## Wednesday, November 1, 2017

### Select Questions-Answers From All Experts Astronomy Forum (Gravity & Orbits)

Question -

Why do planets and other bodies stay in orbit instead of crashing straight
into the Sun or the Earth? (Ex. the moon orbits the Earth, why doesn’t it just crash
into the Earth?)

Planets, other celestial objects, and satellites already in orbit are really in a state of free fall.

Note also an object like an artificial satellite has to be launched into orbit, which
means it must attain a certain escape velocity.(  vesc  )

In the case of  vesc  attained for an object with respect to Earth, we can work it out.
The Earth's radius is 6396 km or call it 6.4 x 106 m.

G, the Newtonian gravitational constant is G = 6.7 x 10-11 Nm2/ kg2

The mass of the Earth is 6 x 1024 kg

Then:

[2MG/ r]1/2  =  [2 (6 x 1024 kg) (6.7 x 10-11 Nm2/ kg2)/ 6.4 x 106 m ]1/2 This comes out to vesc = 11,200 m/s or 11.2 km/s

Now consider a satellite already in orbit around Earth. Why doesn't it just fall?
Well, the first answer is because it is moving too rapidly to do so. The speed of
the satellite with respect to Earth ensures it will stay in orbit. What does this mean?

Although at each second the satellite has an acceleration toward the center of the
Earth (called 'centripetal', denoted  Fc    we note from the diagram below : it has NO vertical or downward velocity . The reason is that it falls
from each position at the same rate the Earth's surface falls away
underneath it.  Basically, the inward directed force  Fc      keeps the
satellite moving in its orbit and also falling from successive tangents denoted
in the diagram by v over bar.  These are given in the diagram as the product:

r  =   (v/r) r

Thus, relative to Earth's surface, the velocity in a VERTICAL (downward)
direction is ZERO, since the distance between the satellite and Earth's
surface remains constant. Hence, we say the satellite - or any orbiting
body - is in a constant state of "free fall'.

Perhaps a better diagram to convey the falling is shown below: Where the T', T" etc. denote successive tangents to the orbit and the 'force' denotes the centripetal (inward directed) force.

Note here if the motion is uniform then the only way that the centripetal
acceleration (ac = v2/ r) arises is via the change in direction of the velocity
vector, v. Thus, the acceleration is: D v/r   or:  (vv)/ r, but the magnitude
of each vector is: |v| = rq/ t = r w. By similar triangles one would obtain:

D v/v  = s/ r  and   D v = v(s/r)  but s = (rq)/ t

So: D v = v(q/ t) = vw

And since: w = v/r then:

a c = D v/ r = vw/ r = = v2/ r

And: GMm/ r 2   =    m v2/ r

Thus we see the gravitational force of attraction (based on Newton's law of
gravitation, e.g. F =  GMm/ r 2)   is what supplies the centripetal force to keep
the object in orbit. This condition of perpetual free fall is what we mean by
being "in orbit".

The same applies to other bodies orbiting larger ones. Thus, a planet
orbiting the Sun is also in a similar state of free fall, with respect to
the surface of the Sun. By the same token its gravitational force of
attraction to the Sun supplies its own centripetal force, in effect a condition of
force balance. So long as this is sustained the object remains in its orbit and
doesn't fall.

While it is being pulled in toward the Sun (by the Sun's gravity) it has a
speed in its orbit large enough so there's no vertical (e.g. downward) velocity
component. Thus, planets orbit in a circle rather than falling into the Sun - which
they would do if their orbital speeds suddenly halted or slowed drastically. This
is much like an Earth satellite would if subjected to atmospheric friction that slows
its velocity.  Then we say its orbit degrades.

In the same exact fashion, the Moon orbits the Earth (free fall) rather
than 'crashing down' into it, because the speed of the Moon in its orbit
is large enough to overcome the inward pull of Earth gravity. The Moon
keeps continually falling in its orbit relative to the Earth - marking out
an orbit or circular path- rather than falling ONTO the Earth.

Two final points need to be made:

1) When bodies far out in space collide with each other, it is usually
because their orbits intersect.

This is the case, for example, with asteroids and planets. For example,
through Earth's history numerous asteroids- some quite large- have
collided with Earth. One of these, 65 million years ago, wiped out the
dinosaurs.

2)When an orbiting body does finally come crashing to its parent body, say
a satellite to Earth, this is because its orbit has destabilized or degraded-
say departing too much from its original circular path. Thus, when the
Skylab crashed to Earth in 1973 it did so because its orbit became too
elongated - too close to Earth at its perigee point.

This meant it encountered friction with the outer regions of Earth's
atmosphere, which slowed its orbital speed, which in turn kept bringing
the low point of the orbit down lower- until it finally burnt up in the
atmosphere. Some theorize powerful solar flares contributed to this by causing
the atmosphere to expand sufficiently to affect the Skylab's orbit.