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Orbital Velocity Calculator: Circular and Elliptical Vis-Viva, Escape Velocity and Period

Most orbital velocity tools only handle the simple circular case, but a real satellite orbit is usually elliptical, moving fastest at perigee and slowest at apogee and a plain circular formula can’t capture that. This vis-viva calculator does both. It computes circular orbital velocity around Earth, the Moon, Mars, or the Sun with gravitational parameters pre-filled or any custom body via a directly editable μ and its elliptical mode takes your apogee and perigee altitudes and returns the true velocity at both points using the full vis-viva equation. Every result comes with a built-in escape velocity calculator reading alongside for direct comparison and an orbital period calculator using Kepler’s third law, shown in minutes, hours or days as appropriate, plus a geostationary quick-reference for a familiar earth benchmark.

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How to Use

This tool has two tabs: Circular Orbit and Elliptical (Vis-Viva).

Step 1: Setting up your central body (shared across both modes)

• Select a Body from the dropdown, Earth, Moon, Mars, and the Sun are included, to pre-fill the Gravitational Parameter μ and Body Radius. Both fields always stay fully editable, so you can select Custom and type any μ for another planet, moon or hypothetical body entirely.

Step 2: Using Circular Orbit mode

• Enter your orbit Value and select whether it should be Interpreted as an Altitude above surface or an absolute Radius from centre.
• Tap Calculate Circular Orbit. The result shows orbital velocity, escape velocity at that same radius for comparison, and the orbital period.
• Alternatively, tap Load Geostationary (Earth) to instantly load the standard ~35,786 km geostationary altitude and see its familiar orbital velocity and 24-hour-class period.

Step 3: Using Elliptical (Vis-Viva) mode

• Switch to the Elliptical (Vis-Viva) tab. Enter your Perigee Altitude and Apogee Altitude, both measured above the body’s surface.
• Tap Calculate Elliptical Orbit. Instead of assuming a circular path, tool applies the full vis-viva equation to give you the true velocity at both perigee, where the satellite moves fastest and apogee, where it moves slowest.

Step 4: Exporting your result

• Use Print / PDF for a clean printable copy or Copy to paste the figures elsewhere.

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Key Features

• Circular orbital velocity for Earth, Moon, Mars, or the Sun, with a directly editable μ for any custom body
• Full vis-viva elliptical mode, giving true velocity at both perigee and apogee, not just an averaged circular estimate
• Escape velocity shown alongside every result for direct comparison at the same radius
• Orbital period from Kepler’s third law, displayed in minutes, hours or days depending on scale
• Geostationary quick-reference button, instantly loading Earth’s ~35,786 km GEO altitude
• Accepts orbit position as either altitude above the surface or absolute radius from the centre
• Export as PDF or copy the result.

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Formula / Logic Used

Circular Orbital Velocity

$$v = \sqrt{\frac{\mu}{r}}$$

Where $\mu$ is the central body’s gravitational parameter and $r$ is the orbital radius, measured from the body’s centre.

Vis-Viva Equation (Elliptical Orbit)

$$v = \sqrt{\mu\left(\frac{2}{r} – \frac{1}{a}\right)}, \qquad a = \frac{r_{apo} + r_{peri}}{2}$$

Evaluating this at $r = r_{peri}$​ gives the fastest velocity in the orbit and at $r = r_{apo}$​ gives the slowest, since $a$, the semi-major axis, stays fixed for the whole ellipse.

Escape Velocity

$$v_{esc} = \sqrt{\frac{2\mu}{r}}$$

Escape velocity at any given radius is always √2 times the circular orbital velocity at that same radius.

Orbital Period (Kepler’s Third Law)

$$T = 2\pi\sqrt{\frac{r^3}{\mu}} \ \text{(circular)}, \qquad T = 2\pi\sqrt{\frac{a^3}{\mu}} \ \text{(elliptical)}$$

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Who Should Use This Tool

Aerospace engineering and physics students learning orbital mechanics, the vis-viva equation and Kepler’s laws for a spaceflight dynamics or astrodynamics course. Also useful for space enthusiasts and hobbyist satellite trackers wanting a quick, correct velocity and period estimate for a given orbit.

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Frequently Asked Questions (FAQs)

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Related Tools

• Rocket Equation Calculator – for the delta-v and propellant side of reaching this orbit
• Mach Number Calculator – for the atmospheric flight phase before reaching orbit
• Lift & Drag Coefficient Calculator – for the aerodynamic phase of an ascent trajectory

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