Goal
Calculate a Newtonian two-body orbital period in seconds and days from semimajor axis and primary and secondary masses.
Kepler's Third Law Two-Body Orbital Period
Calculate a Newtonian two-body orbital period in seconds and days from semimajor axis and primary and secondary masses.
Key facts
- What it does
- Calculate a Newtonian two-body orbital period in seconds and days from semimajor axis and primary and secondary masses.
- Formula
T = 2 pi sqrt(a^3 / (G (M1 + M2))), using G = 6.67430e-11 m^3/(kg s^2), semimajor axis a in metres, and both body masses in kilograms.- You enter
- Semimajor axis · Primary mass · Secondary mass
- Worked example
- An Earth-Sun-scale two-body input gives an ideal period of about 365.26 days, with the exact seconds value returned by the formula.
A clearer path to an answer
From your question to a useful result
This page keeps the calculation transparent: define the goal, enter the matching values, inspect the method, and decide what the result means in your situation.
Inputs
Semimajor axis · Primary mass · Secondary mass
Method
T = 2 pi sqrt(a^3 / (G (M1 + M2))), using G = 6.67430e-11 m^3/(kg s^2), semimajor axis a in metres, and both body masses in kilograms.
Next step
Calculate, review the assumptions below, then compare a related tool when the decision needs more context.
Kepler's Third Law Two-Body Orbital Period
Calculate a Newtonian two-body orbital period in seconds and days from semimajor axis and primary and secondary masses.
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Result
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01
Inputs (3)
- Semimajor axis Ready
- Primary mass Ready
- Secondary mass Ready
02
Formula
T = 2 pi sqrt(a^3 / (G (M1 + M2))), using G = 6.67430e-11 m^3/(kg s^2), semimajor axis a in metres, and both body masses in kilograms.
Bounded, transparent calculation
03
Result
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Formula, assumptions, and example
Formula: T = 2 pi sqrt(a^3 / (G (M1 + M2))), using G = 6.67430e-11 m^3/(kg s^2), semimajor axis a in metres, and both body masses in kilograms.
This calculator evaluates the Newtonian two-body Kepler period for a positive semimajor axis and positive primary and secondary masses. It returns the period in seconds and days; eccentricity, perturbations, relativistic effects, body sizes, and mission planning are outside the ideal orbit model.
- Semimajor axis is a finite positive relative-orbit length in metres, and both primary and secondary masses are finite positive kilograms.
- The orbit is treated as a bound Keplerian two-body problem with total mass M1 + M2 and fixed Newtonian gravitational constant G = 6.67430e-11 m^3/(kg s^2).
- The supplied semimajor axis describes the relative orbit; eccentricity does not enter the period for a fixed semimajor axis in this ideal model.
- Third-body perturbations, tides, oblateness, mass change, relativistic corrections, body radii, launch trajectories, and mission advice are outside the calculation.
Worked example: An Earth-Sun-scale two-body input gives an ideal period of about 365.26 days, with the exact seconds value returned by the formula.
Displayed input contract
- Semimajor axis · minimum 1.0E-6 · maximum 10000000000000000
- Primary mass · minimum 1.0E-6 · maximum 1.0E+31
- Secondary mass · minimum 1.0E-6 · maximum 1.0E+31
The displayed limits are checked before the handler runs. Model-specific domain checks may also reject impossible or non-finite inputs.
Methodology: This calculator follows the WorldCalculate input, formula, precision, and boundary policy. Read the official methodology.
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Answer-first guide
How to use the Kepler's Third Law Two-Body Orbital Period for a real question
Calculate a Newtonian two-body orbital period in seconds and days from semimajor axis and primary and secondary masses. Start with one clearly defined goal, enter values in the units shown, and keep the result attached to the assumptions below.
What this answers
This tool is useful when your question includes Kepler third law, orbital period, semimajor axis. It returns the outputs declared in the calculator contract rather than a live quote, approval, diagnosis, or professional sign-off.
What you enter
Semimajor axis · Primary mass · Secondary mass. Keep the same time period, unit system, and currency wherever the form requires comparable values.
How to check it
Run the worked example first, compare its output with the page's example, then change one input at a time. This makes an unexpected result easier to trace to a unit, boundary, or assumption.
