Chapter 12, Problem 40QP

(a)

To determine

The semi-major axes of the orbits of the comets.

Answer to Problem 40QP

The semi-major axes of the comet Encke is $2.2\text{AU}$, for the comet Halley is $17.94\text{AU}$ and for the comet Hale-Bopp is $185.67\text{AU}$.

Explanation of Solution

The time period of the Earth around the sun is $1\text{year}$ and the distance for the Earth from the sun is measured as $1\text{AU}$.

The semi-major axis of a celestial body around the planet is almost equal to the average orbital distance of the body given by the Kepler’s law.

Write the expression for the Kepler’s third law for the comparison between the time period and orbital distance.

  ${\left(\frac{T_1^2}{R_1^3}\right)}_{Earth}={\left(\frac{T_2^2}{R_2^3}\right)}_{Comet}$        (I)

Here, $T_1$ is the orbital time period of Earth, $T_2$ is the orbital time period of the comet, $R_1$ is the orbital distance of Earth from the sun and $R_2$ is the orbital distance of the comets.

Conclusion:

For comet Encke:

Substitute $1\text{year}$ for $T_1$, $1\text{AU}$ for $R_1$, $3.3\text{years}$ for $T_2$ and $R_{Encke}$ for $R_2$ in equation (I).

  $\begin{array}{c}\left(\frac{{\left(1\text{year}\right)}^2}{{\left(1\text{AU}\right)}^3}\right)=\left(\frac{{\left(3.3\text{year}\right)}^2}{R_{Encke}^3}\right)\\ R_{Encke}={\left(3.3\right)}^{2/3}\\ =2.2\text{AU}\end{array}$

For comet Halley:

Substitute $1\text{year}$ for $T_1$, $1\text{AU}$ for $R_1$, $76\text{years}$ for $T_2$ and $R_{Halley}$ for $R_2$ in equation (I).

  $\begin{array}{c}\left(\frac{{\left(1\text{year}\right)}^2}{{\left(1\text{AU}\right)}^3}\right)=\left(\frac{{\left(76\text{year}\right)}^2}{R_{Halley}^3}\right)\\ R_{Halley}={\left(76\right)}^{2/3}\\ =17.94\text{AU}\end{array}$

For comet Hale-Bopp:

Substitute $1\text{year}$ for $T_1$, $1\text{AU}$ for $R_1$, $2530\text{years}$ for $T_2$ and $R_{Hale-Bopp}$ for $R_2$ in equation (I).

  $\begin{array}{c}\left(\frac{{\left(1\text{year}\right)}^2}{{\left(1\text{AU}\right)}^3}\right)=\left(\frac{{\left(2530\text{year}\right)}^2}{R_{Hale-Bopp}^3}\right)\\ R_{Hale-Bopp}={\left(2530\right)}^{2/3}\\ =185.67\text{AU}\end{array}$

Thus, the semi-major axes of the comet Encke is $2.2\text{AU}$, for the comet Halley is $17.94\text{AU}$ and for the comet Hale-Bopp is $185.67\text{AU}$.

(b)

To determine

The minimum and maximum distance of the comets Halley and Hale-Bopp from the sun.

Answer to Problem 40QP

The minimum and maximum distance of the comet Halley from the sun is $0.592\text{AU}\&35.28\text{AU}$ respectively and for the comet Hale-Bopp is $0.928\text{AU}\&370.41\text{AU}$ respectively.

Explanation of Solution

The maximum and minimum distances reached by the comet from the sun in their orbits are termed as the Aphelion and Perihelion respectively.

Write the expression for the perihelion distance of the body in its orbit.

  $D_P=A\left(1-e\right)$        (II)

Here, $D_P$ is the perihelion distance of body in the orbit, $A$ is the semi-major axis of the comet’s orbit and $e$ is the eccentricity of the orbit.

Write the expression for the aphelion distance of the body in its orbit.

  $D_A=A\left(1+e\right)$        (III)

Here, $D_A$ is the aphelion distance of body in the orbit.

Conclusion:

For Comet Halley:

Substitute $17.94\text{AU}$ for $A$, $0.967$ for $e$ in equation (II).

  $\begin{array}{c}{D}_P=17.94\text{AU}\left(1-0.967\right)\\ =0.592\text{AU}\end{array}$

Substitute $17.94\text{AU}$ for $A$, $0.967$ for $e$ in equation (III).

  $\begin{array}{c}{D}_A=17.94\text{AU}\left(1+0.967\right)\\ =35.28\text{AU}\end{array}$

For Comet Hale-Bopp:

Substitute $185.67\text{AU}$ for $A$, $0.995$ for $e$ in equation (II).

  $\begin{array}{c}{D}_P=185.67\text{AU}\left(1-0.995\right)\\ =0.928\text{AU}\end{array}$

Substitute $185.67\text{AU}$ for $A$, $0.995$ for $e$ in equation (III).

  $\begin{array}{c}{D}_A=185.67\text{AU}\left(1+0.995\right)\\ =370.41\text{AU}\end{array}$

Thus, the minimum and maximum distance of the comet Halley from the sun is $0.592\text{AU}\&35.28\text{AU}$ respectively and for the comet Hale-Bopp is $0.928\text{AU}\&370.41\text{AU}$ respectively.

(c)

To determine

The region of origin of the comets Encke, Halley and Hale-Bopp in the solar system.

Answer to Problem 40QP

The Encke and Halley’s Comet are considered to be originated in the Kuiper belt whereas the Hale-Bopp comet is considered to be originated in Oort cloud region.

Explanation of Solution

The orbital time period of the comet suggests the region of origin of the Comet in the solar system. The comets that have a smaller time period are termed as the Short period comet and the one which have large time period are the long period comet.

The Encke comet has a very small time period of $3.3\text{years}$ and it suggests that the orbit of the comet is not so far. Therefore, it is considered to be originated from the region of inner Kuiper belt just beyond the Neptune.

The Halley comet is a short period comet with the time period of $76\text{years}$ and this suggests that this comet belongs to some region in the Kuiper belt itself. The Hale-Bopp comet has a very long time period and therefore, it is a long period comet and it is considered to be originated in the Oort cloud region beyond the solar system.

Conclusion:

Thus, the Encke and Halley’s Comet are considered to be originated in the Kuiper belt whereas the Hale-Bopp comet is considered to be originated in Oort cloud region.

(d)

To determine

The comet which is the most and least pristine of the three.

Answer to Problem 40QP

The comet Encke is the least pristine and the comet Hale-Bopp is most pristine.

Explanation of Solution

As the comet comes close to the sun, the comet action takes place due to the sublimation of the ice and gases on the comet. This leads to the wear out of the material of the comet. The more the comet comes near the sun; more it loses its material.

The comet Encke is considered to be the least pristine as it has a very short time period and it encounters with the sun most frequently whereas the comet Hale-Bopp is considered to be the most pristine as it comes near the sun after a very long period.

Conclusion:

Thus, the comet Encke is the least pristine and the comet Hale-Bopp is most pristine.