Tidal Deformability in Neutron Star Physics

When two neutron stars spiral toward one another, tidal forces raise bulges on each star, altering the gravitational-wave signal emitted during the inspiral. The ease with which a star deforms under an external tidal field is captured by the tidal deformability, a dimensionless parameter usually denoted Λ. Observations of binary neutron star mergers, especially the landmark event GW170817, have shown that the tidal deformability carries valuable information about the equation of state of ultradense matter. Softer equations of state produce stars that are more compressible and have smaller radii, yielding lower Λ values, whereas stiffer equations of state give larger radii and higher Λ. Measuring Λ therefore provides a rare window into the behavior of matter at densities several times greater than those found in atomic nuclei. This calculator allows users to estimate Λ for a single neutron star by providing its mass, radius, and dimensionless second Love number k₂, which encapsulates details of the star’s internal structure.

The dimensionless tidal deformability is defined by

$\Lambda =\frac{2k{2}_{}}{C^5}$

where C is the compactness parameter

$C=\frac{GM}{c^{2}R}$

In these expressions, M is the mass of the neutron star, R its radius, G Newton’s constant, and c the speed of light. The compactness is a measure of how strongly gravity is concentrated within the star: larger masses or smaller radii increase C. Because Λ scales as C⁻⁵, small changes in compactness produce dramatic differences in tidal response. A star with C = 0.2 and k₂ = 0.1 has Λ ≈ 500, while increasing C to 0.3 drops Λ to roughly 70. The calculator implements these formulas using SI units internally, converting masses from solar masses and radii from kilometers before evaluating C and Λ. The returned values help users explore how various equations of state map to tidal properties.

The Love number k₂ encapsulates details of how the star’s interior distribution of mass responds to tidal perturbations. It is obtained by solving the linearized Einstein equations for a slowly varying quadrupolar perturbation. Although its calculation requires integrating differential equations that depend on the equation of state, typical values for neutron stars lie between 0.05 and 0.15. Softer equations of state generally produce smaller k₂ due to stronger central concentration of mass. By allowing users to input k₂, the calculator accommodates a variety of theoretical models. Researchers often quote Λ for a canonical 1.4 M_☉ neutron star, comparing predictions with constraints from gravitational wave observations and x-ray timing measurements of masses and radii.

To see how the calculator works, consider the example of a 1.4 M_☉ star with a 12 km radius and k₂ = 0.1. The compactness is C ≈ 0.17, leading to Λ ≈ 400. If the radius is reduced to 11 km with the same mass and k₂, the compactness increases to 0.19 and Λ falls to about 250. Conversely, increasing the radius to 13 km boosts Λ to roughly 600. The table below summarizes these illustrative values and highlights the sensitivity of Λ to the stellar radius.

M (M☉)	R (km)	k2	Λ
1.4	11	0.1	≈250
1.4	12	0.1	≈400
1.4	13	0.1	≈600

The implications of measuring Λ extend beyond individual star properties. Gravitational wave signals from binary neutron star mergers encode a combination of the component tidal deformabilities called the weighted tidal parameter. Constraints on this parameter from GW170817 have already ruled out certain stiff equations of state that predict Λ values exceeding around 800 for typical masses. Future observations with improved sensitivity will further refine these limits, narrowing the range of viable models of dense matter. In turn, a better understanding of the equation of state feeds back into nuclear physics, astrophysics, and even interpretations of heavy-ion collision experiments.

From a theoretical perspective, the extreme sensitivity of Λ to compactness arises because tidal forces couple strongly to the star’s outer layers. The polytropic index, crust thickness, and possible phase transitions to exotic forms of matter such as hyperons or quark deconfinement all influence Λ. Some models even entertain the possibility of hybrid stars with quark cores and hadronic envelopes, producing distinctive k₂ and Λ values. Consequently, calculating Λ for a wide grid of masses and radii across different equations of state is a standard practice in neutron star research. This calculator provides a quick, transparent way to explore such parameter space without requiring specialized numerical codes.

Beyond gravitational wave astronomy, tidal deformability plays a role in electromagnetic counterparts to mergers. The disruption of neutron star matter depends on the stars’ compactness; less compact stars tidally shred more easily before merging, affecting the amount of ejecta that powers kilonova emission. By adjusting the inputs, users can experiment with how variations in Λ might influence observable light curves. For instance, a low Λ value suggests a compact star that plunges into its companion with little tidal disruption, whereas a high Λ points to significant mass ejection and possibly brighter kilonovae.

The calculator’s implementation is straightforward. The user-entered mass is multiplied by 1.98847 × 10³⁰ kg to convert from solar masses, while the radius is converted to meters. Newton’s constant 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² and the speed of light 2.99792458 × 10⁸ m/s are used to evaluate the compactness. The result for Λ is dimensionless. After computation, the script displays both C and Λ, noting whether Λ exceeds 1000 (indicative of a very stiff equation of state) or falls below 100 (suggestive of a soft equation). These informal classifications help contextualize the numerical output.

Exploring tidal deformability enriches our appreciation of how fundamental physics manifests in astrophysical settings. By combining gravitational wave data, electromagnetic observations, and theoretical insights encoded through parameters such as k₂, scientists inch closer to deciphering the true nature of matter at supranuclear densities. This calculator serves as a bridge between abstract theory and tangible numbers, enabling students, researchers, and enthusiasts to experiment with the delicate balance between gravity and internal pressure that shapes one of the universe’s most enigmatic objects.