Three checks before you rely on the answer
- Match the question. Confirm that the result means the quantity you need, not a similar-sounding percentage, balance, rate, or estimate.
- Match the inputs. Use the requested units and period, and read each hint before replacing the example values with your own.
- Read the boundary. Review the assumptions and limits. Semimajor axis is a finite positive relative-orbit length in metres, and both primary and secondary masses are finite positive kilograms.
Need a wider view? Browse Science Calculators or compare the related tools below. The WorldCalculate methodology explains how formulas, examples, limits, and revisions are reviewed.
How to use the Kepler's Third Law Two-Body Orbital Period
- Enter Semimajor axis — Finite positive center-to-center relative-orbit semimajor axis in metres. (m).
- Enter Primary mass — Finite positive mass of the primary body in kilograms. (kg).
- Enter Secondary mass — Finite positive mass of the secondary body in kilograms. (kg).
- Choose Calculate and read the result panel.
- Use Download PDF or Download Word to save a result sheet.
Formula
T = 2 pi sqrt(a^3 / (G (M1 + M2))), using G = 6.67430e-11 m^3/(kg s^2), semimajor axis a in metres, and both body masses in kilograms.
This calculator evaluates the Newtonian two-body Kepler period for a positive semimajor axis and positive primary and secondary masses. It returns the period in seconds and days; eccentricity, perturbations, relativistic effects, body sizes, and mission planning are outside the ideal orbit model.
Worked example
An Earth-Sun-scale two-body input gives an ideal period of about 365.26 days, with the exact seconds value returned by the formula.
Assumptions and limits
- Semimajor axis is a finite positive relative-orbit length in metres, and both primary and secondary masses are finite positive kilograms.
- The orbit is treated as a bound Keplerian two-body problem with total mass M1 + M2 and fixed Newtonian gravitational constant G = 6.67430e-11 m^3/(kg s^2).
- The supplied semimajor axis describes the relative orbit; eccentricity does not enter the period for a fixed semimajor axis in this ideal model.
- Third-body perturbations, tides, oblateness, mass change, relativistic corrections, body radii, launch trajectories, and mission advice are outside the calculation.
Who uses this calculator?
- Astronomy students learning Keplerian periods
- Physics learners connecting Newtonian gravity with orbital motion
- Teachers demonstrating semimajor-axis scaling
When is it useful?
- Calculate an ideal two-body orbital period from SI masses and semimajor axis.
- Convert the period from seconds to days for an astronomy worksheet.
- Compare how total mass and orbital scale change a Keplerian period.
Context and background
The model-first approach to science
Science calculators define a system, choose an equation, apply units and constants, and show the substitution. Effects outside that model remain outside the result.
Introductory science problem solving builds from measured quantities and idealized relationships. Those models are valuable for learning and first-pass estimates, while experiments and engineering decisions need additional evidence.
Research and review
How this guide was researched
Researched by Hassan ALRowaie, Founder and editorial researcher at WorldCalculate.
This guide follows the live calculator's declared inputs, formula, worked example, assumptions, validation boundaries, and source-backed methodology. The review date describes editorial review of the calculator explanation; it is not a promise that external facts or rates remain current.
Kepler's third-law period relation connects the size of a bound two-body orbit with the total mass that supplies its Newtonian gravitational parameter. This calculator accepts a positive semimajor axis in metres and positive primary and secondary masses in kilograms. It uses G = 6.67430e-11 m^3/(kg s^2) and evaluates T = 2 pi sqrt(a^3/(G(M1 + M2))). The outputs are orbital period in seconds and days. The page describes an ideal Newtonian Keplerian model, not a launch plan, ephemeris, spacecraft mission, relativistic orbit, or guarantee about a real system. The guide explains the relative semimajor axis, total mass, Earth-Sun-scale example, bounds, finite checks, eccentricity, and model boundary.
The orbital question answered here
The calculator answers a defined astronomy question: how long does one ideal two-body orbit take when the relative semimajor axis and both body masses are supplied? The handler combines the two positive masses, applies the fixed Newtonian gravitational constant, raises the semimajor axis to the third power, and takes the square-root period relation. It reports seconds and a unit conversion to days. It does not determine an orbit from observations, select a launch state, or predict where a body will be at a future date.
