7. Kepler's Laws

Main ideas & concepts

1) Recap: Energy conservation and when it works

• The Law of Conservation of Energy can be expressed as:

  • 1D form: [ K_1 + U_1 = K_2 + U_2 ] (when certain conditions hold)

• Starting from the Work–Energy Theorem (Newton’s laws):

  • Change in kinetic energy equals the work done: [ K_2 - K_1 = \int_{x_1}^{x_2} F(x)\,dx ]

• Key issue:

  • In higher dimensions, work generally depends on the path, so you don’t automatically get a conserved energy law.

2) Path dependence vs path independence

• Work in higher dimensions is written as: [ \int \mathbf{F}\cdot d\mathbf{r}=\int (F_x\,dx+F_y\,dy) ]

• Even if endpoints are fixed ((\mathbf{r}_1\to \mathbf{r}_2)):

  • If the force’s work depends on the path taken, then you cannot simplify work into a potential-energy difference.
• Core idea:
  • A conserved energy law requires that (\int \mathbf{F}\cdot d\mathbf{r}) be path independent.

3) Conservative forces and potentials (the “only if” and “recipe”)

• Sufficient idea: If a force comes from a potential (U(x,y)), defined by: [ F_x=-\frac{\partial U}{\partial x},\qquad F_y=-\frac{\partial U}{\partial y}, ] then work becomes path independent and energy is conserved.

• Claim (framed as necessity):

  • If a force’s work is path independent, it must be derivable from some potential (U) in this manner.
• Recipe to test conservativeness (2D):
  • Compute:
    • (\dfrac{\partial F_x}{\partial y})
    • (\dfrac{\partial F_y}{\partial x})
  • If they are equal, the force is conservative (path independent).

4) Kepler’s laws (as empirical summaries)

Kepler’s three laws of planetary motion:

1. Ellipses: Planets orbit the Sun on elliptical orbits with the Sun at a focus.

2. Equal areas in equal times: The line from planet to Sun sweeps out a constant area rate: [ \frac{dA}{dt}=\text{constant} ]

3. Harmonic relation: The ratio [ \frac{T^2}{r^3} ] is the same for all planets (where (T) is orbital period and (r) is orbit size, e.g., semi-major axis).

Additional notes from discussion:

• Kepler’s laws are not exact in real observations due to:
  • gravitational effects of other planets (e.g., Jupiter),
  • and relativistic corrections (Mercury’s perihelion precession).

5) Newton’s step beyond Kepler

• Newton’s goal: explain Kepler using Newton’s laws, especially:
  • that the force is related to acceleration ((\mathbf{F}=m\mathbf{a})).
• Gravity is motivated by comparing:
  • the Moon’s centripetal acceleration toward Earth
  • to apples accelerating toward Earth → proposing a single force: universal gravitation.

Newton’s universal law of gravitation

• Magnitude for two masses (m) and (M) separated by distance (r): [ F=G\frac{mM}{r^2} ]

• Direction:

  • the force points along the line joining the masses (attractive).

• In the “planet around Sun” setup (Sun at center, planet tiny): [ \mathbf{F}=-G\frac{Mm}{r^2}\,\hat{\mathbf{r}} ]

• Emphasis:

  • It’s a “tremendous leap of faith” that the same laws apply from Earth to the planets/universe.

6) Deriving Kepler’s third law from circular motion

To connect Newton to Kepler, the lecturer first considers circular orbits:

• For a circular orbit of radius (r) and speed (v):

  • centripetal acceleration magnitude: (a=v^2/r)
  • so: [ m\left(\frac{v^2}{r}\right)=G\frac{Mm}{r^2} ]

• Cancel (m) and simplify: [ v^2=\frac{GM}{r} ]

• Use (v=\dfrac{2\pi r}{T}): [ \frac{T^2}{r^3}=\frac{4\pi^2}{GM} ]

• This reproduces Kepler’s third law, identifying the constant in terms of:

  • (G) and (M) (mass of the Sun).

7) Applying orbital formula to real situations (example: geosynchronous satellites)

Using the (T^2/r^3) relation:

• For geosynchronous satellites:
  • orbital period (T=24) hours
• Steps described conceptually:
  • Choose (T) (24 hours) → solve for the required orbital radius (r)
  • then use the orbital condition to find orbital speed/velocity (via the Newton/Kepler relation)
• Satellites must stay in the correct orbit; otherwise the “reflection/coverage” use case fails (communication context).

