2.2ORBITAL MOTION AND

TIDES

In order to stay in a closed orbit, an object has to be within a certain range of velocities:

Too slow Object falls back down to Earth

Too fast Object escapes Earth’s orbit

Orbital Motion

http://hal.physast.uga.edu/~rls/1020/ch5/cannonball.swf

Circular Velocity

An object orbiting Earth is actually falling (being accelerated) toward Earth’s center.Continuously misses Earth

due to its orbital velocity.

To follow a circular orbit, the object must move at circular velocity.

Circular Velocity Formula

Vc = Gm r√

G = gravitational constant; 6.67 x 10-11 m3/kg•s2

m = mass of the central body in question, in kg

r = radius of orbit, in meters

Circular Velocity Example How fast does the moon travel in its orbit around

the Earth? (Answer with 3 sig figs and in m/s) Hint: Earth’s mass is 5.98 x 1024 kg and the radius of the moon’s orbit around the Earth is 3.84 x 108 m.

1020 m/s or 1.02 km/s

GEOSYNCHRONOUS ORBITS

Escape Velocity

The velocity required to escape from the surface of an astronomical body is known as the escape velocity.

Escape Velocity Formula

Ve = 2Gm r

G = gravitational constant; 6.67 x 10-11 m3/kg•s2

m = mass of object, in kg

r = radius of object, in meters

√

Escape Velocity Example Find the escape velocity from Earth.

11,200 m/sor

11.2 km/s

Newton’s Version of Kepler’s 3rd Law

The equation for circular velocity:

The circular velocity of a planet is simply the circumference of its orbit divided by the orbital period:

If you substitute this for V in the first equation and solve for P2, you will get:

Vc = Gm r

V = 2πr P

√

NVK3L

P2 = (4π2) r3

(Gm)

NVK3L This is a powerful formula in

Astronomy because it allows us to calculate the masses of bodies by observing orbital motion.For example, you observe a moon

orbiting a planet and can measure the radius of its orbit, r, and its orbital period, P.

You can now use this formula to solve for m, the total mass of the system.

There is no other way to find the masses of objects in the universe stars, galaxies, other planets.

G = gravitational constant; 6.67 x 10-11 m3/kg•s2

m = mass of the total system, in kg

r = radius of orbit, in meters

P = orbital period, in seconds

NVK3L Example Planet Cooper has a radius of 6840 km.

and a mass of 5.21 x 1025 kg. What is the orbital period of a satellite orbiting just above this planet’s surface?

1910 sec.or

31.8 min.

NVK3L Example Planet Goofball has a radius of 4390

km. and a mass of 3.67 x 1022 kg. What is the orbital period of a satellite orbiting 50 km. above this planet’s surface?

? sec.or

? min.

Tides and Tidal Forces Earth attracts the moon, and the moon attracts Earth.

Tides are caused by small differences in gravitational forces. Oceans respond by flowing into a bulge of water on the side of Earth facing the

moon.

Also, a bulge exists on the side of Earth facing away from the moon since the moon pulls more strongly on Earth’s center than the side facing away.

Tides and Tidal Forces You might wonder … If the moon and

Earth accelerate toward each other, why don’t they smash together?They are orbiting around a common center of

mass: 4708 km. from Earth’s center.

Spring Tides Gravity is universal, so the Sun also produces tides on Earth.

Twice a month, at new moon and full moon, the moon and Sun produce tidal bulges that add together and produce extreme tidal changes. High tide exceptionally high; Low tide exceptionally low.

These are called spring tides.

“Spring” refers to the rapid welling up of water.

Neap Tides At 1st and 3rd quarter moons, the Sun and moon pull at right

angles to each other, and the Sun’s tides cancel out some of the moon’s tides.

These less-extreme tides are called neap tides. “Neap” means lacking power to advance.