The phrase two-body is essential. The formula treats the primary and secondary as the only gravitating bodies in the model and assumes the entered semimajor axis belongs to their relative orbit. A real planetary or satellite system can include other bodies, nonspherical gravity, tides, atmosphere, radiation pressure, and changing mass. The calculator intentionally leaves those effects out. Its result is useful as a Keplerian reference when the input definitions and approximation are made explicit.
- Inputs are a relative semimajor axis and two masses.
- Outputs are period in seconds and days.
- Total mass enters the Newtonian period relation.
- No ephemeris or mission trajectory is generated.
What the semimajor axis represents
The semimajor axis is the long radius of the ellipse associated with the relative orbit. For a circular orbit it equals the radius. For an elliptical orbit it is the average of the longest and shortest diameters divided by the appropriate two-radius interpretation, and it remains the parameter that controls the ideal period relation. The field is labeled in metres and is described as a relative-orbit length. It is not automatically the distance from the primary's center to the secondary at one arbitrary instant.
A real observation may report an instantaneous separation, a barycentric semimajor axis, or a semimajor axis for one body's motion around the center of mass. Those quantities need careful definition before entry. This calculator expects the relative semimajor axis used with M1 + M2 in the stated formula. It cannot inspect an orbital data set or convert between coordinate conventions. Preserving that distinction prevents a correct equation from being applied to a mismatched length.
- The input is a positive relative semimajor axis.
- A circular orbit has a equal to its radius.
- An instantaneous separation is not automatically a.
- Coordinate conventions must be resolved before entry.
Why the two masses are added
For the Newtonian two-body relative motion, the gravitational parameter uses the total mass M1 + M2. The primary and secondary fields are therefore both required even when one body is much lighter. Adding them gives the mass that controls the relative orbital period in this model. The handler finite-checks the sum and requires it to be positive. Positive individual bounds already ensure that ordinary catalog inputs meet this condition, while the explicit aggregate check documents the cross-field domain the formula depends on.
When the secondary mass is negligible compared with the primary, the result approaches the familiar small-satellite approximation around a fixed central mass. The calculator does not silently discard the secondary. Retaining both fields makes the approximation visible and supports binary or comparable-mass examples. It also prevents a visitor from assuming that every period relation uses only a central mass without checking which coordinate and two-body convention the formula requires.
- Total mass is M1 + M2.
- The secondary is not silently ignored.
- The aggregate mass is finite-checked and positive.
- Comparable-mass systems remain within the stated relation.
Deriving Kepler's third-law form
For circular motion, gravitational attraction supplies the required centripetal acceleration. Extending the Newtonian two-body result to a bound ellipse gives the same period dependence on semimajor axis: T squared is proportional to a cubed divided by the total gravitational parameter. Solving for T produces T = 2 pi sqrt(a^3/(G(M1 + M2))). The handler follows this form directly and does not use a fitted astronomical constant. This is why the input units and fixed value of G are visible in the catalog formula.
The square root is applied only after the positive ratio a cubed divided by G times total mass has been validated. The factor 2 pi converts the characteristic angular relation into a complete orbital period. A dimensional check gives seconds squared inside the square root and seconds after it is taken. The derivation is a model explanation, not a claim that all measured orbits are exact Kepler ellipses once perturbations and relativistic effects are considered.
- T^2 scales as a^3 divided by total gravitational parameter.
- The fixed constant is G = 6.67430e-11.
- The positive ratio is square-rooted before multiplying by 2 pi.
- The dimensional result is seconds.
The Earth-Sun-scale worked example
The catalog example uses semimajor axis 149597870700 m, primary mass 1.98847e30 kg, and secondary mass 5.9722e24 kg. These values are Earth-Sun-scale reference inputs. Adding the masses and applying the exact formula gives a period of about 365.26 days; the seconds row contains the unconverted floating-point result for the selected values. The example demonstrates the scale of an astronomical orbit and provides a useful known-answer check without treating rounded body masses as a complete ephemeris data set.
The example should not be read as a forecast of the actual calendar year or a claim that the real Earth-Sun system is an isolated point-mass pair. The entered masses are rounded reference values, and the formula omits perturbations and observational uncertainty. A report should preserve the exact input numbers used by the calculation. If a higher-precision orbital determination is needed, use a data and dynamics model that defines reference frames, epochs, perturbations, and fitted parameters separately.
