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. In a binary inspiral, that stretching feeds back into the orbital motion and leaves a subtle but measurable imprint on the gravitational-wave signal. The quantity this calculator focuses on is the dimensionless tidal deformability Λ, which condenses that response into a single number.

This calculator estimates two closely related quantities from three inputs that matter most in the simple single-star picture: mass, radius, and the dimensionless quadrupolar Love number k₂. It first computes compactness, which tells you how tightly the star's mass is packed into its radius, and then combines that compactness with k₂ to estimate Λ. A star with a larger radius at the same mass is generally less compact and therefore easier to deform, while a smaller star with the same mass is more compact and harder to stretch.

For neutron-star astrophysics, Λ matters because it responds strongly to radius and to the internal density profile. That is why even a modest change in radius can shift the predicted tidal signal by a large factor. Observations such as GW170817 made that sensitivity especially valuable, since they allowed researchers to compare waveform data with theoretical equations of state. This page gives you a fast educational estimate, not a full relativistic stellar-structure solver, but it is a practical way to see how mass, radius, and Love number work together.

Introduction to neutron-star tidal deformability calculations

The neutron-star tidal deformability calculator is most useful when you want a quick sense of how the same mass can hide very different tidal behavior depending on radius and internal structure. The Love number k₂ carries information about that structure, while compactness captures the mass-to-size balance. Together they set Λ, the parameter that gravitational-wave analysts often quote when discussing binary neutron star mergers.

You can think of Λ as a response factor. A large value means the star is comparatively easy to distort by its companion's gravity, while a small value means the star is more tightly bound and resists deformation. That does not imply that a low-Λ star is abnormal; it simply reflects the way gravity, pressure support, and the distribution of dense matter interact. In many papers, a 1.4-solar-mass neutron star is used as a reference point because it is a convenient benchmark for comparing candidate equations of state.

The calculator works in SI units internally even though the form asks for mass in solar masses and radius in kilometers. Converting the user-friendly inputs into kilograms and meters keeps the physics consistent. Once the compactness is known, the page evaluates Λ and reports both values so you can trace the effect from input to output instead of seeing only the final number.

How to use the neutron-star tidal deformability calculator

To use this neutron-star tidal deformability calculator, enter one value in each of the three input fields, then press the compute button. Mass should be given in solar masses, radius in kilometers, and the Love number as a dimensionless decimal. The result box will show compactness C and tidal deformability Λ for the star you specified.

Each input has a specific meaning:

Mass is the gravitational mass of the neutron star, measured in solar masses. Values around 1.2 to 2.0 are common in many astrophysical studies, but the calculator accepts any positive number.

Radius is the circumferential radius in kilometers. Realistic neutron-star models are often discussed in the rough 10 to 14 km range, though the calculator will evaluate any positive radius you supply.

Love number k₂ describes how strongly the star's shape responds to a quadrupolar tidal field. It is dimensionless and often falls somewhere around 0.05 to 0.15 in many models. If you enter 0, the calculator will return Λ = 0 by construction, representing no tidal response in this simplified relation.

After calculation, read the output in two stages. Start with compactness, because it tells you how concentrated the star is. Then look at Λ, which depends on compactness to the fifth power and can therefore swing rapidly when radius or mass changes only slightly. The short note that appears after the number is intentionally qualitative; it is there to orient you, not to label a model as observationally confirmed or ruled out.

Formula for neutron-star compactness and tidal deformability

The formula section in this neutron-star tidal deformability calculator uses the standard compactness relation and the familiar Love-number expression for Λ. Compactness is

and the dimensionless tidal deformability is

Here, G is Newton's gravitational constant, M is the stellar mass, c is the speed of light, and R is the stellar radius. Compactness is dimensionless because the units cancel when SI quantities are used consistently, and Λ is dimensionless for the same reason.

In the script, the mass you enter is multiplied by 1.98847 × 10³⁰ kg to convert solar masses to kilograms, and the radius is multiplied by 1000 to convert kilometers to meters. The constants are G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² and c = 2.99792458 × 10⁸ m/s. After compactness is computed, the calculator evaluates Λ directly. Because Λ scales as C⁻⁵, a small decrease in compactness produces a very large increase in tidal deformability.

