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Orbits and satellites Knowledge-Based Study

Physics · Forces & Newton's Laws · Physics · Circular Motion & Gravitation

Physics topic explanations on Orbits and satellites.

Study notes

  1. An orbit is the gravitationally curved trajectory of an object, such as a satellite or a planet, around a central mass. In a simplified circular orbit of radius r around a body of mass M, the gravitational attraction provides the necessary centripetal force holding the satellite in its path.

    Newton's Law of Universal Gravitation gives the magnitude of the force exerted on a satellite of mass m:

    Fg=GMmr2F_g = \frac{G M m}{r^2}

    Because the orbit is circular, this gravitational force acts as the centripetal force:

    Fc=mv2rF_c = \frac{m v^2}{r}

    Equating these two expressions allows us to analyze the dynamics of the satellite:

    GMmr2=mv2r\frac{G M m}{r^2} = \frac{m v^2}{r}

    Notice that the mass of the satellite, m, cancels from both sides, indicating that all satellites at a given orbital radius experience the same acceleration regardless of their mass.

    Solving the force balance equation for the linear speed v yields the orbital speed:

    v=GMrv = \sqrt{\frac{G M}{r}}

    This relationship shows that orbital speed is inversely proportional to the square root of the orbital radius. Satellites in lower orbits must travel faster than those in higher orbits to maintain a stable circular path.

    The orbital period T is the time required to complete one full revolution of circumference 2πr2\pi r:

    T=2πrv=2πrGMr=2πr3GMT = \frac{2\pi r}{v} = \frac{2\pi r}{\sqrt{\frac{G M}{r}}} = 2\pi \sqrt{\frac{r^3}{G M}}

    Squaring both sides produces Kepler's Third Law for circular orbits:

    T2=(4π2GM)r3T^2 = \left(\frac{4\pi^2}{G M}\right) r^3

    The square of the orbital period is directly proportional to the cube of the orbital radius, where the constant of proportionality depends solely on the mass of the central body.

    Different orbital altitudes and inclinations serve specific scientific and technological purposes:

    • Low Earth Orbit (LEO): Altitudes typically between 160 km and 2000 km, requiring high orbital speeds (approximately 7.8 km/s) and yielding short periods (around 90 minutes). Common for Earth observation and crewed stations.
    • Geostationary Orbit (GEO): An equatorial circular orbit with a period equal to Earth's rotational period (approximately 24 hours), positioned at a radius of roughly 42,200 km from Earth's center. Satellites in GEO remain stationary relative to a point on the equator, making them ideal for telecommunications and weather monitoring.

    The total mechanical energy E of a satellite in a circular orbit consists of kinetic energy K and gravitational potential energy U:

    K=12mv2=GMm2rK = \frac{1}{2}m v^2 = \frac{G M m}{2r}

    U=−GMmrU = -\frac{G M m}{r}

    E=K+U=−GMm2rE = K + U = -\frac{G M m}{2r}

    The negative total energy indicates a bound gravitational system. Increasing the orbital radius increases the total mechanical energy (makes it less negative), which requires adding energy to the satellite.