Orbits of planets and satellites

1. Orbits:

  • In Figure 1, a planet of mass m is in a circular orbit around a star of mass M.

  • Figure 1 A planet is in a circular orbit around the Earth
  • You have already studied circular motion.
  • Now we can combine the equations of circular motion and gravitation to link the speed or time period of a planet’s orbit to its distance from the Sun.
  • The pull of gravity provides the necessary centripetal force to keep the planet in orbit. So, we can write Centripetal force is
  • [math] F_c = \frac{mv^2}{r} \qquad \text{(1)} [/math]
  • And gravitational force is
  • [math] F_g = \frac{GMm}{r^2} \qquad \text{(2)} [/math]
  • Comparison equation 1 and 2 then
  • [math]\frac{GMm}{r^2} = \frac{mv^2}{r}\\
    \frac{GM}{r^2} = \frac{v^2}{r} \\
    v^2 = \frac{GM}{r} \qquad \text{(3)} [/math]
  • From this, you can see that speed of the orbit is faster for small orbits. Figure 2 shows the link between orbital speed and the distance of our eight planets from the Sun.
  • Figure 2 the link between orbital speed and the distance of our eight planets from the Sun
  • We can also link the time period of the orbit (q year for the earth) to the radius of the orbit. The speed of the orbit is linked to the circumference, 2πr , and the time period of the orbit, T, through the equation
  • [math] v = \omega r \\v = (2 \pi f) r \\
    v = \left(2 \pi \frac{1}{T}\right) r \\
    v = \frac{2 \pi r}{T} [/math]
  • And squaring on both side
  • [math]  v^2 = \left(\frac{2 \pi r}{T}\right)^2 \\v^2 = \frac{4 \pi^2 r^2}{T^2} \qquad \text{(4)} [/math]
  • Compare with equation 3 then
  • [math] \frac{GM}{r} = \frac{4 \pi^2 r^2}{T^2} \\ \frac{r^3}{T^2} = \frac{GM}{4 \pi^2} \\
    T^2 = \frac{4 \pi^2}{GM} r^3 [/math]
    [math] T^2 \propto r^3 [/math]
  • Figure 2 also shows the relationship between time period and orbital radius of the planet- the green curve.

2. Energy considerations for an orbiting satellite:

  • When a planet or comet moves in an elliptical orbit, its speed changes, but the total energy of the body stays the same.
  • At point A in Figure 3, the comet is moving at its latest orbital speed, but it has the smallest gravitational potential energy, because it is closest to the Sun.
  • At point B, there is a component of the Sun’s gravitational pull on the comet, which slows it down.
  • When the comet reaches point C, it is at its furthest point from the Sun. It has its lowest kinetic energy at this point, but its highest gravitational potential energy.
  • At point D, the comet is falling back towards the Sun. There is a component of the Sun’s gravitational pull, which speeds it up. The comet’s potential energy is being transferred into kinetic energy, and the comet reaches its maximum speed again at A.
  • Figure 3 Comets move in elongated or eccentric elliptical orbits round the Sun. Note that the Sun’s solar wind always blows the comet’s tail away from the Sun.
  • Kinetic Energy: The energy of motion, dependent on the satellite’s velocity and mass.
  • Potential Energy: The energy of position, dependent on the satellite’s height and mass.
  • Total Mechanical Energy: The sum of kinetic and potential energy, remaining constant if no external forces act.
  • Orbital Energy: A measure of the satellite’s energy, combining kinetic and potential energy.
  • Escape Velocity: The minimum velocity required for an object to escape the gravitational pull of the central body (e.g., Earth).
  • Orbital Velocity: The velocity required for a satellite to maintain a stable orbit.
  • Gravitational Potential Energy: The energy associated with the satellite’s position in the gravitational field.
  • Energy Transfer: Occurs when a satellite’s orbit changes due to external forces (e.g., gravitational influences, thruster firings).
  • Key concepts:
    – Energy conservation: Total mechanical energy remains constant in a closed system.
    – Energy transfer: Energy is exchanged between forms (e.g., kinetic to potential).
    – Orbital stability: Achieved when the satellite’s energy is balanced.
  • Understanding energy considerations is crucial for satellite mission design, orbit maintenance, and maneuver planning.

3. Escape velocity:

  • The minimum velocity an object must have at the surface of a planet to escape the pull of gravity using its own kinetic energy.
  • The Earth has its atmosphere because the molecules of gas, moving in our atmosphere, do not have enough kinetic energy to escape from the pull of gravity at the Earth’s surface.
  • When a fast-moving object leaves the surface of a planet, we can write that
  • [math] \text{Decrease in kinetic energy} =\text{increase in gravitational potential energy} [/math]
    [math] \Delta E_k = \Delta E_p [/math]
  • This assumes that the object is not affected by an atmosphere, and is in free fall this is not a spacecraft with a rocket,
    The equation above can be written as
  • [math] \frac{1}{2} m v_1^2 – \frac{1}{2} m v_2^2 = m \Delta V [/math]
  • If the object is to just escape the pull of the planet, its speed,  will just reach zero at an infinite distance from the planet.
  • The gravitational potential at the surface of a planet is given by:
  • [math] V = – \frac{GM}{r} [/math]
  • So, the change in potential is
  • [math] \Delta V = \frac{GM}{r} [/math]
  • So, the escape velocity for a planet can be calculated using
  • [math] \frac{1}{2} m v^2 = m \Delta V =  \frac{GMm}{r} [/math]
  • Or
  • [math]  v^2 = \frac{GM}{r} [/math]
  • For the Earth, [math] M= 6*10^{24}kg \, \text{and}\, r = 6400 km [/math]So,
  • [math] v^2 = \frac{2 * (6.67 * 10^{-11}) * (6 * 10^{24})}{6.4 * 10^{6}} [/math]
  • [math] v=11200 m.s^{-1} [/math]
  •  Science air molecules travel at approximately 500 m.s-1 on the surface of the Earth, they travel well below the Earth’s escape velocity.

