Satellite motion

Satellite Motion

Review of Weight in Gravitational Fields

• A satellite experiences a gravitational force of 228 N at an altitude of $4.0 imes 10^7$ m above Earth.

• Multiple-choice options to determine the satellite's mass:

  • 23 kg

  • 650 kg

  • 910 kg

  • 1,200 kg

Force Required for Circular Motion

• An object traveling along a circular path experiences a force $F$.

• If the object travels at twice the speed ($2v$), the required force changes to:

  • $rac{1}{2}F$

  • $F$

  • $2F$

  • $4F$

Escape Velocity of Earth

• Find the escape velocity of Earth, denoted by $v_e$, with reference to gravitational force equations.

• Reference to work done, represented by $F imes d$:

  • $W = F imes d$,

  • where $F$ is force and $d$ is displacement.

Orbital Dynamics of Saturn's Moon Titan

• WACE 2019 Question 4 (5 marks):

  • Titan is the largest of Saturn's moons.

  • Orbital radius given: $1.22 imes 10^6$ km.

  • Task: Use Formulae and Data booklet to determine the strength of Saturn's gravitational field at Titan's orbit, expressed in $N ext{ kg}^{-1}$ and $m ext{ s}^{-2}$.

Overview of Satellites

• A satellite is defined as a small body that revolves around a larger body (or mass).

  • Oldest natural satellite: Earth’s Moon.

  • Uses of artificial satellites include:

  • Space observation.

  • Deep space probing.

  • Weather monitoring.

  • GPS, navigation, global communication.

  • Pollution tracking.

  • Spying.

Kepler’s Laws of Planetary Motion

• The motion of planets around the sun is governed by the universal law of gravitation. Kepler’s laws derived from this principle include:

  • 1st Law: Planets move in elliptical orbits with the Sun at one focus.

  • 2nd Law: A planet sweeps out equal areas in equal times.

Kepler’s Third Law

• Definition: The square of a planet's orbital period ($T$) is directly proportional to the cube of the radius ($r$) of its orbit:

  • $T^2 ext{ is proportional to } r^3$

• Implications:

  • Relationship holds true despite variations in planetary sizes and distances.

Additional Insights into Kepler’s Law

• Kepler’s laws derived under the assumption of circular orbits.

• Complete orbits can are typically treated as circular where the elliptical nature has minimal consequence.

• The third law holds for both circular and non-circular motion.

Geosynchronous Orbit

• Definition: An orbit around Earth with a period equal to one sidereal day (approximately 23 hours, 56 minutes, and 4 seconds).

• Features: Can exist in any plane, closer to the equator correlates with greater latitudes.

Geostationary Orbit

• Definition: A specific type of geosynchronous orbit directly above the equator with a period of one sidereal day.

• Characteristics: Maintains the same relative position in the sky from the ground, cuts through the equator.

Polar-Synchronous Orbit

• Definition: An orbit where the satellite passes through the polar axis, allowing it to sweep through all latitudes.

• Features: Longitude varies based on altitude above Earth.

Important Note on Geosynchronous Satellites

• For Earth, $T = 24 ext{ hours}$, where the satellite appears stationary above a single point.

Force and Energy in Satellite Motion

• Gravitational force consulted via Newton's law of gravitation:

  • $F = rac{G imes M_1 imes M_2}{r^2}$ where $G$ is the gravitational constant, $M_1$ and $M_2$ are the masses, and $r$ is the distance.

Orbital Velocity Equation

• Orbital velocity ($v$) can be defined by:

  • $v = rac{GM}{r}$

  • where $GM$ is the product of gravitational constant and the mass, $r$ represents radius of orbit.

Orbital Speed Calculation Examples

• Example: Calculate the orbital speed of a satellite in a stable orbit at $5.00 imes 10^3$ km.