8) Gravitational potential energy and energy conservation globally

Potential energy near Earth (approximation)

• Near Earth, gravitational force is approximately: [ \mathbf{F}\approx -mg\,\hat{\mathbf{y}} ]

• Potential energy (with chosen reference): [ U=mgh ]

• Total energy near Earth: [ \frac12 mv^2+mgh=\text{constant} ]

Potential energy far from Earth (inverse-square gravity)

• For gravity at distance (r) from the center of mass (M):
  • force behaves as (\sim -GMm/r^2) toward the center

• Potential energy is: [ U(r)=-\frac{GMm}{r} ] (with convention (U(\infty)=0))

• Total energy for an orbiting body: [ E=\frac12 mv^2-\frac{GMm}{r}=\text{constant} ]

Reconciling sign differences

• Resolution:
  • Potential energy is defined up to an additive constant.
• Near Earth, the reference point is often chosen so that (U=0) at Earth’s surface (or at a height convention).
• In celestial mechanics, it’s common to choose (U(\infty)=0).
• Shifting the zero level changes numerical values/signs, but energy differences and conservation remain consistent.

9) Bound vs unbound motion (escape velocity)

• If total energy is:
  • negative → object is bound (cannot escape to infinity)
  • zero → borderline case
  • positive → object is unbound (can reach infinity)
• At infinity:
  • (U(\infty)=0)
  • if total energy were negative, the object would require negative kinetic energy to get there, which is impossible.

• Escape velocity (energy method idea):

  • set total energy at infinity to zero: [ \frac12 mv^2-\frac{GMm}{R_E}=0 ]

  • yielding: [ v^2=\frac{GM}{R_E} ]

• Takeaway phrasing:

  • Fire a projectile at escape velocity (or slightly above) to ensure it does not return.

10) Dark matter (application of gravity/Kepler/Newton reasoning)

• Evidence described:
  • Visible matter isn’t enough to explain observed galactic rotation curves.

• Using gravitational/orbital logic:

  • treat enclosed mass as determining orbital speed: [ v^2 r \ \text{estimates enclosed mass} ]

  • observations show (v^2 r) keeps increasing with radius

  • but visible matter doesn’t increase accordingly
  • Conclusion:
  • additional unseen mass is needed → dark matter halos.

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Methodology / instruction-style bullet points

A) How to test whether a force is conservative (path independent)

Given a 2D force field (\mathbf{F}=(F_x(x,y),F_y(x,y))):

• Compute:
  • (\dfrac{\partial F_x}{\partial y})
  • (\dfrac{\partial F_y}{\partial x})
• If: [ \frac{\partial F_x}{\partial y}=\frac{\partial F_y}{\partial x}, ] then the force is conservative, so work depends only on endpoints and a potential (U) exists.

B) How to get the gravitational potential energy for inverse-square gravity

• Assume spherical inverse-square form:
  • (F(r)\propto -1/r^2)
• Use:
  • (\mathbf{F}=-\nabla U)
• With the convention (U(\infty)=0), the standard result is: [ U(r)=-\frac{GMm}{r}. ]

C) How to apply Kepler/Newton to geosynchronous satellites (as described)

• Set desired orbital period:
  • (T=24) hours

• Use: [ \frac{T^2}{r^3}=\text{constant}=\frac{4\pi^2}{GM} ]

• Solve for orbital radius (r)

• Then determine orbital speed/velocity consistent with circular orbit constraints.

D) How to determine escape velocity (energy method)

• Use energy relative to infinity:
  • require threshold escape condition (E=0)
• With:
  • (U(\infty)=0)

• Solve: [ \frac12 mv^2-\frac{GMm}{R_E}=0 ]

• Result: [ v_{\text{escape}}=\sqrt{\frac{GM}{R_E}}. ]

• If you fire slightly faster, you retain positive total energy and escape.

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Speakers / sources featured

• Professor Ramamurti Shankar (main lecturer)
• Students / audience questions (unidentified individuals; multiple brief questions)
• Historical figures referenced as context:
  • Copernicus, Tycho Brahe, Johannes Kepler, Isaac Newton, Edmond Halley
  • Ray Davis (solar neutrino example)
  • Henry Balmer (spectral lines)
  • Niels Bohr
• Conceptual scientific references referenced:
  • Einstein’s general relativity
  • Theory of quarks
  • Dark matter (astronomy/cosmology context)