Neutron Star Tidal Deformability Calculator

Understanding neutron-star tidal deformability and Λ

Neutron-star tidal deformability describes how a compact star stretches when a companion’s gravity tries to pull it out of shape. During a binary inspiral, that stretching feeds back into the orbit and leaves a subtle imprint on the gravitational-wave signal. The dimensionless tidal deformability, Λ, condenses this response into one useful number.

This calculator estimates two related quantities from a star’s mass, radius, and dimensionless quadrupolar Love number k₂. It first calculates compactness, which measures how tightly mass is packed into a radius, then combines compactness with k₂ to estimate Λ. At the same mass, a larger-radius star is less compact and is generally easier to deform; a smaller-radius star is more compact and resists stretching more strongly.

Λ is valuable because it is sensitive to stellar radius and internal density structure. Even a modest radius change can move the predicted tidal signal substantially. Gravitational-wave observations, including GW170817, made this sensitivity especially useful for comparing waveform measurements with possible neutron-star equations of state. This page is an educational estimate rather than a relativistic stellar-structure solver, but it makes the main dependencies easy to explore.

Introduction to neutron-star tidal deformability calculations

The neutron-star tidal deformability calculator is useful when you want to compare how stars of similar mass can have different tidal behavior. The Love number k₂ represents information about the interior response, while compactness captures the mass-to-size balance. Together, they determine Λ, a parameter commonly discussed in binary neutron-star merger analysis.

You can regard Λ as a response factor. A large value means the star is comparatively easy for a companion’s tidal field to distort, whereas a small value means it is more tightly bound. Neither outcome is automatically better or worse: the value reflects gravity, pressure support, and the way dense matter is distributed. A 1.4-solar-mass neutron star is often used as a benchmark because it gives researchers a shared reference for comparing candidate equations of state.

The calculator uses SI units internally even though the form accepts solar masses and kilometers. Converting these convenient inputs to kilograms and meters keeps the compactness calculation consistent. Reporting both compactness and Λ lets you follow the physical chain from mass and radius to the final tidal-response estimate.

How to use the neutron-star tidal deformability calculator

To use this neutron-star tidal deformability calculator, enter all three values and press the compute button. Give mass in solar masses, radius in kilometers, and the Love number as a dimensionless decimal. The result area reports compactness C and tidal deformability Λ for the specified star.

Mass is the gravitational mass in solar masses. Values around 1.2 to 2.0 are common in many studies, although the calculator accepts any positive number. Radius is the circumferential radius in kilometers; many model discussions place it roughly between 10 and 14 km. Love number k₂ describes the quadrupolar shape response to a tidal field and is dimensionless. Many models have values around 0.05 to 0.15. Entering zero returns Λ = 0 by construction in this simplified relation.

Read the result in two stages. Compactness tells you how concentrated the mass is, then Λ shows the tidal consequence. Since Λ depends on the inverse fifth power of compactness, small input changes can produce large changes in Λ. The brief descriptive note in the result is only an orientation aid; it is not a statement that a model is confirmed or ruled out.

Formula for neutron-star compactness and tidal deformability

The neutron-star tidal deformability formula used by this calculator begins with the standard compactness relation:

The calculator then evaluates its displayed dimensionless tidal-deformability relation:

Here, G is Newton’s gravitational constant, M is stellar mass, c is the speed of light, and R is stellar radius. Both compactness and Λ are dimensionless when SI units are used consistently. The script multiplies the entered mass by 1.98847 × 10³⁰ kg and the radius by 1000 m/km. It uses G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² and c = 2.99792458 × 10⁸ m/s.

The fifth-power dependence is the key lesson. Reducing compactness a little, perhaps by increasing radius while holding mass fixed, can increase Λ dramatically. The Love number adds structure information, so stars with equal mass and radius can still differ if their internal density profiles differ.

Worked example: a 1.4 M_☉ neutron star

For a concrete neutron-star tidal deformability example, enter mass 1.4 M_☉, radius 12 km, and Love number k₂ = 0.1. The calculator converts those values to SI units, finds a compactness near 0.17, and then evaluates Λ at roughly 1.3 × 10³.

Now keep mass and Love number fixed while changing radius. At 11 km, the star is more compact and Λ falls; at 13 km, it is less compact and Λ rises sharply. The illustrative comparison below highlights that radius sensitivity.

These values illustrate a trend rather than a detailed stellar model. In a full relativistic calculation, k₂ also changes with the interior profile, so it would not normally remain fixed as radius changes. The example nevertheless gives a useful first intuition for the strong control compactness has over tidal response.

Interpreting neutron-star compactness and Λ results

A compactness near 0.1 indicates a relatively less compact neutron star, while values nearer 0.2 or above indicate a stronger mass concentration. The calculator describes very high values as extremely compact, but that wording does not establish stability or compatibility with a particular equation of state.

Larger Λ values generally mean a star is easier to deform and often correspond to models with larger radii. Smaller values generally correspond to more compact stars and often softer equations of state. In real binary-neutron-star work, the two individual deformabilities are combined into an effective tidal parameter that affects gravitational-wave phase evolution. This calculator intentionally focuses on the single-star building block first.

Limitations and assumptions of this tidal deformability estimate

This neutron-star tidal deformability calculator assumes that k₂ is already known. In research modeling, it is typically derived by solving relativistic stellar-structure and perturbation equations for a selected equation of state. The page is therefore an exploratory tool, not a replacement for a full physical calculation.

The estimate does not include rotation, magnetic fields, temperature, crust microphysics, superfluidity, phase transitions, or equation-of-state uncertainty. It also does not test whether an entered mass, radius, and Love number form a self-consistent neutron-star model. Mathematically valid inputs may still describe no star nature permits.

Use the output for classroom demonstrations, quick comparisons, and intuition building rather than publishable predictions. For research-grade values, compare this estimate with Tolman-Oppenheimer-Volkoff integrations, waveform inference, and detailed equation-of-state calculations. Its continuing value is transparency: you can immediately see how mass and radius change compactness and how the fifth-power scaling amplifies that change in Λ.