Motion of Orbiting Satellites
Orbital motion is governed by gravity together with conservation of energy and angular momentum. For a satellite of negligible mass m orbiting a much more massive body M, a circular orbit satisfies GMm/r²=mv²/r, so v=√(GM/r), K=GMm/(2r), U=-GMm/r, and E=-GMm/(2r). Elliptical orbits conserve total mechanical energy and angular momentum while K and U vary; escape occurs when total mechanical energy reaches zero. A higher circular orbit has lower circular speed but greater (less negative) total energy, so raising an orbit requires net energy input rather than simply 'slowing down.'
Topic sequence · 6 of 6
Learning objectives
Objective 1
Derive circular orbital speed, kinetic energy, total energy, and escape speed from gravitational and conservation models.
Objective 2
Compare which quantities remain constant in circular versus elliptical orbits.
Objective 3
Use gravitational potential energy U=-GMm/r and angular momentum consistently instead of near-surface approximations.
- Topic 2.6 gravitational force and field.
- Topic 2.10 circular motion.
- Topic 3.3 gravitational potential energy.
- Topics 3.4 and 6.4 conservation of energy and angular momentum.
Definitions first
Representations you must read and create
Representation 1
Energy-versus-radius curves showing U=-GMm/r and E_circ=U/2.
Representation 2
Velocity vectors tangent to a circular orbit while gravitational force points radially inward.
Representation 3
Elliptical-orbit sketches marking faster motion near periapsis and slower motion near apoapsis while angular momentum remains constant.
Representation 4
Tables comparing circular, elliptical, and escape-energy conditions.
Core relationships and calculus
Variables: G N·m²/kg²; M,m kg; r m
Use: spherically symmetric masses or point-mass model
Variables: v m/s
Use: circular orbit with m≪M
Variables: U J
Use: zero potential energy chosen at infinite separation
Variables: K J
Use: circular orbit
Variables: E J
Use: total mechanical energy of circular two-body system
Variables: L kg·m²/s
Use: particle angular momentum about central body; v_t tangential component
Variables: v m/s
Use: escape with final speed approaching zero at infinity
For any central gravitational orbit, τ about the central body is zero because r⃗ and F⃗ are parallel, so dL⃗/dt=0. Energy conservation gives ½mv²-GMm/r=E. These two invariants together explain why an elliptical satellite moves faster at smaller r. Circular-orbit formulas are obtained by additionally imposing constant r and radial acceleration v²/r; do not use E=-GMm/(2r) with an arbitrary instantaneous radius in an elliptical orbit.
Start from Newton’s gravitation and circular dynamics to derive v_circ. Substitute that speed into K and combine with U to derive E_circ. For escape, set the total energy at launch equal to the limiting energy at infinity, where U→0 and the minimum escaping case has v→0. The derivations show which formulas are circular-orbit results and which, such as U=-GMm/r and conservation of E and L, are broader.
Use the two-body model in which the satellite mass is negligible compared with the central body unless a problem explicitly requires center-of-mass motion. The course supports exact circular-orbit calculations and conservation-law reasoning for elliptical orbits. Keep E=-GMm/(2r) tied to a circular orbit of radius r; for an ellipse the instantaneous radius changes while total energy stays fixed. A brief thrust is an external interaction that changes the mechanical energy and generally the angular momentum of the selected gravitational system.
Physical examples
Example 1
A lower circular orbit has greater speed and more negative total energy than a higher circular orbit.
Example 2
A satellite in an elliptical orbit speeds up as it falls inward because U decreases while K increases at fixed total energy.
Example 3
A tangential burn changes both orbital energy and angular momentum and therefore changes the orbit rather than simply moving the satellite to a new circular radius instantly.
Example 4
Escape speed is √2 times circular orbital speed at the same radius.
Example 5
Moving from a lower circular orbit to a higher circular orbit requires an increase in total mechanical energy even though the final circular speed is smaller; an orbital transfer therefore cannot be understood by comparing final speeds alone.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
For a circular orbit, derive speed from gravity supplying the required radial acceleration before using energy relations.
Reasoning pattern 2
Compare lower and higher circular orbits using both speed and total energy: lower orbit → faster circular speed but more negative total energy.
Reasoning pattern 3
For an ellipse, use conservation of total mechanical energy and angular momentum rather than substituting the instantaneous r into the circular-orbit energy formula.
Reasoning pattern 4
For escape, set the minimum escaping state to E=0 with v→0 as r→∞, then derive v_esc from the launch radius.
Reasoning pattern 5
After an impulse or thrust, distinguish the immediate post-burn state from a later circularized orbit; energy and angular momentum changes determine the new trajectory.
Deeper calculus and limiting-case check
For an ellipse with semimajor axis a, the broader two-body result E=-GMm/(2a) explains why total energy is constant even though r varies. This is a useful extension, but the core page keeps circular-orbit formulas tied to circular radius.
Model-selection workflow: decide what the problem is before choosing an equation
For Motion of Orbiting Satellites, the fastest reliable route is not equation hunting. Start by identifying the rotating system and constraint, encode it with the energy/angular-momentum representation, choose the work-energy or angular-impulse equation, and then test the rolling/orbit/isolation assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat satellite and circular orbit as physical quantities first and symbols second.
