Download Ellipses in Reality

ELLIPSES IN REALITY
An Example of Ellipses in
Reality
 Ellipses appear in
planetary orbits.
 Although most people
think that planets orbit
stars in a circular
fashion, their orbits are
actually elliptical.
 The object that the
planet is orbiting is one
of the foci of the
ellipse.
Example
 Say that a comet is in
orbit around a star. It is 2
AU from the star when it
passes closest to the star
in its orbit, and is 8 AU
from the star when it is
furthest from the star.
How far is it from the
star when it is at the
point in its orbit shown in
the diagram to the right?
Essentially, what is X?
Solution
 At the point given in the problem, the comet is at
one of the edges of the minor axis of its orbit. We
need to find the length of the minor axis.
 We know that the two vertices and the two foci
are equidistant from the center of the orbit.
 Call the distance from the center to one of the
foci f.
 Call the distance from the center to one of the
vertices b.
Solution
 Using the information given in the problem,
we can form a system of equations involving f
and b.
 The distance from the sun to the comet at its
closest is b – f, so we can write b – f = 2.
 Similarly, the distance from the sun to the
comet at its furthest point is b + f, so we can
write b + f = 8. Solving these two equations
gives us b = 5 and f = 3.
Solution
 In ellipses, the relationship f2 = b2 – a2 holds
when b > a. We know b and f, so we can solve
for a, which is one half of the length of the
minor axis.
32 = 52 – a2
a2 = 16
a=4
Solution
 We can now form a right
triangle between the
center, the sun, and the
comet.
 You may recognize this
as a 3-4-5 right triangle.
This tells us that X = 5.
 The distance from the
sun to the comet is 5 AU
at this point in its orbit.