All Experts Redux: The Basics Of Escape Velocity, Gravity And Orbits

 Question -

Why do planets and other bodies (like artificial satellites) stay in orbit instead of crashing straight into the Sun or the Earth?

Answer:

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.(  v_esc  )

In the case of  v_esc  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 v_esc = 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, firstly 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  F_c  )   - 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  F_c    keeps the

satellite moving in its orbit and also falling from successive tangents denoted in the diagram by v bar.  These are given in the diagram as the product:

w 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, from successive tangents, T, T' T"' etc is shown below:

Note that the successive vectors (from P, P' in upper diagram) denote successive tangents to the orbit identical to the T, T' etc. of Fig. 2. The 'force' denotes the centripetal (inward directed) force  F_c  . 

Note also that  if the motion is uniform then the only way that the centripetal acceleration (a_c = v²/ r) arises is via the change in direction of the velocity vector, v. Thus, the acceleration is: D v/r   or:  (v’ – v)/ 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.

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 - or ellipse - rather than falling into the Sun - which they would do if their orbital speeds suddenly halted or slowed drastically.  In the case of elliptical orbits we have:

Where  the points A and P denote the aphelion (farthest point) and perihelion (closest point) to the Sun (S), respectively. We let   V_A, V_P  be the respective velocities at those orbital extremes. As may be deduced here, points A and P are the only ones in the whole orbit for which the velocities are truly tangential or at right angles to the radius vectors for those positions. Consequently, we can write: 

V = (2π/T) r

where r is the radius vector at the point, and T is the period. 

For the perihelion velocity we have:

V_P = h/ a(1 - e)

where a is the semi-major axis, and e is the eccentricity.

For the velocity at aphelion: 

V_A = h/ a (1 + e)

We see from this that:   V_P   >   V_A

So the planet is always moving faster at perihelion because it is closer to the Sun.

Then the ratio of velocities is:

(V_P/V_A) = (1 + e)/ (1 - e)

In the same exact fashion, the Moon (in its own elliptical orbit)  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 elliptical orbit- rather than falling ONTO the Earth.

Note in the case of the Moon's elliptical orbit,

We refer to the points P and A as the perigee and apogee, with its velocity greater at perigee than at apogee.