Chapter 11, Problem 40Q

(a)

To determine

The radius of the orbit of Mars Global Surveyor spacecraft given that, the orbital period of the Mars Global Surveyor spacecraft is $\text{117 minutes}$. Use table 11-3 for necessary data.

Answer to Problem 40Q

Solution:

$\text{3770 km}$

Explanation of Solution

Given data:

The orbital periodic time for Mars Global Surveyor spacecraft is $117$ minutes.

Mass of Mars is $6.418\times {10}^{23}\text{ kg}$ (provided in table 11-3).

Formula used:

Write the expression for Newton’s form of Kepler’s third law,

$P^2=\left[\frac{4{\pi }^2}{G\left(m_1+m_2\right)}\right]r^3$

Here, $P$ is the sidereal period of the orbit in seconds, $G$ is the universal constant of gravitation, $m_1$ is the mass of the first object in kilograms, $m_2$ is the mass of the second object in kilograms and $r$ is the radius of the orbit in meters.

Explanation:

The spacecraft’s orbital periodic time $P$ is $117$ minutes, covert this into seconds,

$\begin{array}{c}P\text{ = 117 minutes}\left(\frac{\text{60 seconds}}{\text{1 }\text{}\text{minute}}\right)\\ \text{= 7020 seconds}\end{array}$

Recall the expression for Newton’s form of Kepler’s third law,

$P^2=\left[\frac{4{\pi }^2}{G\left(m_1+m_2\right)}\right]r^3$

Substitute $M$ for $\left(m_1+m_2\right)$ (as mass of the spacecraft is very small as compared to the mass of Mars) and rearrange the above expression for $r$,

$r\text{ = }\sqrt[\text{3}]{\frac{GM{P}^2}{{\text{4π}}^{\text{2}}}}$

Here, M is the mass of Mars.

Substitute $6.67\times {10}^{-11}{\text{Nm}}^2{\text{kg}}^{-2}$ for $G$, $6.418\times {10}^{23}\text{kg}$ for $M$ and $7020$ seconds for $P$,

$\begin{array}{c}r\text{ = }\sqrt[\text{3}]{\frac{GM{P}^2}{{\text{4π}}^{\text{2}}}}\\ \text{= }\sqrt[\text{3}]{\frac{\left(\text{6}{\text{.67×10}}^{\text{-11}}\text{N}\cdot {\text{m}}^{\text{2}}\text{/}\text{}{\text{kg}}^{\text{2}}\right)\left(\text{6}{\text{.418×10}}^{\text{23}}\text{ kg}\right){\left(\text{7020 }\text{}\text{}\text{s}\right)}^{\text{2}}}{\text{4}\cdot {\left(3.14\right)}^2}}\\ \text{= 3770 km}\end{array}$

Conclusion:

Hence, the radius of Mars Global Surveyor spacecraft is $3770$ km.

(b)

To determine

The average altitude of Mars Global Surveyor spacecraft above the surface of Mars given that, it was in the orbit of Mars for $\text{117}$ minutes. Use table 11-3 for necessary data.

Answer to Problem 40Q

Solution:

$\text{373 km}$.

Explanation of Solution

Given data:

The orbital periodic time for Mars Global Surveyor spacecraft is $\text{117 minutes}$.

Formula used:

Write the relation between the radius and the diameter of Mars,

$r\text{ = }\frac{\text{diameter of Mars}}{\text{2}}$

Here, $r$ is the radius of Mars.

Explanation:

Recall the expression for the radius of the orbit,

$r\text{ = }\frac{\text{diameter of Mars}}{\text{2}}$

Substitute $6794$ km for the diameter of Mars,

$\begin{array}{c}r=\frac{\text{6794 km}}{\text{2}}\\ \text{= 3397 km}\end{array}$

From part (a), radius of Mars is 3770 km.

The radius of orbit of Mars Global Surveyor spacecraft above the surface is $3397$ km. Therefore, the average altitude of spacecraft above the Martian surface is,

$\begin{array}{c}h=\text{ 3770 km}-\text{3397 km}\\ \text{= 373 km}\end{array}$

Conclusion:

Hence, the average altitude of Mars Global Surveyor spacecraft is more than the radius of Mars by 373 km.

(c)

To determine

The reason for the possibility of observing the entire surface of the planet from the orbit of the Mars Global Surveyor spacecraft passing through the poles of Mars.

Answer to Problem 40Q

Solution:

The orbital plane of Mars Global Surveyor spacecraft remains fixed whereas, Mars rotates continuously on its orbit. This makes it possible for the spacecraft to observe the entire surface of Mars.

Explanation of Solution

Introduction:

Polar orbits are those orbits, in which when an object (usually satellites) orbits those objects (satellites) that are able to pass through both the poles of the object (usually planets) that is being orbited.

Explanation:

Satellites in polar orbits are able to observe the entire surface of the planet as the orbital plane in which the motion of the satellite is fixed, remains the same. Therefore, while Mars is rotating about its axis, spacecrafts at the polar orbit like Mars Global Surveyor are able to observe the entire surface of the planet, Mars in this case.

Conclusion:

The orbital plane of the Mars Global Surveyor spacecraft is fixed with respect to the orbital plane of Mars, so, it is possible for it to observe the entire surface of Mars.