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- According to Newton's Law of Gravity, the acceleration due to gravity at the earth's surface is $\displaystyle g = \frac{GM}{r^2}$ where $M$ is the mass of the earth and $r$ is the radius of the earth. Using $g \approx 9.81 m/sec^2$ and $r \approx 6.37 \times 10^6 m$, approximate the mass of the earth.
- We are unable to stand on the surface of the sun to measure the acceleration of its gravity. But we can find the mass using Kepler's third law, that $\displaystyle \frac{T^2}{a^3} = \frac{4\pi^2}{GM}$, where $T$ is the orbital period, $a$ is the semi-major axis, and $M$ is the mass of the sun. For earth's orbit, $T \approx 3.156 \times 10^7 sec$ and $a \approx 1.496 \times 10^{11} m$. What is the approximate mass of the sun?
- Europa orbits Jupiter with semi-major axis about $6.709 \times 10^8 m$ and period $3.068 \times 10^5 sec$. What is the approximate mass of Jupiter?
- Phobos orbits Mars with semi-major axis about $9.376 \times 10^6 m$ and period $2.755 \times 10^4 sec$. What is the approximate mass of Mars?
- The European Space Agency recently landed the Philae probe on Comet 67P/Churyumov-Gerasimenko (67P/C-G). Comet 67P/C-G is a short period comet with an orbital period of approximately $2.03 \times 10^8 sec$. It has a perihelion of approximately $1.86 \times 10^{11} m$. What is its aphelion?
- One group of short period comets typically have orbits that take them from aphelion at about the distance of Jupiter from the Sun to the inner solar system for perihelion. This gives them a typical semi-major axis of around $10^{11} m$ to $5 \times 10^{11} m$. What would the periods be for such comets?
- Observing that the fossil record shows major extinction events tend to
happen about every 25 million years, it has been theorized that perhaps
there is a red or brown dwarf star orbiting the sun, called
*Nemesis*, that has an orbit that disturbs trans-Neptunian bodies every 25 million years or so, leading to much higher chances of cometary impacts on the inner planets. What would the semi-major axis be for an orbit with a period of 25 million years (remember that $\mu$ is given above in units of seconds, not years)? Note: astronomical surveys have looked for such an object and not found anything, so the Nemesis hypothesis is currently considered incorrect. - Speed is the magnitude of the velocity vector. Show that at perihelion and aphelion, speed is $c/r$ where $c$ is the angular velocity. Hint: there is something special about perihelion and aphelion that will simplify the calculations.
- Halley's comet has a period of about $2.38 \times 10^9 sec$, a perihelion of about $8.766 \times 10^{10} m$, and aphelion of about $5.25 \times 10^{12} m$ (you could actually deduce this last value from the first two using Kepler's third law). What is the speed of Halley's comet at perihelion? Note: it is usually easiest to find $c$ by remembering the area of the orbit is $\displaystyle \frac{cT}{2}$ from our derivation of Kepler's second law.
- Halley's comet has a period of about $2.38 \times 10^9 sec$, a perihelion of about $8.766 \times 10^{10} m$, and aphelion of about $5.25 \times 10^{12} m$ (you could actually deduce this last value from the first two using Kepler's third law). What is the speed of Halley's comet at aphelion? Note: it is usually easiest to find $c$ by remembering the area of the orbit is $\displaystyle \frac{cT}{2}$ from our derivation of Kepler's second law.
- Suppose a comet is observed with a perihelion of $1.30 \times 10^{11} m$ and moving with a speed of $4.50 \times 10^4 m/sec$ at perihelion. What is the period of the orbit of the comet? Hint: recall that orbits take the form $\displaystyle r = \frac{1}{GM/c^2 - A\cos(\theta)}.$ With this data it is easiest to solve for $c$ using the speed data and then solve for $A$. Then use these values to find the area of the orbit and from that the period.
- Suppose a comet is observed with a perihelion of $7.50 \times 10^{10} m$ and moving with a speed of $5.00 \times 10^4 m/sec$ at perihelion. What is the period of the orbit of the comet? Hint: recall that orbits take the form $\displaystyle r = \frac{1}{GM/c^2 - A\cos(\theta)}.$ With this data it is easiest to solve for $c$ using the speed data and then solve for $A$. Then use these values to find the area of the orbit and from that the period.
- Suppose a comet has a perihelion of $5.50 \times 10^{10} m.$ What speed would the comet need to have at perihelion to be on a "parabolic" orbit and escape the solar system?
- Suppose a comet has a perihelion of $8.00 \times 10^{10} m.$ What speed would the comet need to have at perihelion to be on a "parabolic" orbit and escape the solar system?
- Suppose Bennett's Law of Gravity says that the gravitational force between two masses is given by $\displaystyle -\frac{GMm}{r}$ (where the denominator is $r$ instead of $r^2$). Show the Kepler's second law still holds for this model.
- With Bennett's Law of Gravity from the previous problem, the key equation in "Writing an Equation for the Orbit" becomes $$ r'' - r(\theta')^2 = - \frac{GM}{r}. $$ Show that if you start from here and make the substitution $u = 1/r$ and write the equation as a differential equation for $u$ as a function of $\theta$, you get a non-linear equation.
- More generally, show that the law of gravitational force $\displaystyle -\frac{GM}{r^n}$ leads to a linear equation for $u$ as a function of $\theta$ only when $n=2$ or $n=3$.
- From Kepler's first law, we know that if we have an equation for $u = 1/r$ as a function of $\theta$, then the solution must have the form $u = B - A\cos(\theta)$, since that is the general form for an ellipse with a focus at the origin and aphelion at $\theta = 0$. Show that this is a solution for the law of gravitational force $\displaystyle -\frac{GM}{r^n}$ only when $n=2$.
- By definition, an ellipse consists of all points with the property that the sum of the distances from the point to two foci is a fixed quantity. Pick the coordinate system so the foci are $(f,0)$ and $(-f,0)$ and the sum of the distances be $2a$, and let $b^2 = a^2 - f^2$. Show the equation of the ellipse is $$ \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1$$
- Show that the polar equation $\displaystyle r = \frac{ep}{1-e\cos(\theta)}$ becomes $$\left(\frac{x-f}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1 $$ with $\displaystyle f = \frac{e^2p}{1-e^2}$, $\displaystyle a = \frac{ep}{1-e^2}$, and $\displaystyle b = \frac{ep}{\sqrt{1-e^2}}$.
- Show that $b^2 + f^2 = a^2$ for the values of $a$, $b$, and $f$ in problem 20. Note: problems 19-21 show that $$ r = \frac{ep}{1-e\cos(\theta)}$$ is the equation of an ellipse with a focus at the origin, since you now have the equation in problem 20 is that of the ellipse from problem 19, only translated by $f$, the distance from the focus to the origin.

You may use $G \approx 6.67 \times 10^{-11} m^3 kg^{-1} sec^{-2}$ for the gravitational constant in the following problems.

For the next problems you may find it easier to use the

You may find it interesting to compare your answers to problems 9 and 10 to the mean speed of the Earth in orbit, which is about $30,000 m/sec$. Note that the Earth's orbit is close to circular (eccentricity 0.0167) so the speed only varies a few percent over the year.

Answers to the odd numbered exercises.

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©2010 Andrew G. Bennett