- #1

gelfand

- 40

- 3

## Homework Statement

A comet moves around a stat in ##xy## plane along elliptical orbit, described by

$$

0.16 x^2 + y^2 = 4

$$

where ##x, y## are in ##AU##

1) Sketch the comet in the ##x,y## coordinate system denoting all orbit parameters

2) Find the semi major and minor axes of the orbit in ##AU##

3) Assuming the star is located at ##x < 0##, find the eccentricity of the orbit

and the ##x, y## coordinates of the star

4) If the orbital speed of this comet at the perihelion is ##100 ## km/s, find

its orbital speed at aphelion

5) Using the energy conservation approach find the approximate mass of the star

to an order of magnitude.

## Homework Equations

Eccentricity ##e = \sqrt{1 - \frac{b^2}{a^2}}##f ## = \sqrt{a^2 - b^2}## , I'm not sure what this is other than the distancebetween where the semi-major and minor axis meet and the location of the star.

## The Attempt at a Solution

For part 1) I have drawn a sketch.For part 2) I note that the given expression is $$

0.16 x^2 + y^2 = 4

$$

Plugging ##x = 0## into this gives ##y^2 = 4## so ##y = 2##. Then setting ##y = 0##

gives ##x^2 = \frac{4}{0.16}## so ##x = 5##.

This gives us our semi-major axis at ##5## and the semi-minor at ##2##, both in ##AU##.

Part 3) Using the formula for eccentricity as

$$

e = \sqrt{1 - \frac{4}{25}} = \frac{\sqrt{21}}{5} \approx 0.917

$$

The ##x, y## coordinates of the star are found using ##f = \sqrt{a^2 - b^2}## which

is

$$

f = \sqrt{25 - 4} = \sqrt{21} \approx 4.583

$$

From this we note that the location of the star is ##(-4.583, 0)## to 3 decimal

places.

part 4)

I know that the perihelion has the fastest speed - but I'm not sure how to go

about finding the speed of the aphelion given this information.

Should I be setting up some kind of differential equation? Or is there an

appropriate formula?

part 5)

I'm not sure how to consider this either. If the energy is conserved then I have

the definition

$$

W + PE_0 + KE_0 =

PE_f + KE_f + \text{Energy(Lost)}

$$

Given that there's no work being done in the motion and there's no energy lost

due to friction we have

$$

PE_0 + KE_0 =

PE_f + KE_f

$$

I'm not sure what to do with this though, in terms of finding the mass.

There's no 'at rest' for this system is there, in the way that this expression

is often used.

The potential energy would be a result of the gravitational field, which is

$$

\text{Gravitational force } = \frac{-GMm}{x}

$$

Where ##x## is the distance between the orbiting body and the star, ##G## is the

gravitational constant, ##M## is the mass of the star (what we want) and ##m## is

the mass of the orbiting body (which we don't have either).Thanks