Choose a representation
Build Energy-versus-radius curves showing U=-GMm/r and E_circ=U/2.. If the prompt supplies Velocity vectors tangent to a circular orbit while gravitational force points radially inward., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is F_g = GMm/r². State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Use the two-body model in which the satellite mass is negligible compared with the central body unless a problem explicitly requires center-of-mass motion. The course supports exact circular-orbit calculations and conservation-law reasoning for elliptical orbits. Keep E=-GMm/(2r) tied to a circular orbit of radius r; for an ellipse the instantaneous radius changes while total energy stays fixed. A brief thrust is an external interaction that changes the mechanical energy and generally the angular momentum of the selected gravitational system. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Energy-versus-radius curves showing U=-GMm/r and E_circ=U/2. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | F_g = GMm/r² | spherically symmetric masses or point-mass model. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | v_circ = √(GM/r) | circular orbit with m≪M. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | For any central gravitational orbit, τ about the central body is zero because r⃗ and F⃗ are parallel, so dL⃗/dt=0. Energy conservation gives ½mv²-GMm/r=E. These two invariants together explain why an elliptical satellite moves faster at smaller r. Circular-orbit formulas are obtained by additionally imposing constant r and radial acceleration v²/r; do not use E=-GMm/(2r) with an arbitrary instantaneous radius in an elliptical orbit. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A satellite of mass m moves in a circular orbit of radius r around a much more massive body M. Set gravity equal to the required radial force: GMm/r²=mv²/r, giving v²=GM/r. Then K=½mv²=GMm/(2r). With U=-GMm/r, total energy is E=K+U=-GMm/(2r). The negative sign means the orbit is bound. For escape from the same radius, require ½mv_esc²-GMm/r=0, so v_esc=√(2GM/r)=√2 v_circ. Notice that satellite mass cancels from both speed formulas. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 6.6
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship F_g = GMm/r² should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Energy-versus-radius curves showing U=-GMm/r and E_circ=U/2. and Velocity vectors tangent to a circular orbit while gravitational force points radially inward.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use an orbital simulation or numerical integration of d²r⃗/dt²=-GM r⃗/r³. Launch the same test mass with several tangential speeds at one radius, record r(t), speed, E and L, and classify bound elliptical, circular, and escape trajectories. Verify that E and L remain constant to numerical uncertainty when only gravity acts.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Using mgh for orbital-scale gravitational potential energy. | Return to the energy/angular-momentum representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using E=-GMm/(2r) at an arbitrary point of an elliptical orbit; that expression is the circular-orbit result at radius r. | Return to the work-energy or angular-impulse equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming speed is constant in every orbit; it is constant only for a circular orbit in this model. | Return to the energy/angular-momentum representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Treating escape speed as the speed of a circular orbit; it is √2 larger at the same radius. | Return to the work-energy or angular-impulse equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Forgetting that a brief external thrust changes the satellite system’s energy and angular momentum. | Return to the energy/angular-momentum representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Concluding that a higher circular orbit must have lower total energy because its circular speed is lower. The higher orbit actually has greater (less negative) total mechanical energy. | Return to the work-energy or angular-impulse equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Treating an instantaneous tangential burn as if it places the satellite immediately into a new circular orbit; after the burn the state generally belongs to a transfer orbit unless the new velocity is exactly the circular value at that radius. | Return to the energy/angular-momentum representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 2.10 supplies circular-motion dynamics used to derive v_circ. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 3.3 supplies the general gravitational potential U=-GMm/r. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 6.4 explains why angular momentum stays constant under a central gravitational force. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write F_g = GMm/r² and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 6.6 mastery ladder
Interactive circular-orbit and escape check
Enter central mass and orbital radius. The tool compares circular speed, escape speed, and specific orbital energies for the ideal m≪M model.
Experimental connection
Use an orbital simulation or numerical integration of d²r⃗/dt²=-GM r⃗/r³. Launch the same test mass with several tangential speeds at one radius, record r(t), speed, E and L, and classify bound elliptical, circular, and escape trajectories. Verify that E and L remain constant to numerical uncertainty when only gravity acts.
Connections across Physics C
Connection 1
Topic 2.10 supplies circular-motion dynamics used to derive v_circ.
Connection 2
Topic 3.3 supplies the general gravitational potential U=-GMm/r.
Connection 3
Topic 6.4 explains why angular momentum stays constant under a central gravitational force.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Using mgh for orbital-scale gravitational potential energy.
Avoid this
Using E=-GMm/(2r) at an arbitrary point of an elliptical orbit; that expression is the circular-orbit result at radius r.
Avoid this
Assuming speed is constant in every orbit; it is constant only for a circular orbit in this model.
Avoid this
Treating escape speed as the speed of a circular orbit; it is √2 larger at the same radius.
Avoid this
Forgetting that a brief external thrust changes the satellite system’s energy and angular momentum.
Avoid this
Concluding that a higher circular orbit must have lower total energy because its circular speed is lower. The higher orbit actually has greater (less negative) total mechanical energy.
Avoid this
Treating an instantaneous tangential burn as if it places the satellite immediately into a new circular orbit; after the burn the state generally belongs to a transfer orbit unless the new velocity is exactly the circular value at that radius.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Why is orbital energy negative?
With U=0 at infinity, a bound orbit has insufficient total energy to separate the bodies to infinity, so E<0.
Does a satellite need thrust to stay in circular orbit?
Not in the ideal model. Gravity continuously bends the velocity vector and supplies the centripetal acceleration.
Why does a satellite move faster near a planet in an elliptical orbit?
Total energy and angular momentum are conserved; decreasing r makes U more negative and K larger.
Is escape speed independent of satellite mass?
Yes in the m≪M model because the satellite mass cancels from the energy equation.