That steep scaling is exactly why neutron-star radius estimates matter so much. A slightly puffier star can be far easier to deform, and that difference shows up in the gravitational-wave phase evolution. The Love number contributes additional structure information by encoding how the star's interior responds to the external tidal field, so two stars with the same mass and radius can still differ if their internal profiles are not the same.

Worked example for a 1.4 M_☉ neutron star

For a concrete neutron-star tidal deformability example, suppose the mass is 1.4 M_☉, the radius is 12 km, and the Love number k₂ is 0.1. Enter those three values and submit the form. The calculator converts the inputs into SI units, computes compactness, and then evaluates Λ.

With those inputs, the compactness is about 0.17. Plugging that compactness into the deformability formula with k₂ = 0.1 gives a Λ value of roughly 1.3 × 10³. That is a helpful benchmark because it sits in the broad range often discussed for canonical neutron stars in realistic equations of state.

Now hold mass and Love number fixed while changing the radius. If you reduce the radius to 11 km, the star becomes more compact and Λ falls noticeably. If you increase the radius to 13 km, the star becomes less compact and Λ rises sharply. The table below illustrates that radius sensitivity and shows why astrophysical constraints on neutron-star size are so valuable.

The values are illustrative rather than definitive. They are meant to show the trend, not to stand in for a detailed stellar-structure model. In real calculations, k₂ also changes with the interior profile, so a full relativistic treatment would not usually keep it fixed while only the radius changes. Even so, this simple example is a good way to build intuition for how strongly compactness controls tidal response.

Interpreting compactness and Λ for neutron stars

A compactness value near 0.1 suggests a neutron star that is relatively less compact, while values closer to 0.2 or higher indicate a stronger concentration of mass. The calculator flags very high compactness values as extremely compact, but that note is descriptive only. It does not determine whether the star is stable, realistic, or compatible with any particular equation of state.

For Λ, larger values usually correspond to stars that are easier to deform and often to equations of state that predict larger radii. Smaller values correspond to more compact stars and often to softer equations of state. The script adds a brief note when Λ is above 1000 or below 100. Those labels are intentionally informal, so they are useful for quick orientation but not as hard observational thresholds.

In binary neutron-star studies, researchers often combine the deformabilities of both stars into an effective tidal parameter that shapes the gravitational-wave phase evolution. This calculator does not compute that binary quantity. It focuses on the single-star Λ so you can understand the building block before moving on to more advanced merger analyses.

Limitations and assumptions of this tidal deformability estimate

This neutron-star tidal deformability calculator is intentionally simple. It assumes that the Love number k₂ is already known and can be entered directly. In actual neutron-star modeling, k₂ is usually derived rather than observed outright: one solves relativistic stellar-structure and perturbation equations for a chosen equation of state, then extracts the Love number from that model. That makes the page an exploratory tool, not a replacement for a full physical calculation.

The relation used here also treats the star as a single isolated object described only by mass, radius, and k₂. It does not include rotation, magnetic fields, finite temperature, crust microphysics, superfluidity, phase transitions, or uncertainty in the equation of state. It also does not check whether a chosen combination of mass, radius, and Love number corresponds to a self-consistent neutron-star model. You can enter mathematically valid numbers that may not describe any star nature would actually permit.

Treat the result as a quick estimate rather than a publishable prediction. It is fine for classroom demonstrations, back-of-the-envelope comparisons, and intuition building, but it is not a substitute for peer-reviewed waveform inference, Tolman-Oppenheimer-Volkoff integrations, or detailed equation-of-state studies. If you need research-grade values, use this calculator as the first step and then compare its output with full relativistic modeling.

Even with those limitations, the calculator remains useful because it makes the scaling transparent. You can immediately see how increasing mass at fixed radius raises compactness, how increasing radius at fixed mass lowers compactness, and how both changes feed into Λ through a steep fifth-power dependence. That transparency is often exactly what students and non-specialists need before they move on to the more technical literature.