Exam Notes on Ellipses, Kepler's Laws, and Habitable Zones

Ellipses

• Eccentricity:

  • Value between 0 (circle) and 1 (flat line).
  • 0: Perfect circle.
  • As it approaches 1, it becomes flatter.
  • Eccentricity must be between 0 and 1. If not, there's an error.

• Axis:

  • Major Axis: Longest axis of an ellipse, from furthest point to furthest point.
  • Minor Axis: Shortest axis of an ellipse.
  • Semimajor Axis: Half of the major axis.
    • Importance: Average distance between the orbiting body and the host star.
    • The orbiting body sometimes is further or closer than this average.

Kepler's Laws of Planetary Motion

• Deals with the motions of planets.

• First Law:

  • Each planet's orbit has a star at one of the foci.
  • The second focus is a point in space determined by gravity.

• Understanding Planetary Motion Requires Understanding:

  • Newton's Third Law: For every action, there is an equal and opposite reaction.
  • Law of Gravitational Forces.

• Newton's Third Law Explained

  • If an object puts a force on another, the second object puts an equal force back.
  • Example: Punching a wall. The wall exerts an equal force back, but the effect is greater on the hand due to differences in mass/ fragility..
  • Mosquito and truck example: Forces are the same, effects differ due to mass.
  • The sun pulls on the planet, and the planet pulls on the sun.
  • This is why stars have a wobble.

• Gravity

  • Property of mass.
  • More mass = greater gravitational force.
  • Everything has a gravitational pull but often is insignificant to cause acceleration.
  • There's a gravitational pull between all particles of matter in the universe, regardless of size or distance.
  • If the forces are small, acceleration won't occur, but the force is still there.

• Elliptical Orbits

  • The star is at one focus, not the center.
  • Planets do not orbit in perfect circles.
  • Sometimes closer, sometimes farther from the star.
  • Some planets have more elliptical orbits than others.

• Perihelion and Aphelion

  • Perihelion: Point where the planet is closest to its star.
  • Aphelion: Point where the planet is furthest from its star (mnemonic: Aphelion is AWAY).

Calculating Eccentricity

• Formula provided to find eccentricity.

• Earth's Orbit

  • Has an eccentricity.
  • Meaning Earth gets closer and further from the sun.

Kepler's Second Law

• An imaginary line joining a planet and a star sweeps equal space during equal time intervals.
• Planets don't move at a constant speed.
• Analogous to inside vs. outside corner in F1 racing.
• Equal area is covered in equal time.
• When closer to the star, the planet moves faster.

Perihelion: Planet is moving faster.
Aphelion: Planet is moving slower.

*Speed varies, time to get from point A to point B does not.

• Explanation
  • Stronger gravitational pull when closer means faster movement to maintain orbit.

Aphelion vs. Perihelion

• Table summary:
  • Aphelion:
    • Furthest from the sun.
    • Less gravity.
    • Slower velocity.
  • Perihelion:
    • Closest to the sun.
    • More gravity.
    • Faster velocity.

Kepler's Third Law

• The square of the orbital period is directly proportional to the cube of the semimajor axis.

• $mass * t^2 = r^3$ Where,

  • $t$ can also be called $p$ (period).
  • $r$ can also be called $a$ (astronomical units).

• Implications

  • Planets further from the sun take longer to orbit.

  • Because of less gravity.

  • Formula:

• Assuming the sun is the center of the system.

• Example Calculation

  • Finding the distance of an imaginary planet, that takes 8 earth years to orbit a star:
  • $1 * 8^2 = r^3$
  • $64 = r^3$
  • $r = \sqrt[3]{64} = 4 AU$
    One astronomical unit (AU) is the distance from the sun to Earth.

• Habitable Zones

  • Also known as Goldenrod Zone.
  • The region where liquid water can exist.
  • Water is the first building block of life and is found in many places in the universe.
  • Life most likely needs liquid water to move to the next step.

• Habitable Zone Conditions

  • Not too hot, not too cold.
  • Must have liquid water over long periods of time.
  • Position varies with star's luminosity and temperature.