Orbital Velocity⚠ unverified

Understanding the Parameters

• The gravitational parameter $\mu = GM$ — the product of Newton's constant and the primary's mass. It is known far more precisely than $G$ or $M$ separately (because it is what actually governs orbits, measured directly from spacecraft tracking): Earth's $\mu = 3.986\times10^{14}\ \text{m}^3/\text{s}^2$. Using $\mu$ instead of $GM$ is the universal convention in astrodynamics.

• Semi-major axis $a$ — the energy variable — this is the deep content. The orbit's total specific energy depends only on $a$: $\varepsilon = -\mu/2a$ (see Vis Viva Energy). So $a$ alone fixes how fast the body goes at any given $r$. Two orbits crossing the same point at the same $r$ but with different $a$ have different speeds — the bigger orbit (larger $a$) is faster there, because it carries more energy.

• Radius $r$ — the position variable — as the body moves around an ellipse, $r$ changes but $a$ does not, so the speed rises and falls: maximum at perigee ($r = a(1-e)$), minimum at apogee ($r = a(1+e)$). The equation captures the continuous speed-up and slow-down that Kepler saw as his second law.

• The three regimes in one formula:

• $r = a$ (circle): $v = \sqrt{\mu/r}$, the Circular Orbit Velocity.
• $a \to \infty$ (parabola, marginal escape): $1/a \to 0$, giving $v = \sqrt{2\mu/r}$, the Escape Velocity.

• $a < 0$ (hyperbola, excess energy): $-1/a > 0$ adds to the speed — the body arrives with leftover velocity ("hyperbolic excess"), the basis of interplanetary departures.

• What it does not give — direction. Vis-viva returns only the magnitude of velocity. The flight-path angle (the tilt of velocity relative to local horizontal) needs the angular momentum separately, as in Eccentricity Vector.

Derivation (Approaching a Proof)

Vis-viva is conservation of energy for the two-body problem, made specific (per unit mass). Two quantities are conserved on a Keplerian orbit: the specific mechanical energy $\varepsilon$ and the specific angular momentum. Use energy.

The specific orbital energy at any point is kinetic plus (negative) gravitational potential:

$$\varepsilon = \frac{v^2}{2} - \frac{\mu}{r}.$$

This is Vis Viva Energy, and it is the same value everywhere on the orbit (energy is conserved). Now evaluate it at a convenient point to pin its value in terms of $a$. Take perigee and apogee, where velocity is purely tangential, and use conservation of both energy and angular momentum between them; the algebra collapses to the compact and famous result that the total energy depends only on the semi-major axis:

$$\varepsilon = -\frac{\mu}{2a}.$$

(For a circle this is immediate: $\varepsilon = v^2/2 - \mu/r$ with $v^2 = \mu/r$ gives $\varepsilon = \mu/2r - \mu/r = -\mu/2r = -\mu/2a$.) Set the two expressions for $\varepsilon$ equal:

$$\frac{v^2}{2} - \frac{\mu}{r} = -\frac{\mu}{2a}.$$

Solve for $v^2$:

$$\frac{v^2}{2} = \frac{\mu}{r} - \frac{\mu}{2a} \quad\Longrightarrow\quad v^2 = \frac{2\mu}{r} - \frac{\mu}{a} = \mu\left(\frac{2}{r} - \frac{1}{a}\right),$$

and taking the square root gives the vis-viva equation:

$$v = \sqrt{\mu\left(\frac{2}{r} - \frac{1}{a}\right)}. \qquad\blacksquare$$

The physical reading: the $2\mu/r$ term is (twice) the local potential depth — how much kinetic energy the body has "fallen into" — and the $-\mu/a$ term is (twice) the fixed total energy that caps it. Their difference is the kinetic energy per unit mass.

Dimensional check. $\mu$ is $\text{m}^3/\text{s}^2$ and $2/r-1/a$ is $\text{m}^{-1}$: $$\mu\left(\frac{2}{r}-\frac1a\right) = \frac{\text{m}^3}{\text{s}^2}\cdot\frac{1}{\text{m}} = \frac{\text{m}^2}{\text{s}^2},$$ and the square root gives $\text{m}/\text{s}$ ✓.

History and Development

• Kepler's laws (1609–1619). Johannes Kepler, working from Tycho Brahe's observations, found orbits are ellipses (first law), sweep equal areas in equal times (second law — the varying speed vis-viva encodes), and obey $T^2 \propto a^3$ (third law — see Orbital Period). His laws were empirical; the why awaited Newton.

• Newton's synthesis (1687). Isaac Newton's Principia derived all three of Kepler's laws from the inverse-square law of gravitation and his laws of motion, showing conic-section orbits are the general solution of the two-body problem.

• Vis viva. The name is a fossil of a 17th–18th century dispute: Gottfried Leibniz called $mv^2$ the vis viva ("living force"), contesting Descartes's momentum $mv$ as the true measure of motion. The conserved combination of $\tfrac12 v^2$ and potential that resolves the orbit problem inherited the name, and it stuck in astrodynamics even after "kinetic energy" replaced "vis viva" elsewhere.

• The workhorse of spaceflight. Vis-viva is the single most-used equation in mission design: every burn calculation (see Hohmann Transfer Dv1, Hohmann Transfer Dv2) evaluates it at two points and differences the speeds. It is exact for the two-body model; real missions add perturbations (Nodal Precession) and multi-body effects on top.

Related Concepts: Vis Viva Energy, Circular Orbit Velocity, Escape Velocity, Orbital Period, Hohmann Transfer Dv1, Hohmann Transfer Dv2, Eccentricity Vector, Gravitational Force

Notes: The vis-viva equation — one relation for all conics: $r=a$ → circular $\sqrt{\mu/r}$; $a\to\infty$ → escape $\sqrt{2\mu/r}$; $a<0$ → hyperbolic. Energy depends only on $a$ ($\varepsilon=-\mu/2a$); $r$ sets position. Returns speed magnitude only (not direction). $\mu=GM$ mislabelled dimensionless (is m³/s²). The $a\le0$ guard returns 0, suppressing valid hyperbolic ($a<0$) speeds.