- Semimajor axis is 149597870700 m.
- The primary is about 1.98847e30 kg.
- The secondary is about 5.9722e24 kg.
- The ideal result is about 365.26 days.
Seconds and days are two scales of one period
The primary period result is in seconds because SI units make the gravitational formula direct. The second result divides by 86400 seconds per day. It is the same period at a more readable calendar scale, not a separate orbital calculation. A reverse check multiplies the days value by 86400 and recovers the seconds value apart from display rounding. Keeping both rows helps a learner connect textbook SI arithmetic with the time unit commonly used in astronomy.
A day in this conversion is the fixed 86400-second unit used by the calculator. It is not a sidereal day, a solar-day observation, or a calendar convention. If a source uses another time scale, the conversion must be stated outside the handler. The page does not adjust for leap years, observation epochs, or the difference between orbital period and a calendar recurrence. The output label stays orbital period so it cannot be confused with a date prediction.
- Seconds are the direct SI result.
- Days equal seconds divided by 86400.
- The two rows represent one period.
- Calendar and sidereal-time interpretation is outside the conversion.
Eccentricity and the fixed semimajor axis
In the ideal Keplerian two-body model, the period depends on semimajor axis and total mass, not directly on eccentricity. Two ellipses with the same semimajor axis and total mass have the same period even if one is more elongated. This is why eccentricity is not an input here. The statement assumes the orbit remains a bound Kepler ellipse and that the same semimajor-axis convention is used. It does not mean that eccentricity has no effect on speed, distance, or the path shape during a revolution.
At different points in an eccentric orbit, the instantaneous separation and speed vary. The calculator does not return either quantity; it returns the complete-cycle period associated with a. Substituting a current separation for a can therefore produce a different and incorrectly interpreted result. If a problem asks for periapsis speed, apoapsis distance, or a time-of-flight segment, a separate orbital equation and additional inputs are required.
- Fixed a and total mass determine the ideal period.
- Eccentricity changes path shape and speed distribution.
- Instantaneous separation is not a substitute for a.
- Segment time-of-flight needs another model.
Catalog bounds and finite arithmetic
The semimajor-axis field accepts 0.000001 through 1e16 m. Each mass accepts 0.000001 through 1e31 kg. The default Earth-Sun-scale values are inside those bounds, while the lower endpoints keep every denominator positive and the upper endpoints keep the axis cube, gravitational parameter, period ratio, and day conversion finite in the JavaScript number range. These are conservative computational limits for a general calculator. They are not claims about the smallest orbit, largest body, or any physically stable configuration across the full interval.
The engine finite-checks the total mass, a cubed axis, the gravitational parameter, the ratio under the square root, the seconds result, and the days result. It also rejects a nonpositive aggregate before taking a square root. This layered protection matters because a finite input can still create an overflow in a derived expression if future bounds change. The current metadata and handler are aligned for ordinary endpoints, but a finite result remains an explicit requirement rather than an assumption.
- Axis bounds are 1e-6 to 1e16 m.
- Each mass bound is 1e-6 to 1e31 kg.
- All intermediate period quantities are finite-checked.
- Bounds are software limits, not stability guarantees.
Positive masses and aggregate validation
Both primary and secondary masses must be strictly positive finite numbers. A zero or negative mass is outside this Newtonian orbital contract, and a text value is not converted. The handler checks each field before adding them. It then checks total mass explicitly, even though the individual lower bounds make a nonpositive sum impossible for accepted ordinary inputs. This aggregate check records the actual denominator condition and protects the formula if the field contract is later reused or modified.
The semimajor axis also must be strictly positive. A zero axis would collapse the relative orbit and produce a zero period that does not describe the positive-size bound orbit intended by this page. Negative length has no meaning here. Values just outside the bounds are rejected rather than made positive with an absolute value. Preserving sign and domain errors is important in astronomy because silently repairing a mass or length can produce a plausible but unrelated period.
- Both masses are strictly positive.
- Total mass must be positive and finite.
- Semimajor axis is a positive length.
- Invalid signs are rejected rather than repaired.
