Conic Sections & Vis-Viva

Orbit Geometry and the Energy Equation

Aerospace EngineeringOrbital MechanicsFree preview

⏱️ About 30 min

Conic Sections & Vis-Viva — illustration

Decorative illustration.

How can a single, elegant equation tell you the exact speed of a satellite at any point along its orbit?

💡

The big idea: All gravitational orbits trace conic sections, and the vis-viva equation directly connects a satellite's speed to its current position and the orbit's semi-major axis.

🎯 By the end, you'll be able to

Classify orbits based on eccentricity (\(e < 1\), \(e = 1\), \(e > 1\)).
Apply the vis-viva equation to calculate orbital speed at any radius.
Calculate specific orbital energy using the semi-major axis.

📎 Helpful to know first

Completion of the Two-Body Problem lesson and knowledge of basic algebra.

Orbits as Conic Sections

The solutions to the two-body equation of motion are conic sections. The shape of the orbit is determined by its eccentricity \(e\). A circle has \(e = 0\), an ellipse has \(0 < e < 1\), a parabola has \(e = 1\), and a hyperbola has \(e > 1\).

Specific Orbital Energy

Specific orbital energy \(\varepsilon\) is the sum of kinetic and potential energy per unit mass. It is a constant for an orbit and is directly related to the semi-major axis \(a\).

\[ \varepsilon = -\dfrac{\mu}{2a} \]

The Vis-Viva Equation

By equating the specific orbital energy at any point in the orbit to its definition, we derive the vis-viva equation. This is one of the most useful formulas in orbital mechanics, allowing us to calculate the speed \(v\) at any distance \(r\) given the semi-major axis \(a\).

\[ v^2 = \mu\left(\dfrac{2}{r} - \dfrac{1}{a}\right) \]

🔑 Conservation of Energy

Vis-viva is essentially a statement of conservation of energy. As a satellite falls closer to the primary body (decreasing \(r\)), its velocity \(v\) must increase to keep the total orbital energy constant.

🎮 Vis-Viva Explorer LIVE

Predict first: Change the radius and semi-major axis to observe how orbital velocity varies.

Interactive visualization of the vis-viva equation across an orbit.

📝 Worked example: An elliptical orbit has a semi-major axis \(a = 10000\) km. What is the orbital speed when the satellite is at radius \(r = 8000\) km?

1. Identify parameters: \(\mu = 398600\) km\(^3\)/s\(^2\), \(a = 10000\) km, \(r = 8000\) km.
2. Plug into vis-viva: \(v = \sqrt{398600 \times (2/8000 - 1/10000)}\).
3. Simplify the bracket: \((2/8000 - 1/10000) = 0.00025 - 0.0001 = 0.00015 = 1.5 \times 10^{-4}\).
4. Calculate \(v = \sqrt{398600 \times 1.5 \times 10^{-4}} = \sqrt{59.79} = 7.732\) km/s.

✓ The orbital speed at that point is about 7.732 km/s.

✏️ Practice: Calculate the orbital speed for an elliptical orbit where \(a = 9000\) km and \(r = 7000\) km.

Solution

Hint: Plug the values directly into the vis-viva equation: v = sqrt(mu*(2/r - 1/a)).
Answer: v = 8.342 km/s.

Check your understanding

1. What eccentricity defines a parabolic orbit?

A parabolic orbit has an eccentricity exactly equal to 1, representing the escape trajectory boundary.

2. What happens to the vis-viva equation if the orbit is circular (\(r = a\))?

If \(r = a\), the term \((2/r - 1/a)\) simplifies to \(1/r\), giving \(v^2 = \mu/r\), which is the circular orbit speed.

✅ Key takeaways

Orbits are categorized by eccentricity into circles, ellipses, parabolas, and hyperbolas.
Specific orbital energy depends only on the semi-major axis.
The vis-viva equation allows computation of speed at any point in the orbit.

➡️ Now that we know how to calculate orbits, we need a standard way to describe their orientation and shape in 3D space using classical orbital elements.

Want to test yourself on this? Try the Aerospace Engineering Aptitude test →