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In a central gravitational field, the radius vector sweeps out equal areas in equal times; areal velocity is constant.
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Formula
dA/dt and h in m2/s; A in m2; t in s
| Symbol | Meaning | Unit |
|---|---|---|
| Area swept by the radius vector | m2 | |
| Time | s | |
| Specific orbital angular momentum |r×v| | m2/s |
\frac{dA}{dt}=\frac{h}{2}=\mathrm{konstant}Gravity acts as a central force, so torque about the force centre is zero and specific angular momentum h=r2dφ/dt is conserved. Swept area per time is h/2. This applies to Newtonian two-body orbits; perturbations from other bodies, relativistic effects or noncentral forces can break the constancy. An ellipse moves faster at periapsis than at apoapsis.
Gravity acts as a central force, so torque about the force centre is zero and specific angular momentum h=r2dφ/dt is conserved. Swept area per time is h/2. This applies to Newtonian two-body orbits; perturbations from other bodies, relativistic effects or noncentral forces can break the constancy. An ellipse moves faster at periapsis than at apoapsis.
The relation \frac{dA}{dt}=\frac{h}{2}=\mathrm{konstant} connects the quantities defined in the variable list. Its form follows from the underlying astronomical model; it is not a universal rule outside the stated assumptions.
Defining relation
This is the documented defining relation. Rearrange it only when the requested quantity can be isolated unambiguously.
Check the worked example for Kepler’s equal-area law and areal velocity: For a planet with h=4.46·1015 m2/s, dA/dt=2.23·1015 m2/s; in 30 days it sweeps about 5.78·1021 m2.
Substitute quantities in the documented order, keep units consistent and compare the result with the check stated in the example.
Applying an approximate relation to real measurements without checking its model range.
Check the measurement definition, assumptions and units before interpreting the result physically.
Formula cluster
Quantify motion along elliptical orbits and apply Kepler’s second law.
For a planet with h=4.46·1015 m2/s, dA/dt=2.23·1015 m2/s; in 30 days it sweeps about 5.78·1021 m2.
Quantify motion along elliptical orbits and apply Kepler’s second law.
Here is how to work through a typical Kepler’s equal-area law and areal velocity (dA/dt = h/2 = konstant) task step by step:
Task
Check the worked example for Kepler’s equal-area law and areal velocity: For a planet with h=4.46·1015 m2/s, dA/dt=2.23·1015 m2/s; in 30 days it sweeps about 5.78·1021 m2.
Solution path
Substitute quantities in the documented order, keep units consistent and compare the result with the check stated in the example.
The equation describes a specific relation among astronomical quantities; it supports a reliable conclusion only when measurements match the variables’ definitions. Start by identifying each quantity, unit and reference frame. The relation \frac{dA}{dt}=\frac{h}{2}=\mathrm{konstant} organizes those values mathematically, while the physical model determines whether it applies to the situation. A number alone does not establish that. Record assumptions, signs and any approximations. This keeps separate what follows directly from calculation and what requires interpretation. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
The relevant limits are stated in the subject background: Gravity acts as a central force, so torque about the force centre is zero and specific angular momentum h=r2dφ/dt is conserved. Swept area per time is h/2. This applies to Newtonian two-body orbits; perturbations from other bodies, relativistic effects or noncentral forces can break the constancy. An ellipse moves faster at periapsis than at apoapsis. Read them before substitution because ideal geometry, small angles, constant properties or a particular cause of motion may be part of the derivation. Measurement uncertainty is also distinct from model validity: a precise measurement does not make an unsuitable approximation correct. Check that quantities use consistent conventions. If the assumptions fail, use a more suitable model or explicitly report the result as an estimate rather than an exact measurement. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Use the documented worked example as a calculation check: For a planet with h=4.46·1015 m2/s, dA/dt=2.23·1015 m2/s; in 30 days it sweeps about 5.78·1021 m2. Carry units through the arithmetic rather than handling numbers alone. Do they combine into the unit of the requested quantity? A dimensional mismatch often reveals a conversion error or swapped variables. If possible, substitute the result back into a rearranged original relation. Finally compare its order of magnitude with familiar astronomical scales; plausibility is not proof, but it is a useful error check. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Read the exponents and placement of quantities in \frac{dA}{dt}=\frac{h}{2}=\mathrm{konstant}: direct proportionality behaves differently from an inverse or power relation. Mentally double one input at a time and infer how the requested quantity changes. In observational data, a small relative uncertainty can matter greatly when a relation uses a small angle, a difference or an inverse quantity. Distinguish this sensitivity from measurement precision: the equation describes error propagation, not how well an instrument measures. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
State the numerical value with its unit, the input measurements and the main model assumption. If using an approximation, label the result as an estimate and say which effects were ignored. Avoid conclusions beyond the equation: a geometric or photometric quantity is not automatically a complete description of a star or galaxy. When measurement errors are known, do not report more precision than the data support. This concise record makes both calculation and interpretation traceable and reproducible. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Quanta exam set
Curated exam set for "Kepler’s equal-area law and areal velocity":
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Which formula describes Kepler’s equal-area law and areal velocity?
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How do you rearrange Kepler’s equal-area law and areal velocity for Defining relation?
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Which common mistake happens with Kepler’s equal-area law and areal velocity?
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