3.2.2 · D5Orbital Mechanics & Astrodynamics
Question bank — Conservation of energy and angular momentum in gravitational field
True or false — justify
A satellite in an elliptical orbit moves at constant speed.
False. Only is constant. As shrinks toward perihelion, drops so (and speed) rises; speed is maximum at perihelion, minimum at aphelion.
Angular momentum is conserved because gravity is weak.
False. It has nothing to do with strength. is conserved because gravity is central (), making the torque .
Total mechanical energy of a bound orbit is negative.
True. A bound orbit has (since ). The negative sign means the body sits in a potential well and lacks the energy to reach infinity.
Two orbits with the same semi-major axis but different eccentricities have different total energies.
False. depends on alone; eccentricity never appears. Same ⇒ same , no matter how squashed the ellipse.
The orbit is always confined to a single flat plane.
True. Since is a fixed vector, both and stay perpendicular to it forever, so the motion cannot leave the plane through the centre perpendicular to .
Kepler's 2nd law (equal areas in equal times) is a separate law you must add on top of energy and momentum conservation.
False. It is angular momentum conservation: , so a constant directly forces equal areas in equal times.
If you double the speed at a given radius, the total energy doubles.
False. ; doubling quadruples the term while is unchanged, so does not simply double.
An orbit with is a circle.
False. means — a parabola (the borderline escape trajectory). A circle is a bound orbit with .
Along the whole orbit, is conserved.
False. trades continuously with . Only their sum is conserved; near perihelion is large, near aphelion it is small.
Spot the error
"As a planet falls toward the Sun, becomes more negative, so energy is being lost."
Error: energy is only redistributed, not lost. As drops, rises by exactly the same amount, keeping constant.
"The planet speeds up near the Sun, so its angular momentum increases there."
Error: is constant. Near the Sun is small, so grows to compensate — the tangential rate changes but stays fixed.
"A more eccentric orbit is 'wilder', so it must carry more energy."
Error: energy depends on , not . A near-circular and a highly elongated orbit with the same have identical energy.
"To find escape speed, set ."
Error: escape corresponds to , i.e. , giving . Setting describes a body at rest, which is the opposite situation.
" — both terms are attractive."
Error: the term is a repulsive centrifugal barrier (positive, rising as ); only is truly attractive. Their opposition creates the potential well minimum.
"At perihelion the whole speed is radial, so there."
Error: at a turning point , so the whole speed is tangential: . That is precisely why turning points are the easy place to use .
"Vis-viva says ."
Error: the correct form is — the and are swapped in the wrong version, which would even give wrong signs.
Why questions
Why does a central force guarantee a flat (planar) orbit?
Because is conserved and constant in direction; and must always lie perpendicular to that fixed , which pins the whole motion into one plane.
Why is potential energy defined with a minus sign and referenced to infinity?
We set and define , the work done against gravity to bring the mass in. Since gravity pulls inward, this gives , a well you must climb out of.
Why does the vis-viva equation contain both and but not or explicitly?
Because it comes from combining with ; energy already hides and inside , so eliminating leaves only (where you are) and (the orbit's size).
Why is energy conserved for gravity but not, say, for a body sliding with friction?
Gravity is conservative — its work depends only on start and end positions, so a potential with exists. Friction's work depends on the path length, so no such potential exists and mechanical energy leaks away as heat.
Why does the effective potential have a minimum, and what does that minimum represent?
The repulsive centrifugal term dominates at small and the attractive dominates at large ; their tug-of-war produces a dip. The bottom of the dip is the radius of a stable circular orbit for that .
Why can you determine the size of an orbit () from energy alone, without knowing the shape?
Because links energy and one-to-one; the shape (eccentricity) is set separately by . So fixes size, fixes shape.
Edge cases
What is the total energy and semi-major axis for a body launched at exactly escape speed?
exactly, and formally (from ). The path is a parabola — it just barely reaches infinity, arriving with zero speed.
What happens to the "orbit" if ?
With the centrifugal barrier vanishes and there is no tangential motion; the body falls straight along a radial line into (or straight out from) the centre — a degenerate, purely radial "orbit."
For a hyperbolic flyby, what are the signs of and ?
(unbound, more than enough energy to escape) and , since can only be positive if is negative — a formal bookkeeping value, not a physical length.
At the turning points of an ellipse, what fraction of the speed is radial?
Zero — at perihelion and aphelion , so the velocity is entirely tangential, . This is the extremal radius, where momentarily stops changing.
As on a bound orbit, does the body ever actually reach infinity?
No. A bound orbit has , so there is a finite maximum radius (aphelion) where would have to go negative to go further — impossible. The body turns back.
In the limit , what does the ellipse become, and what stays true?
It becomes a circle with everywhere; the speed becomes constant, and vis-viva collapses to . Even here, and still hold.
If a rocket burn changes only the direction of velocity (not speed) at some point, does the total energy change?
No — depends on speed and radius , not on direction. Redirecting velocity at fixed and fixed leaves unchanged but generally alters and thus the orbit's shape.
Recall Self-test protocol
Go back through every item and, for the ones you missed, write the governing equation (, , vis-viva, or ) beside your wrong reasoning. Ninety percent of these traps dissolve the instant you name which conservation law is doing the work.