Kepler's Laws and Newton's Universal Gravitation

Kepler’s Laws of Planetary Motion

• First Law: Planets move in elliptical orbits with the sun at one focus.
• Second Law: A line drawn between the sun and a planet sweeps out equal areas during equal intervals of time. This implies that the area swept per unit time ($L/2m$) is constant.
• Third Law: The square of a planet’s orbital period is proportional to the cube of the semimajor-axis length ($T^2 \propto a^3$).
• Circular Orbits: A circular orbit is considered a special case of an elliptical orbit.

Newton’s Law of Gravity

• Universal Attraction: Isaac Newton posited that gravity is a universal attractive force between all objects in the universe.
• Inverse-Square Force: Gravitational force magnitude is calculated as $F = \frac{Gm_1m_2}{r^2}$.
• Gravitational Constant ($G$): A universal constant valued at $G = 6.67 \times 10^{-11}\,N \cdot m^2/kg^2$. Gravity is a relatively weak long-range force.
• Principle of Equivalence: The assertion that gravitational mass ($m_{grav}$) is equal to inertial mass ($m_{inert}$).

Acceleration Due to Gravity ($g$)

• Surface Gravity: The free-fall acceleration on a planet's surface is $g_{surface} = \frac{GM}{R^2}$.
• Decrease with Altitude: At height $h$ above sea level, acceleration is $g = \frac{g_{earth}}{(1 + \frac{h}{R_e})^2}$, where $g_{earth} = 9.83\,m/s^2$ and $R_e = 6.37 \times 10^6\,m$.
• Weightlessness: Astronauts in the International Space Station are weightless because they are in a state of free fall, not because gravity is absent.

Gravitational Potential Energy ($U_G$)

• Definition: For two masses separated by distance $r$, $U_G = -\frac{Gm_1m_2}{r}$.
• Zero Point: Potential energy is defined as zero at infinity ($r = \infty$).
• Conservation of Energy: In an isolated system, the sum of kinetic and potential energy ($K + U_G$) remains constant as masses approach or recede.

Escape Speed and Orbital Mechanics

• Escape Speed ($v_{escape}$): The minimum speed required to leave a planet forever: $v_{escape} = \sqrt{\frac{2GM}{R}}$. For Earth, this is approximately $11,200\,m/s$.
• Circular Orbit Speed: The speed for a satellite in circular orbit is $v = \sqrt{\frac{GM}{r}}$. Speed is independent of the satellite's mass.
• Conservation of Angular Momentum: Because gravity exerts no torque, a satellite's angular momentum ($L$) is conserved, causing it to move faster when closer to the central body.
• Orbital Energetics: For circular orbits, the kinetic energy relates to potential energy as $K = -\frac{1}{2}U_G$, and total mechanical energy is $E_{mech} = \frac{1}{2}U_G$.

Questions & Discussion

• QuickCheck 13.1: The force of Planet Y on Planet X is equal in magnitude to the force of Planet X on Planet Y.
• QuickCheck 13.2: Doubling the distance between two objects reduces the gravitational force to one quarter; a $1,000,000\,N$ force becomes $250,000\,N$.
• QuickCheck 13.4: If Planet Y has twice the mass and twice the radius of Planet X, its surface gravity is half as strong as Planet X's ($4\,m/s^2$ vs $8\,m/s^2$).
• QuickCheck 13.7/13.8: Satellites in the same radius orbit at the same speed; satellites in smaller orbits travel faster than those in larger orbits.
• QuickCheck 13.9: In an elliptical orbit, the satellite's speed is higher when it is closer to the central mass ($v_A > v_B$).