4. Synchronous orbits:

  • A synchronous orbit is a type of orbit where a satellite’s orbital period matches the rotational period of the central body (e.g., Earth).
  • This means the satellite remains stationary in the sky, relative to a fixed point on the Earth’s surface.
  • Types of synchronous orbits:
    1.  Geosynchronous Orbit (GEO): Orbital period matches Earth’s rotational period (24 hours). Satellites in GEO are stationary relative to a fixed point on the equator.
    2. Sun-synchronous Orbit (SSO): Orbital period matches the time it takes the Earth to rotate once relative to the Sun (about 24 hours). Satellites in SSO pass over the same point on the Earth at the same local solar time.
    3. Polar Sun-synchronous Orbit (PSSO): A type of SSO where the satellite passes over the polar regions.
  • Characteristics:
    – Stationary or near-stationary in the sky
    – Constant visibility from a fixed point on the Earth
    – Orbital period matches rotational period of the central body
    – Used for various applications like communication, weather forecasting, and Earth observation
  • Advantages:
    – Continuous coverage of a specific region
    – Simplified communication and data transmission
    – Improved observation and monitoring capabilities
  • Figure 4 Geosynchronous orbit
  • One of the most useful orbits for satellites is the geosynchronous orbit.
  • In this case the satellite is placed in an orbit above the Earth’s equator, at such a height that it takes exactly one day to complete an orbit.
  • The orbit is synchronized with the Earth’s rotation, so that it remains in the same place above the Earth’s surface.
  • This means that our satellite dishes, for example, can be aligned with a satellite, which always lies in the same position relative to Earth.

5. Low orbits and geostationary orbits:

  • ⇒ Low earth orbit:

  • Low Earth Orbits (LEO) are circular orbits around Earth with an altitude of approximately 160 to 2,000 kilometers (100 to 1,243 miles). Satellites in LEO have an orbital period of about 90 to 120 minutes, which means they complete one rotation around Earth in less than two hours.


    Figure 5 Low earth orbit

  • Characteristics of LEO:
    1. Low altitude: Close proximity to Earth’s surface.
    2. Short orbital period: Complete one rotation in less than two hours.
    3. High inclination: Often polar or near-polar orbits.
    4. Limited visibility: Satellites are only visible from a specific region for a short time.
  • Applications of LEO:
    1. Earth observation: Study Earth’s surface, climate, and natural resources.
    2. Remote sensing: Gather data on Earth’s surface, atmosphere, and oceans.
    3. Communication: Enable data transmission between two points on Earth.
    4. Scientific research: Conduct space-based research in various fields.
    5. Technology demonstration: Test new space technologies and systems.
    6. Earth imaging: Capture high-resolution images of Earth’s surface.
    7. Disaster response: Provide critical information during natural disasters.
    8. Environmental monitoring: Track environmental changes and pollution.
  • Some examples of satellites in LEO include:
    1. International Space Station (ISS)
    2. Hubble Space Telescope
    3. Planet Labs’ Dove constellation
    4. NASA’s Terra, Aqua, and Aura satellites
    5. European Space Agency’s Copernicus program

  • ⇒ Geostationary orbit:

  • Geostationary Orbits are circular orbits around Earth with an altitude of approximately 35,786 kilometers (22,236 miles) above the equator.
  • Satellites in GEO have an orbital period of 24 hours, which matches Earth’s rotational period, allowing them to remain stationary in the sky.
  • Characteristics of GEO:
    1. High altitude: Farthest orbit from Earth’s surface
    2. Synchronous with Earth’s rotation: Orbital period matches Earth’s rotational period
    3. Stationary in the sky: Appears fixed over a single point on the equator
    4. Constant visibility: Visible from a fixed point on Earth 24/7
    5. Equatorial plane: Orbits in the same plane as Earth’s equator
  • Applications of GEO:
    1. Telecommunications: Satellite TV, radio broadcasting, and data transmission
    2. Weather forecasting: Monitoring weather patterns and storm tracking
    3. Navigation: Augmenting GPS signals for improved accuracy
    4. Earth observation: Studying Earth’s climate, land use, and natural resources
    5. Scientific research: Conducting space-based research in various fields
    6. Military communications: Secure communication for military operations
  • Advantages of GEO:
    1. Stationary in the sky: Simplifies communication and data transmission
    2. Constant visibility: Enables continuous monitoring and observation
    3. High altitude: Reduces atmospheric interference and gravity influences
  • Some examples of satellites in GEO include:
    1. Intelsat
    2. Inmarsat
    3. SES
    4. Eutelsat
    5. NASA’s TDRS (Tracking and Data Relay Satellite)
    6. European Space Agency’s MSG (Meteosat Second Generation)
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