Exam 1 Review: Galileo, Escape Speed, and Orbital Speed

General Exam Information & Review

• Exam Logistics:

  • Exam is scheduled for Friday at $03:00$ PM in class.

  • Required materials: Pencil and a scientific calculator.

  • Resources: Review "topics for exam one" and "equations for exam one" files on Canvas.

  • Content: All units for the exam are covered in the "topics for exam one" file. All relevant equations are also provided there.

• Unit 18 & Homework 4:

  • Unit 18 is the final unit required for Exam 1 and Homework 4.

  • Recommendation: Study homeworks in 'study mode' and review all units included in the exam.

• Today's Class Structure:

  • Finish Unit 18 with examples.

  • Short quiz at the end of class.

• Wednesday's Class:

  • Review material.

  • Opportunity for students to ask questions.

Review of Solar System & Galileo's Discoveries

• Planets & Ecliptic Plane:

  • Pluto is mentioned as being an outlier, not on the ecliptic plane where most other planets orbit.

• Heliocentric vs. Geocentric Models - Galileo's Key Observations:

  • Jupiter's Moons: Galileo observed four of Jupiter's moons (Europa, Callisto, Io, Ganymede) orbiting Jupiter.

    • Significance: This was a significant finding, indicating that not everything orbited the Earth, challenging the geocentric model.

  • Phases of Venus: This was considered the most important indication supporting the heliocentric model.

    • Observation: Venus exhibits a full cycle of phases, similar to Earth's moon (e.g., crescent Venus, full Venus).

    • Explanation: These phases can only be explained if Venus orbits the Sun, allowing different amounts of its illuminated surface to be visible from Earth as it moves around its trajectory. The geocentric model, with planets moving in epicycles around Earth, cannot account for the observed full range of Venus's phases.

Unit 18: Escape Speed and Orbital Speed

• Introduction to Speeds: Today's focus is on calculating escape speed and orbital speed using the principles of gravitational force.

Escape Speed ($V_{ ext{escape}}$)

• Definition: The minimum speed an object needs to achieve to escape the gravitational pull of a celestial body (e.g., a planet, moon) and never return.

• Intuition from Videos:

  • Comparing a spacecraft launching from the Moon (Eagle) versus a rocket launching from Earth (Saturn V).

  • The Earth launch requires significantly more fuel and produces much greater sound and thrust, indicating a much larger escape speed is required compared to the Moon.

• Factors Influencing Escape Speed:

  • Mass ($M$): The mass of the object one wishes to escape from.

  • Radius ($R$): The radius of the object (or the distance from its center to surface where the escape attempt begins).

    • Reasoning: Gravitational force is inversely proportional to the square of the distance ($1/r^2$). The radius defines the starting distance from the object's center.

• Formula: The escape speed is given by: $V_{ ext{escape}} = rac{ ext{2GM}}{ ext{R}}$ Where:

  • $G$ is the gravitational constant.

  • $M$ is the mass of the celestial body you are escaping from.

  • $R$ is the radius of the celestial body (distance from its center to surface).

• Proportionalities:

  • $V<em>{ ext{escape}}$ is directly proportional to the square root of the mass ($V</em>{ ext{escape}} imes ext{M}$).

  • $V<em>{ ext{escape}}$ is inversely proportional to the square root of the radius ($V</em>{ ext{escape}} imes rac{ ext{1}}{ ext{R}}$).

• Derivation (Optional for advanced students): The formula can be derived from basic principles of gravitational force ($F = rac{ ext{GM}<em>{ ext{1}} ext{M}</em>{ ext{2}}}{ ext{r}^{ ext{2}}}$) and Newton's second law ($F = ext{ma}$), but this algebra is skipped in class to avoid confusion. Similarly, Kepler's Third Law can be derived.

Escape Speed Examples & Implications

• Metaphor/Analogy: Think of escape speed as a "fence" that an object needs to climb.

• Hypothetical Example 1: Earth becomes $25$ times more massive (radius constant).

  • Question: What happens to $V_{ ext{escape}}$? Increase or decrease? By how much?

