Vis Viva Energy⚠ unverified

Understanding the Parameters

• Why "specific" (per unit mass) — dividing total energy $\tfrac12 mv^2 - \mu m/r$ by the mass $m$ removes it entirely. A pebble and a space station on the same orbit have the same specific energy. This is why astrodynamics works in per-unit-mass quantities throughout: the orbit does not care how heavy the satellite is.

• The sign is everything — a body is gravitationally bound only if its kinetic energy is less than the depth of the potential well:

• $\varepsilon < 0$: kinetic $<$ potential depth → ellipse (returns). Circles and all closed orbits.

• $\varepsilon = 0$: kinetic exactly balances → parabola, the escape threshold; the body coasts to infinity arriving with zero speed. This is where Escape Velocity comes from ($v^2/2 = \mu/r$).

• $\varepsilon > 0$: leftover kinetic energy → hyperbola, escapes with speed to spare.

• Energy fixes the size: $\varepsilon = -\mu/2a$ — the most useful identity in the subject. All orbits with the same energy have the same semi-major axis $a$, regardless of shape (eccentricity). A burn that changes the orbit's energy changes its $a$; a burn that only rotates the velocity (a pure plane change, see Inclination Change Delta V) leaves $\varepsilon$ and $a$ untouched.

• The kinetic and potential trade around the orbit — as a satellite falls toward perigee, potential energy ($-\mu/r$, more negative) converts to kinetic ($v^2/2$, larger); toward apogee the reverse. Their sum $\varepsilon$ stays fixed — the essence of an orbit as perpetual energy exchange.

Derivation (Approaching a Proof)

This is the specific (per-unit-mass) mechanical energy of a body in an inverse-square gravitational field. Start from the total mechanical energy of a mass $m$ moving at speed $v$ at distance $r$ from a primary of gravitational parameter $\mu = GM$:

$$E = \underbrace{\tfrac12 m v^2}_{\text{kinetic}} + \underbrace{\left(-\frac{\mu m}{r}\right)}_{\text{potential}}.$$

The gravitational potential energy $-\mu m/r$ comes from integrating the gravitational force (Gravitational Force) done in bringing the mass from infinity to radius $r$:

$$U(r) = -\int_\infty^r \left(-\frac{\mu m}{r'^2}\right)dr' = -\frac{\mu m}{r},$$

negative because gravity does positive work pulling the mass inward, and zero at infinity by convention. Dividing the total energy by the mass $m$ gives the specific energy:

$$\varepsilon = \frac{E}{m} = \frac{v^2}{2} - \frac{\mu}{r}. \qquad\blacksquare$$

Connecting to the semi-major axis. Because both energy and angular momentum are conserved on the orbit, $\varepsilon$ can be evaluated once and holds everywhere. Evaluate at perigee/apogee (velocity purely tangential) and eliminate the speed with angular-momentum conservation; the result is the compact identity

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

Equating this to $v^2/2 - \mu/r$ and solving for $v$ reproduces the vis-viva equation — so this page and vis-viva are two faces of one conservation law: one solved for energy, the other for speed.

Dimensional check. $$\frac{v^2}{2} - \frac{\mu}{r} = \frac{\text{m}^2}{\text{s}^2} - \frac{\text{m}^3/\text{s}^2}{\text{m}} = \frac{\text{m}^2}{\text{s}^2} - \frac{\text{m}^2}{\text{s}^2} = \frac{\text{m}^2}{\text{s}^2} = \frac{\text{J}}{\text{kg}}. \checkmark$$ Both terms are energy per unit mass, and $\text{J/kg} = \text{m}^2/\text{s}^2$.

History and Development

• Vis viva and the conservation of energy. The term traces to Leibniz (1680s), who argued $mv^2$ (his vis viva, "living force") was conserved in mechanical processes — an early, partial glimpse of energy conservation, formalised only in the 19th century. Orbital mechanics kept the name for the conserved energy-like quantity.

• Newton and the potential. The $-\mu/r$ potential is the integral of Newton's inverse-square gravitation (Principia, 1687); the recognition that total energy governs whether an orbit is bound came with the 18th–19th century development of analytical mechanics (Euler, Lagrange, Hamilton).

• Escape and the space age. The $\varepsilon = 0$ threshold is the mathematical statement of escape velocity, the number Tsiolkovsky, Goddard, and the mid-20th-century rocket pioneers had to beat. Specific energy became the natural bookkeeping for spaceflight because the "$-\mu/2a$" identity turns every trajectory design into simple energy arithmetic.

Related Concepts: Orbital Velocity, Escape Velocity, Circular Orbit Velocity, Orbital Period, Gravitational Force, Potential Energy, Kinetic Energy

Notes: Specific energy = per unit mass (J/kg = m²/s²); body mass cancels. Sign classifies the orbit: $<0$ ellipse, $=0$ parabola/escape, $>0$ hyperbola. Value fixes size: $\varepsilon=-\mu/2a$. Same conservation law as the vis-viva equation (this solved for energy, that for speed). $\mu=GM$ mislabelled dimensionless (is m³/s²). Potential negative (zero at infinity).