Self-Gravitating Bose–Einstein Condensates as Astrophysical Objects

Boson stars are hypothetical astrophysical objects composed not of baryonic matter but of scalar or vector bosons occupying the same quantum state. Much like laboratory Bose–Einstein condensates, a large number of particles share a single macroscopic wavefunction. However, the gravitational self‑interaction balances the quantum pressure or self-interactions, yielding a compact object without the need for degeneracy pressure. Such configurations arise in models of dark matter, including axions and other light scalar fields, and have been studied since the pioneering work of Kaup and Ruffini & Bonazzola in the 1960s. Our calculator provides quick estimates of the characteristic mass and radius for two limiting regimes: noninteracting bosons stabilized purely by uncertainty pressure and bosons with repulsive contact interactions described by a scattering length $a_s$.

For noninteracting boson stars, the equilibrium configuration can be derived from the Einstein–Klein–Gordon system. In the Newtonian limit applicable to dilute configurations, the Gross–Pitaevskii–Poisson equations govern the system. Dimensional analysis and numerical solutions show that the maximum mass scales inversely with the boson mass squared: $M$_max≈0.633M_Pl2m. The corresponding radius at maximum mass is roughly $R\approx 6.03\frac{M}{}$_Plmc⁻¹ in natural units. These formulas highlight the extreme lightness required of the constituent bosons for astrophysically relevant masses; for $m\approx 10-10 eV$ the maximum mass is of order a few solar masses.

When self‑interactions are present, the structure changes dramatically. A quartic repulsive interaction with coupling $\lambda$ or, equivalently, scattering length $a_s$ introduces an additional pressure term. In the Thomas–Fermi limit where this pressure dominates over quantum gradients, the equilibrium resembles an n=1 polytrope with radius independent of total mass. The maximum mass and radius scale as $M$_max≈0.06√λM_Pl3m2 and $R\approx 0.27\surd \lambda \frac{M}{}$_Plm2, respectively. Translating to scattering length via $\lambda =\mathrm{8\pi }{a}_{s}m$, one finds that even tiny self-interactions can inflate the star to galactic sizes while raising the maximum mass.

Our calculator implements both regimes with seamless interpolation. When the scattering length input is zero, it uses the noninteracting formulas above to report the maximum mass $M_{\max }$ and corresponding radius. If a nonzero $a_s$ is specified, it converts the scattering length from femtometers to natural units, computes the effective coupling λ, and applies the self‑interaction expressions. The outputs include the radius in kilometers and the mass in solar masses for easy astronomical interpretation. Because the scaling relations are approximate, the results provide order‑of‑magnitude guidance rather than precise values from full numerical solutions.

The notion of boson stars gained renewed interest with the advent of ultralight dark matter models. For axions with masses around 10⁻²² eV, the de Broglie wavelength on galactic scales suppresses small‑scale structure, potentially resolving some discrepancies of cold dark matter. In such scenarios, self‑gravitating solitonic cores resembling boson stars can form at the centers of halos. Their masses of ~10⁷ M_⊙ and radii of ~1 kpc emerge naturally from the scaling with $m-2$. By adjusting the boson mass in our calculator, one can explore whether a given particle candidate could produce astrophysically interesting structures.

Beyond dark matter, boson stars provide a theoretical laboratory for strong-field gravity. They lack an event horizon, offering alternative explanations for compact objects that might mimic black holes. Gravitational wave astronomy could potentially distinguish boson stars from black holes through differences in inspiral and ringdown signals. The mass–radius relation plays a crucial role in modeling such signatures. For example, a noninteracting boson star with mass 1 M_⊙ would have a radius of tens of kilometers, comparable to a neutron star but composed of an exotic condensate.

The table below compares illustrative values for noninteracting and self-interacting cases. The dramatic difference underscores how even minute self-couplings can dramatically alter the astrophysical properties of boson stars.

m (eV)	as (fm)	Mmax (M⊙)	R (km)
1e-10	0
1e-10	1
1e-5	0

When interpreting the outputs, it is essential to recognize the approximations involved. Relativistic corrections become important near the maximum mass, and numerical solutions reveal a more intricate stability structure with multiple branches. Rotating boson stars, vector boson configurations, and mixed fermion–boson systems each introduce additional parameters. Nevertheless, the simple scaling relations implemented here convey the essential dependence on fundamental quantities and provide a launching point for deeper investigation.

From a theoretical standpoint, boson stars showcase the rich interplay between quantum mechanics and gravity. The ground-state wavefunction balances gravity and repulsion or quantum pressure much like atomic orbitals balance the Coulomb attraction and kinetic energy. The formation of such objects in the universe depends on processes such as gravitational cooling, mergers of smaller clumps, or collapse of overdense regions. Numerical simulations indicate that scalar field halos can relax into solitonic cores via wave interference, a phenomenon sometimes called “granular relaxation.”

Detecting boson stars would revolutionize our understanding of dark matter and compact objects. Possible signatures include anomalous microlensing, distinctive gravitational waveforms, or unusual electromagnetic spectra if the bosons couple weakly to photons. The mass–radius relationship is central to predicting these signals. By experimenting with different particle masses and interaction strengths in the calculator, users can gauge which scenarios yield objects of observable size and mass.

In conclusion, boson stars bridge particle physics, cosmology, and astrophysics by linking microscopic parameters of hypothetical bosons to macroscopic celestial objects. The calculator presented here embodies this connection, translating a boson’s mass and scattering length into tangible astrophysical quantities. Whether used for educational purposes or as a rough estimate in research, it highlights the surprising consequences that can arise when quantum fields gravitate.