1. Math in the News
Issue 84
Comet ISON Approaches
In this issue we look at the approaching ISON Comet.

2. Learn more about Comet ISON from this video.
http://youtu.be/A1yH_DuC88M

3. Comet ISON
• As ISON
approaches the
sun, it will lose
mass.
• As it emerges from
around the sun,
there are three
possible outcomes
for the comet’s
trajectory.

4. Scenario 1
• ISON emerges
from the sun’s
orbit intact
(diminished in
size) and
continues on its
journey.

5. Scenario 2
• ISON doesn’t
survive its
encounter with the
sun and is
vaporized entirely.

6. Scenario 3
• ISON fragments
and emerges from
the sun’s orbit as a
scattered
collection of small
comet fragments.

7. Sketching the Path
• Let’s assume
Scenario 1 and
look at the
trajectory that the
comet might take.

8. A Parabolic Path
• The path of
Comet ISON can
be approximated
with a parabola.
• A parabola is the
locus of points
equidistant from
the focus and a
segment
perpendicular to
the directrix.

9. A Parabolic Path
• At any point on
the parabola the
distance to the
focus is
congruent to the
vertical distance
to the directrix.

10. A Parabolic Path
• The path of the comet is approximated by the
shape of the parabola, and the sun is at the
focus of this parabola.

11. A Parabolic Path
• The perihelion distance can be used to find
the equation of the parabola in vertex form.

12. A Parabolic Path
y = a(x − h) + k
2
• This is the vertex form of the equation for a
parabola, where (h, k) define the coordinates
of the vertex.

13. A Parabolic Path
y = ax
2
• If you assume the vertex of the parabola is at
the origin, the equation is simplified.

14. A Parabolic Path
1
a=
4f
• To solve for a, use this equation, where f is the
focal distance, or in this case the perihelion
distance.

15. A Parabolic Path
1
a=
4f
• NASA estimates Comet ISON will come within
730,000 miles from the sun. Let’s express this
distance in Astronomical Units (AU). (1 AU is
the distance from Earth to the sun.)

16. A Parabolic Path
1 AU
f
=
93,000,000 miles 730,000 miles
f ≈ 0.0079 AU
1
a=
4f
1
=
4 • 0.0079
≈ 31.65
• NASA estimates Comet ISON will come within
730,000 miles from the sun. Let’s express this
distance in Astronomical Units (AU).

17. A Parabolic Path
y = 31.65x
2
• This is the equation of the parabola, and it is
scaled for Astronomical Units.

18. A Parabolic Path
y = 31.65x
2
• The graph of the parabola shows where it
intersects the Earth’s orbit (y = 1 AU) as it
leaves the sun’s orbit.

19. A Parabolic Path
y = 31.65x
2
• NASA estimates that Comet ISON may come
as close as 0.5 AU to Earth on its way away
from the sun. Assuming it does, it should be a
dramatic event in the night sky!