In an elliptical orbit, the part of velocity that’s perpendicular to the radius (called transverse velocity) changes due to two main reasons:
1. Conservation of Angular Momentum
Angular momentum stays constant in a stable orbit. Since angular momentum depends on both distance (r) and perpendicular velocity (v⊥), when the distance changes, the velocity must adjust to keep it balanced.
- Closer to the planet (smaller r) → Perpendicular velocity increases.
- Farther from the planet (larger r) → Perpendicular velocity decreases.
2. Gravity’s Changing Effect
Gravity pulls directly toward the planet, but in an elliptical orbit, the changing distance affects speed:
- At closest approach (periapsis) → Fastest perpendicular speed.
- At farthest point (apoapsis) → Slowest perpendicular speed.
Energy Plays a Role Too
As the orbiting object moves:
- Toward the planet → Gains speed (kinetic energy increases).
- Away from the planet → Loses speed (kinetic energy decreases).
Summary
The perpendicular velocity changes to keep angular momentum constant while balancing energy. This is why elliptical orbits have varying speeds, unlike circular orbits where speed stays the same.
In an elliptical orbit, the velocity perpendicular to the radius (often called the transverse or tangential velocity) changes primarily due to the gravitational force acting as a central force directed towards the focus (usually occupied by a massive body like a planet or star).
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Tangential Velocity Variations in Elliptical Orbits
The tangential velocity in an elliptical orbit changes due to the conservation of angular momentum. It is highest at periapsis and lowest at apoapsis, varying smoothly as the object moves along the orbit.
Tangential Velocity Graph
Conclusion
Here’s the corrected plot of tangential velocity versus true anomaly. The curve now shows a smooth and accurate transition from the highest velocity at periapsis (0°) to the lowest velocity at apoapsis (180°) without any numerical issues.
Tangential Velocity vs. Angle in an Elliptical Orbit
In an elliptical orbit, the tangential velocity (the part of velocity perpendicular to the radius) changes depending on the object's position (angle) in the orbit.
Key Points:
- At the closest point (periapsis, angle = 0°): Tangential velocity is fastest.
- At the farthest point (apoapsis, angle = 180°): Tangential velocity is slowest.
- At 90° and 270°: The speed is between the fastest and slowest points.
Why Does This Happen?
Because of conservation of angular momentum:
- When closer to the planet, the object speeds up to balance the stronger pull of gravity.
- When farther away, it slows down because gravity's pull weakens.
Simple Rule:
"Closer = Faster, Farther = Slower"
Tangential Velocity vs. True Anomaly in Elliptical Orbit
This curve shows how tangential velocity changes as an object moves through its elliptical orbit:
- Periapsis (0°): Maximum velocity (closest approach)
- Apoapsis (180°): Minimum velocity (farthest point)
- The transition is smooth and symmetrical
Kepler's Second Law: The Law of Equal Areas
Statement: A line connecting a planet to the Sun sweeps out equal areas in equal times.
[Image: Elliptical orbit with shaded areas showing equal area sectors]
(Visual: A planet moves faster when closer to the Sun, creating equal-area "slices" of its orbit over equal time intervals.)
What It Means:
- ☀️ Planets speed up when closer to the Sun (at perihelion)
- 🐢 They slow down when farther away (at aphelion)
- 📐 The "area swept per day/week/month" is always the same
Why It Happens:
This is a consequence of conservation of angular momentum. The Sun's gravity acts as a central force, causing this speed variation while maintaining balanced orbital areas.
Real-World Example:
Earth moves fastest in January (perihelion) and slowest in July (aphelion), but the area swept by the Earth-Sun line each month stays constant.
Key Takeaway: Kepler's Second Law proves that orbits aren't perfect circles, and orbital speed constantly adjusts to maintain this equal-area rule.