• Worked Example: Calculate the orbital speed of the Moon at $r = 3.84 imes 10^5$ km from Earth's center with: $M_{Earth} = 5.97 imes 10^{24}$ kg; $G = 6.67 imes 10^{-11} ext{ N m}^2/ ext{kg}^2$.

• ISS Altitude Calculation: Given an average altitude of 412.5 km, calculate its period of revolution.

• Spaceship Equation: A spaceship orbits the Moon at 1460 m/s; to find altitude above the Moon's surface given $M_{Moon} = 7.35 imes 10^{22}$ kg and $R_{Moon} = 1.74 imes 10^{6}$ m.

Ganymede as a Case Study

• Definition Details:

  • Mass: $1.66 imes 10^{23}$ kg

  • Orbital radius: $1.07 imes 10^5$ km

  • Orbital period: $6.18 imes 10^6$ s (7.15 Earth days).

• Tasks using Kepler’s laws for calculations include:

  • a. Calculate the orbital radius of Europa, a moon of Jupiter with a period of 3.55 Earth days.

  • b. Calculate the mass of Jupiter using Ganymede’s orbital data.

  • c. Calculate Ganymede's orbital speed.

Probe Measurement Analysis

• Probe measuring the diameter of a new planet: $5.63 imes 10^6$ m, with field strength $3.44 ext{ N kg}^{-1}$:

  • a. Calculate the mass of the new planet.

  • b. Determine the altitude where field strength is $1.00 ext{ N kg}^{-1}$.

  • c. Calculate speed of the probe at that altitude.

Satellite Case Study: Callisto

• Callisto (a moon of Jupiter): mass $1.08 imes 10^{23}$ kg,

  • Orbital radius: $1.88 imes 10^6$ km.

  • Period: $1.44 imes 10^6$ s

• Tasks similar to Ganymede's but focused on Callisto regarding calculations using Kepler's third law.

Key Questions for Students:

1. Choose which statement about satellites is correct.

2. Analyze how the mass increase affects orbital properties of a satellite.

3. Evaluate gravitational field strength and forces on a satellite in orbit.

4. Comparison of orbital radii and periods of Saturn's moons.

5. Evaluate characteristics of a geostationary satellite.

Physics in Action: Satellite Utilization

• Natural satellites have existed for billions of years; they include planets and asteroids.

• Earth has one natural satellite, the Moon, while larger planets like Jupiter have over sixty.

• Since 1957, approximately 15,946 artificial satellites have launched, with about 9,900 still functional.

Classification of Satellites

• Low orbit: 180 km - 2,000 km, e.g., Hubble Space Telescope.

• Medium orbit: 2,000 km - 36,000 km, e.g., GPS satellites.

• High orbit: >36,000 km, e.g., Optus satellites.

Operational Characteristics of Satellites

• Orbits fall into categories:

  • Equatorial orbits - remain above the equator.

  • Polar orbits - travel over poles; capture global coverage.

  • Inclined orbits - between equatorial and polar.

Gravitational Interaction of Satellites

• Satellites in free-fall experience centripetal acceleration equivalent to local gravitational field strength.

• No engines or propulsion maintain orbit; gravitational attraction serves this purpose.

Examples of Valuable Satellites for Australia:

• Himawari-8: Geostationary meteorological satellite, launched 7th October 2014.

  • Operating altitude: 35,786 km; captures real-time weather data.

• Hubble Space Telescope (HST): Joint NASA and ESA venture, launched 25th April 1990,

  • Functions in low-Earth orbit to improve clarity of astronomical imaging.

• NOAA-19: Launched February 2009, operating in near-polar orbit to assist weather forecasts and climate monitoring.

Features of Different Orbits

• Eccentric Orbit: Not all are circular; altitudes vary, easier to achieve.

• Low Earth Orbit: Benefits for rapid communication and less orbital decay.

• Geostationary Orbit: Satellite remains fixed; useful for communication and weather monitoring.

• Downsides of Orbits: Low orbits face atmospheric resistance; geostationary satellites experience transmission delays due to distance.

Key Orbital Features Table