larger than in the noninteracting case. This illustrates how even weak repulsive self-interactions can inflate boson-star configurations from compact-object scales to halo-like sizes.

Intuitive picture: how parameters affect size and mass

For users less familiar with the underlying field theory, a simple way to think about the trends is:

• Lighter bosons (smaller m) are more spread out quantum mechanically, so more particles can occupy a larger region before gravity wins. This tends to increase both the maximum mass and radius.
• Stronger repulsive interactions (larger a_s and thus larger λ) add extra pressure. This also allows heavier, larger configurations to remain stable compared to the noninteracting limit.
• Heavier bosons and weaker interactions push the system toward very compact objects, where simple scaling relations become less reliable and fully relativistic solutions are needed.

Comparison with other astrophysical objects

Use this table to situate the calculator outputs among familiar compact objects or dark-matter halo scales. For example, if your chosen parameters yield a mass of ∼10⁶ M_☉ and a radius of a few parsecs, you are effectively modeling a bosonic core that could sit at the center of a dwarf galaxy.

Assumptions and limitations

The scaling relations implemented in this calculator are intentionally simple and are best interpreted as order-of-magnitude estimates, not precision predictions. Key assumptions include:

• Idealized, isolated, spherically symmetric configurations: rotation, magnetic fields, and environmental effects (e.g. surrounding baryons or tidal fields) are neglected.
• Single bosonic species: the model assumes one scalar field with mass m and contact interaction characterized by a single scattering length a_s.
• Regime of validity: formulas correspond to either the nonrelativistic, dilute limit or the Thomas–Fermi self-interacting regime. Extremely high central densities or strong relativistic effects are not fully captured.
• Parameter ranges: the tool is most physically meaningful for ultralight bosons (roughly m between 10⁻²⁴ eV and 1 eV) and small, positive scattering lengths. Extreme values may formally return a result but fall outside the domain where the approximations are trustworthy.
• No stability or formation history: the calculator does not test dynamical stability, formation channels, or cosmological evolution. It only reports characteristic mass–radius scalings.

For quantitative work, especially near relativistic or strongly self-gravitating regimes, a full numerical solution of the Einstein–Klein–Gordon system with your chosen potential is required. The present tool is meant as a fast way to build intuition, scan parameter space, or prepare back-of-the-envelope estimates for research on compact objects and Bose–Einstein condensate dark matter.

Practical tips for using the form

• Use the boson mass field to explore different dark-matter candidates (e.g. axion-like particles, fuzzy dark matter, scalar fields).
• Set the scattering length to 0 to recover the noninteracting limit; small positive values probe how repulsive interactions change the results.
• If the calculator returns extremely large or extremely small radii, treat them as indicative trends rather than precise numbers, and check whether your parameters lie within a physically motivated range.