SCIENCE · INVESTIGATE AND EXPLAIN

Use angular momentum as well as energy to reconstruct a trajectory

In an isolated inverse-square point-mass field, specific orbital energy ε=v²/2−μ/r and specific angular momentum h=r v_t remain constant. Specific means per unit orbiting mass. Negative energy corresponds to a bound ellipse, zero to the parabolic limit and positive to an unbound hyperbola, but energy alone does not determine the ellipse shape or closest approach.

Learning outcome: Hold speed fixed while changing the radial/tangential split. Reconstruct energy, angular momentum, eccentricity and turning radii; verify both energy and angular momentum at an apsis and explain what a missing finite apoapsis means.

Starting recap

Retain relative versus barycentric geometry, time versus angle, fitted versus instantaneous velocity and complete energy/angular-momentum boundaries.

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Try

At v=1 and θ=30°, ε=−0.5, h=0.5 and e=√0.75≈0.866025. The semimajor axis is 1, periapsis≈0.133975 and apoapsis≈1.866025. At the same speed with θ=90°, h=1 and e=0: the orbit is circular at radius 1.

Open the experiment

Predict

Can two trajectories with the same starting speed and radius have different closest approaches?

Your prediction and reason

Experiment

Hold speed fixed while changing the radial/tangential split. Reconstruct energy, angular momentum, eccentricity and turning radii; verify both energy and angular momentum at an apsis and explain what a missing finite apoapsis means.

Use energy and angular momentum together to classify a trajectory

Worked teaching model · synthetic values · Original point-mass μ=1, initial radius 1. Specific energy ε=v²/2−1; h=v sin θ; e=√(1+2εh²); p=h² and periapsis p/(1+e). Negative energy has a finite apoapsis; positive energy does not. No collision radius, atmosphere or actual launch calculation.

Speed at radius 1: 1 normalized speed Velocity angle from outward radial direction: 30 degrees

Reset keeps your prediction, captures and practice history.

Energy determines boundness, while angular momentum helps determine the turning radii.

The speed alone does not fix the orbit shape: changing the radial/tangential split changes angular momentum at unchanged energy. This point-mass model has no physical surface; a small mathematical periapsis is not evidence an actual body could pass safely.

Orbit class: Bound ellipse (including circular special case); finite apoapsis.

Original conditional quantities and explicit units
Quantity	Value	Unit
Specific orbital energy	-0.5000000	length²/time²
Specific angular momentum	0.5000000	length²/time
Eccentricity	0.8660254	ratio
Semilatus rectum	0.2500000	length
Periapsis radius	0.1339746	length
Initial radial speed	0.8660254	length/time
Semimajor axis	1	length
Apoapsis radius	1.866025	length

These original models are not measured orbit records or operational mission instructions. Retain the frame, assumptions and missing evidence before interpreting a real system.

Energy determines boundness, while angular momentum helps determine the turning radii.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the orbital evidence

2. Connect geometry, force and conservation

The original record fixes μ=1 and starting radius 1. Speed v and its angle θ from the outward radial direction set h=v sin θ and initial radial speed v cos θ. The eccentricity is √(1+2εh²), semilatus rectum is h² and periapsis is h²/(1+e). For negative energy, semimajor axis is −1/(2ε) and apoapsis is h²/(1−e). Positive-energy records have no finite apoapsis, so the ledger omits that quantity instead of reporting a fake number.

Changing the direction at fixed speed changes angular momentum without changing energy. Two trajectories can therefore have the same semimajor axis but different eccentricities and turning radii. A tangent velocity at the starting radius is an apsis condition, but it can be periapsis or apoapsis depending on whether the speed is above or below the local circular speed.

The model treats the attractor as a point and omits collisions, atmospheres and additional gravitating bodies. A small mathematical periapsis is not a safe physical passage. The controls deliberately avoid the exactly parabolic speed and purely radial singular case; the general classification is explained without pretending the displayed grid exhausts every possible orbit.

3. Work the original case

4. Predict and investigate

Can two trajectories with the same starting speed and radius have different closest approaches?

Apply the evidence to a changed case

At v=0.5 and θ=90°, calculate energy, h, eccentricity and both turning radii. Is the starting radius periapsis or apoapsis?

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Energy is −0.875, h=0.5 and e=0.75. Periapsis is 1/7 and apoapsis 1; the starting tangent point is apoapsis because its speed is below the local circular value 1. Semimajor axis is 4/7.

Source and boundaries

Primary mechanism reference. Checked 30 October 2026. Original prose, numerical cases and diagrams; source context is not a claim that these fictional records were observed. No actual mission operation or instrument aiming is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

At v=0.5 and θ=90°, calculate energy, h, eccentricity and both turning radii. Is the starting radius periapsis or apoapsis?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Separate a radial-velocity amplitude from an instantaneous sample Next: Distinguish reaching a radius from joining its circular orbit

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