Everything in Orbit
Orbital Velocity Orbital velocity is the speed at which a planetary body moves in its orbit around another body. If orbits were circular, this velocity would be constant. However, from discussion of Kepler s Laws, we know that planetary bodies orbit the sun (or their planet) in elliptical paths. Therefore, when the planetary body is closer to the sun in their orbit (perihelion), it actually moves faster than it does when it is farther away due to gravitational force
Equation of orbital velocity: v 2 =GM(2/r 1/a) where: G is the gravitational constant (G=6.67x10 11 N/m 3 kg), M is the mass of the sun (or central body), r is the body s distance from the sun, a is the semi major axis If the orbit is circular, r=a, and the equation reduces to v 2 =GM/R The orbits of most of the planets, especially the inner planets are nearly circular, and can be treated as such
Another thing about Orbital Velocity Why don t objects spiral into the sun? The body has a particular orbital velocity, thus it has a certain inertia (a force that tries to keep the body going in a straight line). The body s inertia exactly equals the gravitational force. If for some reason the body slowed down or lost energy, it would crash into the Sun
Application: satellites We can put satellites in any orbit around the Earth, but inertia must equal gravitational force to keep it in orbit That means closer satellites must orbit the Earth faster because there is a larger gravitational force to overcome For example: weather satellites orbit at a distance of 35,786 km and have an orbital velocity (assuming circular orbit) of 11,300kmph But other satellites (i.e., Landsat) orbit the Earth at a distance of 800km with an orbital velocity of 80,317kmph
Escape Velocity If we launched a satellite with so much speed, inertia would overcome gravity and the satellite would achieve escape velocity Escape velocity = speed required to project a body completely free of a primary body (Hartmann 1999) Equation: V=(2GM/R) 1/2 where: G=gravitational constant M=Earth or primary body mass R=Earth or primary body radius
Planetary Escape Velocity Primary Body Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Escape Velocity 4.4 km/sec 10.4 km/sec 11.2 km/sec 5.0 km/sec 59.5 km/sec 35.5 km/sec 21.3 km/sec 23.5 km/sec 1.1 km/sec
Applications of Escape Velocity Meteorites Atmospheres Voyager: to the edge of the Solar System
Orbital Energy Total orbital energy of an orbiting body is conserved. This means: orbiting body will not spontaneously fly off into space! orbiting body will follow same path around central body, with same orbital velocity unless perturbed in some way
Tidal Forces The Earth Moon System Tidal forces between the Earth and Moon are an application of orbital energy the Moon will NOT follow the same orbital path forever because it is perturbed by gravitational forces between the two planetary bodies
Earth Tides Gravitational attraction greater on side that faces the moon Earth stretches along Earth Moon line Oceans fluid and can move in response to this stretching, creates tidal bulges, or high and low tides
Two special tides: spring tides and neap tides. Spring tides: tidal force of Moon and Sun on Earth add to get stronger tides Neap tides: tidal force of Moon and Sun cancel to get weaker tides
Consequences of Tides Force of oceans flexing creates a lot of friction Over the history of the Earth, this friction has actually slowed the Earth s rotation down Conservation of angular momentum: Earth loosing momentum, Moon gains it! Result: Moon is spinning out into a larger orbit!
Moon Tides Interaction with Earth also created bulge on Moon, but Moon solidified, so bulge is frozen in Moon Long axis of Moon always pointed toward Earth In order to keep this orientation, Moon rotates once during the time it orbits the Earth once: synchronous rotation Consequence: Moon always has the same side toward Earth
Tidal Forces The Jupiter Io System Tidal effects on Io creates a tidal bulge gravitational perturbation from other Galilean satellites causes Io s orbit to be more elliptical Io pulled differently depending on distance from Jupiter This flexing is enough to heat Io s interior, called tidal heating Result: Io is most volcanically active planetary body!
Tidal Forces Saturn s Rings Gravitational forces increase the closer you are to a body There s a critical limit within which tidal forces overcome the satellites own gravity, tearing it apart This limit is known as Roche s Limit
Phases of the Moon
Eclipses Two types: Solar and lunar eclipses Solar eclipse: Moon s path moves in front of Sun, blocking it out Lunar eclipse: Earth moves between Sun and Moon, appears red from sunlight filtering through Earth s atmosphere
Eclipses
Morning Star, Evening Star Because Venus is closer to the Sun, it can never appear to be far away from the Sun in the sky. We can t see it in the day because Sunlight blocks it out, but we can see it as the Sun either sets or rises Mercury behaves the same way, but Venus is much brighter
Lagrangian Points Three body problem, interaction between Moon s and Earth s gravity In such a system, there are 5 points where the gravitational forces balance each other A body at one of those points would be stable and stationary Trojan asteroids: in L4 and L5 spots for Sun Jupiter system
Seasons due to tilt of Earth s axis and elliptical orbit For example, winter in Northern Hemisphere: closer to sun, but axis pointed away from sun Seasons