  • Result: $V_{ ext{escape}}$ increases by a factor of $5$.

    • Explanation: Since $M$ is in the numerator and under a square root, if $M$ becomes $25M$, then $ext{M}$ becomes $ext{25}$
      $ext{M}$. The square root of $25$ is $5$.

  • NASA's Reaction: Sad.

    • Reason: A larger escape speed means it's much harder to launch rockets (a taller fence to climb), requiring more energy and fuel to reach the necessary speed.

• Hypothetical Example 2: Earth's radius increases by a factor of $16$ (mass constant).

  • Question: What happens to $V_{ ext{escape}}$? Increase or decrease? By how much?

  • Result: $V_{ ext{escape}}$ decreases by a factor of $4$.

    • Explanation: Since $R$ is in the denominator and under a square root, if $R$ becomes $16R$, then $ext{1 / 16R}$ becomes $rac{ ext{1}}{ ext{4}}$
      $ext{1 / R}$. The square root of $16$ is $4$. Hence, it decreases by a factor of $4$.

    • Numerical Illustration: If initial $V_{ ext{escape}}$ was $100$, it would become $25$ ($100 / 4$).

  • NASA's Initial Reaction: Happy.

    • Reason: A smaller escape speed means it would be easier and require less energy to launch rockets (a shorter fence to climb).

  • Further Implication: Atmosphere Retention: This scenario has significant consequences for a planet's atmosphere.

    • Mechanism: Earth's atmosphere is held by gravitational forces. If $V_{ ext{escape}}$ becomes too small, gas particles in the atmosphere could achieve sufficient velocity to "jump the fence" and escape into space.

    • NASA's Overall Reaction: Not very happy. While rocket launches would be easier, the loss of the atmosphere would be detrimental.

• Escape Speed as an Indicator for Atmosphere:

  • The calculation of escape speed does not involve atmospheric properties. However, $V_{ ext{escape}}$ is a crucial indicator of whether a planet will retain an atmosphere.

  • Low Escape Speed: Planets with low escape speeds (e.g., Moon, Mercury) typically have no atmosphere because gas particles easily escape their gravitational pull.

  • High Escape Speed: Planets with high escape speeds (e.g., Earth, Jupiter, Saturn) typically have thick atmospheres because gas particles find it difficult to escape their strong gravitational pull (a very tall fence).

Orbital Speed (Circular Speed) ($V_{ ext{circular}}$)

• Definition: The speed required for an object to maintain a stable orbit around a central body after having escaped its immediate gravitational pull.

• Formula: The orbital speed for a circular orbit is: $V_{ ext{circular}} = rac{ ext{GM}}{ ext{r}}$ Where:

  • $G$ is the gravitational constant.

  • $M$ is the mass of the central celestial body (the one being orbited).

  • $r$ is the distance from the center of the central body to the orbiting object (altitude + radius of the central body).

• Proportionalities:

  • $V<em>{ ext{circular}}$ is directly proportional to the square root of the central body's mass ($V</em>{ ext{circular}} imes ext{M}$).

  • $V<em>{ ext{circular}}$ is inversely proportional to the square root of the orbital radius ($V</em>{ ext{circular}} imes rac{ ext{1}}{ ext{r}}$).

• Intuition & Examples:

  • Closer Orbits: Objects orbiting closer to the central body (smaller $r$) require higher orbital speeds.

    • Example: The International Space Station (ISS) orbits very close to Earth, completing an orbit in approximately $1.5$ hours, moving very fast.

  • Farther Orbits: Objects orbiting farther from the central body (larger $r$) require lower orbital speeds.

    • Example: The Moon, Earth's natural satellite, orbits far from Earth and takes approximately $28$ days (about a month) to complete one orbit, moving much slower than the ISS.

Trajectories (Briefly Mentioned)

• Different velocities after launch can lead to various trajectories (e.g., parabolic, hyperbolic).

• This topic will not be on the exam but is a complex area of study in rocketry and interplanetary travel.

Upcoming Activity

• A short quiz will be administered at the end of class on the covered material.