Problem Set 7
In this problem set, you will use observational data to understand the orbit of a comet. Remember that
comets, asteroids and Kuiper belt objects are discovered because as the Earth moves around the sun, things
that are close to us seem to move relative to far-away stars. This is known as parallax. Below are two
discovery images for a comet. The two images were separated by a time of 1 hr. The scale on the image is
a unit called ‘arcseconds’, which are 1/3600th of a degree.
Use the images below to calculate the angle, in arcseconds, by which the comet appears to
have moved from one image to the next.
The Earth travels around the sun at a speed of 30 km/s.
Calculate the total distance, l, the Earth will have traveled in 1 hr.
The distance to the comet (d, in astronomical units), the length traveled by the Earth (l, in kilometers), and
the angle by which the comet appears to have moved (θ, in arcseconds) are related with the equation
θ=
l
× 0.00138
2d
Calculate the distance, d to the comet.
Figure 1: These are two images of a comet taken 1 hour apart. Each box is 60 arcseconds high and 60
arcseconds across
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In problem set 2, you learned that the closest approach distance (called the perihelion distance), d,
between a planet (or comet, in this case) and the sun is
d = a × (1 − e)
where a is the semi-major axis of the comet and e is the eccentricity (how non-circular the orbit is).
If the comet is observed over time to have an eccentricity of 0.9, and the images above were
taken when the comet was at perihelion, what is the semi-major axis, a, of the comet’s orbit,
in AU?
What is the period of the comet’s orbit, in years? (hint: Kepler’s 3rd law might come in
handy)
Now, we are going to create a scaled-down version of the comet’s orbit. Let 1 AU be equal to 1/8 of an inch.
We learned in problem set 2 that the distance between the foci of an ellipse is 2 × e × a.
Calculate the distance between the foci of your scaled-down ellipse (in inches!).
On a separate page, draw two points separated by the distance you just calculated
Calculate the size of your semi-major axis, in inches, and then multiply by 2.
Now get two pins and a piece of string (sound familiar?). Place the pins on the points you drew, and tie the
string so that its total length is what you just calculated (2 × the semi-major axis, in inches).
Using the pins and string, draw the orbit of the comet, below.
Is this orbit more or less circular-looking than Mercury’s orbit?
Add the sun to your picture.
Draw four dots on your orbit — one where the comet is closest to the sun, one where it is
farthest, and two points somewhere in-between. At each of these dots, draw and label the
direction of the dust tail and the ion tail of the comet.
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