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and astronomy. It relates to the rate at which area is swept by a planet.
Actually it is a measure of twice the area swept in a unit of time. Chapter 25
will explain this term in more detail. For circular orbits, we will see that areal
velocity is related to the radius of the orbit (distance to the Sun), the mass of
the Sun, and the Gravitational Constant. In later chapters we will see that
aereal velocity is related to the semilatus rectum of the elliptical orbit.
necessary to know how to compare the velocity of different planets whose
hodographs are drawn on the same diagram. First we will do this for circular orbits,
here in Chapter 26. Here is the paradox encountered: We know that for circular
orbits within a solar system the velocity is less when the radius is greater. But when
we represent the hodographs of both orbits on the same diagram, the larger orbit will
of course have a larger hodograph circle. This would seem to indicate that the
velocity is greater. Since we know that the velocity is actually less, we realize that
we must scale the hodographs. We must progressively scale the velocity to be less
than it seems to be as the radius becomes greater. Chapter 26 is our first
introduction to this kind of scaling technique in Orbits Explained. This scaling
technique is so powerful that it will lead us to the a priori proof of Kepler's Third Law
and to the Energy Equation for orbits.
hodograph and derive an equation from it which reveals an exact mathematical
relationship between the square of total velocity and the following properties -
tangential velocity, radius to the Sun, and semimajor axis length. In physics, the square
of velocity is related to the energy possessed by a moving mass. That energy is called
kinetic energy. So deriving an equation for total velocity squared from the hodograph is
a step towards understanding the energy of a planet in motion.
call "b" which is at the tip of the minor axis. This is truly a special place. In the next two
chapters it will reveal to us how to scale the hodographs for elliptical orbits. In other
words we will refine our scaling technique. The special thing about position b is that it is
at a distance from the Sun that is equal to the length of the semimajor axis. We can
exploit this fact by examining hypothetical planets in the same solar system whose orbits
have the same semimajor axis length but different eccentricities. Since the hypothetical
planets are at equal distances from the Sun, the gravitational force on them must be
equal. Chapter 28 helps us visualize what happens at b.
orbits have equal semimajor axis lengths but different eccentricities. By
inspection we see that the hodograph misleads us. It tells us to expect that
when the planet is at position b, the tangential velocity is proportional to the
semilatus rectum. This is false. But it will show us how to scale the hodograph
for elliptical orbits as we will see in Chapters 30 and 31.
measured in radians, the sine of the angle equals the angle itself for tiny angles.
In Chapter 31 we will need a similar proof for tangents of tiny angles so we
conjured one up. It is presented here in Chapter 30.
the tiny angle swept by two hypothetical planets with orbits of equal semimajor
axis length when they are at position b. One of the hypothetical planets is in an
elliptical orbit. The other is in a circular orbit. When analyzed for a tiny instant of
time we see that these hodographs of the hypothetical planets yield an equation
relating the change in angle of the circular orbit compared to the change in angle
for the elliptical orbit. The equation will turn out to be false and help us to find the
proper scaling method for elliptical orbits of equal semimajor axis length.
equation from Chapter 31 leads us to expect the force to vary for the hypothetical
planets according to the squares of their semiminor axis lengths when the planets are
at position b. Since the distance to the Sun is equal for all these hypothetical planets
at position b ( it is equal to their common semimajor axis length) the force must also be
the same. And so we see the correction that must be made in terms of scaling for
elliptical orbits of equal semimajor axis lengths whose hodographs are drawn on the
same diagram.
for elliptical orbits. We use the findings from Chapter 32 to teach us that the scaling
method for the hodograph of elliptical orbits yields a true relationship between
tangential velocity and semiminor axis length. We learn that tangential velocity is
proportional to semiminor axis length for orbits of equal semimajor axis length when
the planet is at position b. This phenomenal relationship leads directly to the a priori
proof for Kepler's Third Planetary Law regarding the periods of elliptical orbits.
drawn on the same diagram. The analysis yields a more general scaling factor that
can be applied to hodograph comparisons for orbits of equal or unequal semimajor
axis lengths. We will apply this scaling method in Chapter 35 to learn about escape
velocity.
the same diagram. One orbit is a circle. The other is the extremely eccentric
orbit with a semimajor axis length approaching infinity. We use our scaling
method from Chapter 34 to learn about the velocity of the planet in the
eccentric orbit at perihelion and show that this velocity is equal to escape
velocity. We develop a formula comparing this escape velocity at perihelion to
the velocity of a hypothetical planet in a circular orbit with a radius equal to the
perihelion distance.
36 to 43
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