Newtonian two-body and bound-orbit assumptions
The relation is Newtonian: gravity is represented by a fixed G and the bodies are treated as point masses for the relative motion. It assumes a bound Keplerian orbit with a meaningful semimajor axis. At strong gravity, high speeds, close separations, or extreme mass ratios, relativistic and finite-body effects can matter. The handler does not receive enough information to test those regimes. It evaluates the declared formula and leaves the physical validity check to the context in which the inputs were selected.
The word bound also matters. A positive semimajor axis in the catalog arithmetic does not by itself prove that a real object is gravitationally captured or that the specified state has the energy of an ellipse. The page takes a semimajor axis as an already established orbital parameter. It does not solve for eccentricity, velocity, orbital energy, capture conditions, or escape. Those questions need additional observations or initial-state fields and must not be inferred from the period output.
- The model is Newtonian.
- Bodies are idealized as a two-body point-mass system.
- A bound Keplerian orbit is assumed.
- Capture and initial-state conditions are not calculated.
Perturbations, body sizes, and real systems
Real orbital periods can be affected by third-body gravity, oblateness, tides, atmospheric drag, radiation pressure, mass exchange, and relativistic corrections. A body also has a radius and may rotate, deform, or interact with a central surface. None of these effects appears in the three fields. The ideal period can still serve as a baseline, but a difference between it and an observed period is not automatically a software error. It may reflect the model's deliberately omitted physics or the definitions of the supplied orbital parameters.
A real satellite or spacecraft problem may also need launch velocity, inclination, orientation, epoch, maneuvers, atmosphere, and ground-track constraints. The calculator does not plan a transfer, determine reentry, assess collision risk, or guarantee an orbit. A period is one scalar property of a defined orbit, not a mission plan. Keeping this boundary next to the result prevents a familiar astronomical formula from being read as operational guidance.
- Third bodies and nonspherical gravity are omitted.
- Tides, drag, radiation, and mass change are omitted.
- Body radius and surface interaction are not modeled.
- No launch, transfer, collision, or mission advice is returned.
Scaling and independent checks
The formula shows that period scales as the three-halves power of semimajor axis and as the inverse square root of total mass. Increasing a by a factor of four increases the ideal period by a factor of eight when mass stays fixed. Increasing total mass by a factor of four halves the period. These are useful checks for a worksheet or code review. They assume that the orbit remains within the same two-body model and that the changed parameter can be interpreted without changing the other definitions.
A dimensional and reciprocal check is also valuable. Compute G(M1 + M2), divide a cubed by that quantity, take the square root, multiply by 2 pi, and then divide by 86400 for days. If a result differs by a large factor, inspect metres versus kilometres, kilograms versus solar-mass units, or a period-versus-frequency confusion. The handler does not perform those source-unit conversions, so the preparation record must make them visible.
- T scales as a^(3/2) at fixed total mass.
- T scales as total mass^(-1/2) at fixed a.
- Check the gravitational parameter and square-root path.
- Inspect source units before changing the formula.
How to report a Kepler period
A reproducible report should list the semimajor axis in metres, primary and secondary masses in kilograms, G, and the statement that a is the relative-orbit semimajor axis. Show the total mass, the period formula, the seconds result, and the division by 86400 for days. If the values are reference estimates, label them as such and preserve their precision. Do not report only the days value because a reader then cannot tell whether the input used a solar-mass approximation, a different time conversion, or a different length convention.
The report should say ideal Newtonian two-body period. It should not call the number an ephemeris, a future position, a launch requirement, or a guarantee that a real orbit is stable. If the result informs a spacecraft or planetary analysis, add perturbation, reference-frame, epoch, and uncertainty work outside this calculator. The page's value is a transparent baseline that clearly identifies what was calculated and what remains to be established.
- Record a, M1, M2, G, and all SI units.
- State that a is a relative semimajor axis.
- Report seconds and the 86400-second day conversion.
- Label the result as an ideal Newtonian baseline.
Final scope checklist
Before accepting the result, confirm that semimajor axis and both masses are finite positive numbers within the catalog bounds. Check the total mass, the cube and square-root operations, and the conversion from seconds to days. Verify that the example's Earth-Sun-scale result is about 365.26 days when the exact catalog inputs are used. These checks validate the numerical relation. They do not prove that an observed object follows an isolated Kepler ellipse or that the entered semimajor axis belongs to the intended relative coordinate.
