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- Some slides regarding
- Newtonian Gravity and Orbital Motion
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- The force of gravity is mutual and attractive between two objects:
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- First Law (Inertia)
- A body at rest remains at rest, and a body in motion continues to move
in a straight line with a constant speed unless and until an external
unbalanced force acts upon it. That is, an object with no net force
acting upon it has a constant velocity.
- Second Law (acceleration)
- The rate of change of momentum of a body is directly proportional to
the impressed force and takes place in the direction in which the force
acts. Nonrelativistically, the force acting on an object is its
acceleration times its mass. F=ma.
- Third Law (Action and Reaction)
- Whenever A exerts a force on B, B is simultaneously exerting a force of
the same magnitude on A, in the opposite direction.
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- 1609 Published two laws showing:
- K1 Planets orbit the sun in ellipses, with the Sun at one focus
- K2 Motion is faster when they are near the Sun, in such a way
that a line from the planet to the sun sweeps out equal areas
in equal times
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- Ellipse defined by two constants
- semi-major axis a 1/2 length of major axis
- eccentricity e 0=circle, 1 = line
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- 1619 Publishes third law, showing
that there is a relationship orbital period and semi-major axis:
- Exact relationship is P2 µ a3 .
- Outer planets orbit more slowly than inner ones
- Example: Earth P = 365 days, a = 1.00 AU.
Mars p = 687 days, a =
1.524 AU
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- Momentum keeps the planets moving – you do not need some force to do
this.
- Gravity provides the force which makes orbits curve
- Gravity of Sun curves orbits of Planets
- Gravity of Earth curves orbit of moon (and also makes objects on earth
fall downward)
- “Conservation of Angular Momentum” explains why motion is faster when
closer to the sun.
- The inverse square law of gravity explains P2 µ a3 and the details of
why the orbits are ellipses.
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- Centripetal acceleration (v2/r) caused by Gravity
- Period found by
- Kepler’s 3rd Law just comes from this
- Given P and a (and G) we can find the mass of a planet or star, and in
building an alien solar system, Kepler’s laws must be valid!
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- This concept was invented in the 1940s by the science fiction writer,
Arthur C. Clarke. There is an
orbit around the Earth for which the period is exactly 1 day:
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- These are minimum energy trajectories used to travel between two orbits
(e.g., traveling between Earth and Mars).
- We’ll talk about these again as we talk about space travel.
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- The acceleration experienced due to rotation is, mathematically, v2/r. The force then of the artificial
gravity, the weight of an object, is mv2/r. The weight of an object on Earth is
mg, where g is the “surface gravity” which is 9.8 m/s2. (Be careful about weight, a force, and
mass which is not).
- To keep people from getting sick, the rotation rate shouldn’t be faster
than about once per minute or so.
- Imagine playing “space ball” in a rotating cylinder in space – you can
get some crazy passes.
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- Surface Gravity
- The acceleration due to gravity experienced on the surface of a
planet. It can be found using
Newton’s Law of Gravitation and F=ma.
For the Earth, it is g = MEarthG/REarth2
and is 9.8 m/s2.
- Escape Velocity is the velocity needed to overcome the surface gravity
and escape from a planet. For
Earth it is about 11 km/s.
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- Escape velocity from a surface at radius R:
- Imagine a collapsing object like a star.
As R shrinks (but M is fixed), Vescape gets larger and
larger
- At some point VEscape= c
(speed of light)
- Happens at the “Schwarzschild radius”:
- Not even light can escape from within this radius, which gives us a
“black” hole
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Notes
