Chapter 15, Problem 44Q

(a)

To determine

The orbital period Sun grazing comets whose aphelion distance is $100\text{au}$.

Answer to Problem 44Q

Solution:

The orbital period and lifetime of Sun grazing comets are $353.55\text{years}$ and $35355\text{years}$, respectively.

Explanation of Solution

Given data:

The aphelion distance is $100\text{au}$ and the comet can only survive $100$ perihelion passages.

Formula used:

Write the expression for the orbital period from the Kepler law.

$T=a^{3/2}$

Here, the orbital time period is $T$ and the semi major axis is $a$.

Write the expression for the lifetime.

$L=T\times t$

Here, the lifetime is $L$ and the life of the comet is $t$.

Explanation:

Calculate the semi major axis of the orbit.

$a=\frac{y}{2}$

Here, the aphelion distance is $y$.

Substitute $100\text{au}$ for $y$.

$\begin{array}{c}a=\frac{100\text{au}}{2}\\ =50\text{au}\end{array}$

Write the expression for the orbital period.

$T=a^{3/2}$

Substitute $50\text{au}$ for $a$.

$\begin{array}{c}T={\left(50\text{ au}\right)}^{3/2}\\ =353.55\text{years}\end{array}$

The comet can survive for 100 perihelion passages. Therefore, the value of $t$ is $100$.

Write the expression for the lifetime.

$L=T\times t$

Substitute $353.55\text{years}$ for $T$ and $100$ for $t$.

$\begin{array}{c}L=\left(353.55\text{years}\right)\left(100\right)\\ =35355\text{years}\end{array}$

Conclusion:

Hence, the orbital period and lifetime of Sun grazing comets is $353.55\text{years}$ and $35355\text{years}$, respectively.

(b)

To determine

The orbital period and lifetime of Sun grazing comets whose aphelion distance is $1000\text{au}$.

Answer to Problem 44Q

Solution:

The orbital period and lifetime of Sun grazing comets are $11180.34\text{years}$ and $1118034\text{years}$, respectively.

Explanation of Solution

Given data:

The aphelion distance is $1000\text{au}$ and the comet can only survive $100$ perihelion passages.

Formula used:

Write the expression for the orbital period.

$T=a^{3/2}$

Here, the orbital time period is $T$ and the semi major axis is $a$.

Write the expression for the lifetime.

$L=T\times t$

Here, the lifetime is $L$ and the life of the comet is $t$.

Explanation:

Calculate the semi major axis of the orbit.

$a=\frac{y}{2}$

Here, the aphelion distance is $y$.

Substitute $1000\text{au}$ for $y$.

$\begin{array}{c}a=\frac{1000\text{au}}{2}\\ =500\text{au}\end{array}$

Write the expression for the orbital period.

$T=a^{3/2}$

Substitute $500\text{au}$ for $a$.

$\begin{array}{c}T={\left(500\text{ au}\right)}^{3/2}\\ =11180.34\text{years}\end{array}$

Write the expression for the lifetime.

$L=T\times t$

Substitute $11180.34\text{years}$ for $T$ and $100$ for $t$.

$\begin{array}{c}L=\left(11180.34\text{years}\right)\left(100\right)\\ =1118034\text{years}\end{array}$

Conclusion:

Hence, the orbital period and lifetime of Sun grazing comets are $11180.34\text{years}$ and $1118034\text{years}$, respectively.

(c)

To determine

The orbital period and lifetime of Sun grazing comets whose aphelion distance is $10000\text{au}$.

Answer to Problem 44Q

Solution:

The orbital period and lifetime of Sun grazing comets are $353553.4\text{years}$ and $1118034\text{years}$, respectively.

Explanation of Solution

Given data:

The aphelion distance is $10000\text{au}$ and the comet can only survive $100$ perihelion passages.

Formula used:

Write the expression for the orbital period.

$T=a^{3/2}$

Here, the orbital time period is $T$ and the semi major axis is $a$.

Write the expression for the lifetime.

$L=T\times t$

Here, the lifetime is $L$ and the life of the comet is $t$.

Explanation:

Calculate the semi major axis of the orbit.

$a=\frac{y}{2}$

Here, the aphelion distance is $y$.

Substitute $10000\text{au}$ for $y$.

$\begin{array}{c}a=\frac{10000\text{au}}{2}\\ =5000\text{au}\end{array}$

Write the expression for the orbital period.

$T=a^{3/2}$

Substitute $5000\text{au}$ for $a$.

$\begin{array}{c}T={\left(5000\right)}^{3/2}\\ =353553.4\text{years}\end{array}$

Write the expression for the lifetime.

$L=T\times t$

Substitute $353553.4\text{years}$ for $T$ and $100$ for $t$.

$\begin{array}{c}L=\left(353553.4\text{years}\right)\left(100\right)\\ =35355340\text{years}\end{array}$

Conclusion:

Hence, the orbital period and lifetime of Sun grazing comets are $353553.4\text{years}$ and $35355340\text{years}$, respectively.

(d)

To determine

The orbital period and lifetime of Sun grazing comets whose aphelion distance is $100000\text{au}$.

Answer to Problem 44Q

Solution:

The orbital period and lifetime of Sun grazing comets are $11180339.9\text{years}$ and $1118033990\text{years}$, respectively.

Explanation of Solution

Given data:

The aphelion distance is $100000\text{au}$ and the comet can only survive $100$ perihelion passages.

Formula used:

Write the expression for the orbital period.

$T=a^{3/2}$

Here, the orbital time period is $T$ and the semi major axis is $a$.

Write the expression for the lifetime.

$L=T\times t$

Here, the lifetime is $L$ and the life of the comet is $t$.

Explanation:

Calculate the semi major axis of the orbit.

$a=\frac{y}{2}$

Here, the aphelion distance is $y$.

Substitute $100000\text{au}$ for $y$.

$\begin{array}{c}a=\frac{100000\text{au}}{2}\\ =50000\text{au}\end{array}$

Write the expression for the orbital period.

$T=a^{3/2}$

Substitute $50000\text{au}$ for $a$.

$\begin{array}{c}T={\left(50000\text{ au}\right)}^{3/2}\\ =11180339.9\text{years}\end{array}$

Write the expression for the lifetime.

$L=T\times t$

Substitute $11180339.9\text{years}$ for $T$ and $100$ for $t$.

$\begin{array}{c}L=\left(11180339.9\text{years}\right)\left(100\right)\\ =1118033990\text{years}\end{array}$

Conclusion:

Hence, the orbital period and lifetime of Sun grazing comets are $11180339.9\text{years}$ and $1118033990\text{years}$, respectively.