# 15.3 SATELLITES AND ORBITS Anything that circles a planet is called a **satellite**. While the Moon is Earth's natural satellite, humans have launched thousands of artificial ones for purposes such as GPS, weather forecasting, and exploring the far reaches of space. This section explains the requirements for placing a satellite into a stable and maintaining it there. The precise speed known as **orbital velocity** is examined, which a satellite must have to avoid either falling back to Earth or flying off into space. ### 1. Satellites: Natural and Artificial A **satellite** is any object that orbits a planet due to the force of gravity, maintaining a stable path around it. - **Natural Satellites**: Natural bodies orbiting a planet, dwarf planet, or minor planet (e.g., moons). Six of the major planets in our solar system possess natural satellites. - **Artificial Satellites**: Objects intentionally placed into orbit around the Earth or other celestial bodies by humans. The first artificial satellite was launched in 1957. **Purposes**: Communication, military operations, scientific research, weather monitoring, and navigation. **Examples**: - *International Space Station (ISS)*: The largest artificial satellite, capable of supporting a crew and visible with the naked eye. It serves as a space laboratory. - *Voyager 1*: The farthest man-made object, launched in 1977, which has ventured into interstellar space, carrying a golden record with messages from Earth. --- ### 2. Satellite Orbits and Launching **Types of Orbits** (as illustrated in Fig. 15.6): - **Equatorial Orbit (1)**: Lies in the plane of the equator. - **Inclined Orbit (2)**: At an angle to the equator. - **Polar Orbit (3)**: Passes over the Earth's poles. - **Geostationary Orbit (4)**: A specific type of equatorial orbit where the satellite completes a full round trip in 24 hours, making it appear stationary above a fixed point on Earth. This is discussed further in . **Launching Process**: The launching process involves moving the satellite to a high altitude and accelerating it to a sufficiently high tangential speed with the help of a spacecraft (e.g., rockets), as shown in Fig. 15.7. **Critical Speed Considerations**: - **Speed too low**: The satellite will fall back to Earth. - **Speed too high**: The satellite will either move in an elliptical orbit or escape Earth's gravity (escape speed), never to return. - **Adjusted speed**: The satellite will move in a stable circular (or nearly circular) orbit indefinitely. Circular orbits are typically used as they require the least energy. --- ### 3. Orbital Velocity **Definition**: The minimum velocity required for a satellite to orbit the Earth or another celestial body at a specific altitude. **Derivation**: When a satellite moves with velocity $v_o$ in a circle of radius $r$ from the center of Earth, it has centripetal acceleration $a_c = \frac{v_o^2}{r}$. Since this centripetal acceleration is supplied by gravity, $a_c = g$. Therefore: $$g = \frac{v_o^2}{r}$$ This gives $v_o^2 = gr$, leading to the orbital velocity formula: $$v_o = \sqrt{gr}$$ **General Formula (using Newton's Law of Universal Gravitation)**: Considering the orbital radius $r$ as the sum of the radius of Earth $R_E$ and the height $h$ above the surface ($r = R_E + h$), the orbital velocity can also be written as: $$v_o = \sqrt{\frac{GM_E}{R_E + h}}$$ Where: - $G$ is the gravitational constant - $M_E$ is the mass of the Earth (or the larger gravitating body) - $R_E$ is the radius of the Earth - $h$ is the altitude of the satellite above the Earth's surface **Relationship with Distance**: The equation shows that a satellite moves faster when it is closer to the gravitating body (smaller $r$) and slower when it is further away (larger $r$). **Geostationary Orbit Achievement**: If the speed decreases appropriately with height, an orbit can be achieved where the satellite completes a full round trip in 24 hours, appearing stationary above a fixed point. --- ## Worked Examples **Example 1**: Calculate the orbital velocity of the International Space Station (ISS) at an average altitude of about 400 km above the Earth's surface. Given: $h = 400 \text{ km} = 400,000 \text{ m}$, $G = 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$, $M_E = 6 \times 10^{24} \text{ kg}$, $R_E = 6.4 \times 10^6 \text{ m}$ **Solution**: Using the equation for orbital velocity: $$v_o = \sqrt{\frac{GM_E}{R_E + h}}$$ $$v_o = \sqrt{\frac{6.67 \times 10^{-11} \times 6 \times 10^{24}}{6.4 \times 10^6 + 400,000}}$$ $$v_o \approx 7.66 \text{ km s}^{-1}$$ At this speed, the ISS completes 16 orbits in 24 hours. **Example 2**: The mass of the Sun is $1.99 \times 10^{30} \text{ kg}$, and the radius of the Earth's orbit around the Sun is $1.5 \times 10^{11} \text{ m}$. Calculate the orbital velocity of the Earth. Given: $M_s = 1.99 \times 10^{30} \text{ kg}$, $r = 1.5 \times 10^{11} \text{ m}$, $G = 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$ **Solution**: $$v_o = \sqrt{\frac{GM_s}{r}}$$ $$v_o = \sqrt{6.67 \times 10^{-11} \times \frac{1.99 \times 10^{30}}{1.5 \times 10^{11}}}$$ $$v_o \approx 30 \text{ km s}^{-1}$$ **Example 3**: Calculate the orbital speed of a satellite orbiting the Earth at an altitude equal to Earth's radius ($h = R_E$). Given: $G = 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$, $M_E = 6 \times 10^{24} \text{ kg}$, $R_E = 6.4 \times 10^6 \text{ m}$ **Solution**: Using $v_o = \sqrt{\frac{GM_E}{R_E + h}}$ and setting $h = R_E$: $$v_o = \sqrt{\frac{GM_E}{R_E + R_E}} = \sqrt{\frac{GM_E}{2R_E}}$$ $$v_o = \sqrt{\frac{6.67 \times 10^{-11} \times 6 \times 10^{24}}{2 \times 6.4 \times 10^6}} = \sqrt{\frac{40.02 \times 10^{13}}{12.8 \times 10^6}}$$ $$v_o \approx 5591.57 \text{ m s}^{-1} \approx 5.59 \text{ km s}^{-1}$$ --- | Concept | Description | Formula | | :--- | :--- | :--- | | **Satellite** | Object orbiting a planet due to gravity | N/A | | **Artificial Satellite** | Man-made object placed in orbit for various purposes | N/A | | **Orbital Velocity** | Minimum speed for a stable orbit | $v_o = \sqrt{gr}$ | | **Orbital Velocity** | General formula considering altitude | $v_o = \sqrt{\frac{GM_E}{R_E + h}}$ | | **Geostationary Orbit** | Satellite completes orbit in 24 hours, appearing stationary over a fixed point on Earth | N/A |