The honest conclusion is limited: the Newtonian two-body Kepler third-law period was evaluated for the entered semimajor axis and masses. No eccentricity fit, perturbation model, relativistic correction, body-contact analysis, launch trajectory, mission plan, or safety decision was produced. If the actual question includes those features, stop at this baseline and define the added dynamics separately. A clear model boundary makes the seconds and days outputs useful without turning them into unsupported orbital promises.
- Confirm positive finite a, M1, and M2.
- Check total mass and unit conversion.
- Distinguish a period baseline from an ephemeris.
- Keep real-orbit and mission decisions outside this model.
Reference frames, epochs, and observed periods
An observed orbital period is attached to a measurement definition, reference frame, and time span. A catalog semimajor axis can come from a fit whose epoch, coordinate origin, and data treatment are important. This calculator does not ask for any of those metadata because it assumes that the three numeric fields already describe the relative Keplerian orbit. Comparing its output with an observation therefore requires more than matching a number. The analyst must check that the measured cycle and the modeled a refer to the same physical pair and convention.
The page also does not fit data or estimate uncertainty. If repeated observations produce slightly different periods, the difference may reflect noise, perturbations, parameter covariance, or a changing system. The calculator can evaluate each selected parameter set, but it cannot decide which set is preferred or whether a discrepancy is statistically meaningful. Keep the observation record, epoch, reference frame, fitted uncertainties, and model selection outside the pure arithmetic. This separation makes the ideal period a transparent baseline instead of an unsupported claim about a real orbit.
- Observed periods require a reference frame and epoch.
- The handler assumes a is already established.
- No orbital fit or uncertainty estimate is returned.
- Compare measurements only after definitions agree.
Uncertainty in a period estimate
The period is a central value for the entered semimajor axis and masses. The handler does not propagate uncertainty, choose significant figures, or fit a posterior distribution. Because the axis enters with a three-halves power, an uncertainty in a can have a larger relative effect on the period than an equal relative uncertainty in total mass, whose exponent is negative one-half. A reported comparison should preserve the input uncertainties and calculate an interval with a method appropriate to the source data.
The calculator remains useful as a transparent reference calculation. State whether each mass and the semimajor axis is measured, fitted, rounded, or illustrative, then avoid presenting extra decimal places as extra knowledge. If an observed period differs from the central result, inspect the parameter definitions, epoch, reference frame, perturbations, and uncertainty before changing G or the formula. This page produces no confidence level, residual, preferred orbit, or decision threshold for an astronomical system.
- The outputs are central values without propagated intervals.
- Semimajor-axis uncertainty has a three-halves scaling effect.
- Input precision does not establish physical certainty.
- No fit, residual, confidence level, or decision threshold is returned.
Frequently asked questions
What is the Kepler's Third Law Two-Body Orbital Period?
Calculate a Newtonian two-body orbital period in seconds and days from semimajor axis and primary and secondary masses.
What is the formula for the Kepler's Third Law Two-Body Orbital Period?
T = 2 pi sqrt(a^3 / (G (M1 + M2))), using G = 6.67430e-11 m^3/(kg s^2), semimajor axis a in metres, and both body masses in kilograms. This calculator evaluates the Newtonian two-body Kepler period for a positive semimajor axis and positive primary and secondary masses. It returns the period in seconds and days; eccentricity, perturbations, relativistic effects, body sizes, and mission planning are outside the ideal orbit model.
What do I need to use this calculator?
Enter Semimajor axis, Primary mass, Secondary mass, then choose Calculate.
What are the limits of this calculator?
Semimajor axis is a finite positive relative-orbit length in metres, and both primary and secondary masses are finite positive kilograms. The orbit is treated as a bound Keplerian two-body problem with total mass M1 + M2 and fixed Newtonian gravitational constant G = 6.67430e-11 m^3/(kg s^2). The supplied semimajor axis describes the relative orbit; eccentricity does not enter the period for a fixed semimajor axis in this ideal model. Third-body perturbations, tides, oblateness, mass change, relativistic corrections, body radii, launch trajectories, and mission advice are outside the calculation.
Methodology
This calculator is part of the WorldCalculate library. Its formula, example, assumptions, input bounds, and output formatting follow the official methodology.
Use this calculator as part of a bigger plan
These WorldCalculate collections connect this tool with related questions while keeping each calculation separate and transparent.