91Ó°ÊÓ

Given data:

• The spherical planet has mass $\text{M}$.
• The spherical planet has volume $\mathrm{V}$.
• The spherical planet has radius $\mathrm{R}$.
• The spherical planet has diameter $\mathrm{D}=2\mathrm{R}$.

The acceleration due to gravity of a planet is the ratio of the product of the Universal Gravitational constant and mass of the planet to the square of their radius.

The equation of the density of the planet is expressed as-

$$\begin{gathered} \mathrm{ÒÏ}=\frac{\mathrm{M}}{\mathrm{V}} \\ \mathrm{M}=\mathrm{ÒÏV} \end{gathered}$$

Here,$\mathrm{ÒÏ}$is the density of the planet,$\mathrm{M}$ is the mass of the planet and$\mathrm{V}$ is the volume of the planet.

The equation of the acceleration due to gravity is expressed as-

$\mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{R}}^2}$

Here,$\mathrm{g}$ is the acceleration due to gravity, $\mathrm{G}$is the gravitational constant and $\mathrm{R}$is the radius of the spherical planet.

For$\mathrm{M}=\mathrm{ÒÏV}$ and $\mathrm{D}=2\mathrm{R}$,

$\begin{array}{rcl}\mathrm{g}& =& \frac{\mathrm{GÒÏV}}{{\left({\displaystyle \frac{\mathrm{D}}{2}}\right)}^2}\\ & =& \frac{4\mathrm{GÒÏV}}{{\mathrm{D}}^2}...........................................\left(1\right)\\ & & \end{array}$

From the expression of the acceleration due to gravity, the graph of $\mathrm{g}$as a function $\mathrm{D}$of has been described below-

This graph is applicable for all the eight major planets if the average density of the planets is constant. However, if the average density of the planet changes, then the graph will also change.

Thus, the graph of$\mathrm{g}$ versus$\mathrm{D}$ will change with the change in the average density.

From the equation $\left(1\right)$the equation of the average density of the planets is expressed as-

$\mathrm{ÒÏ}=\frac{{\mathrm{gD}}^2}{4\mathrm{GV}}............................\left(2\right)$

As the planets are spherical in nature, then the equation of the volume of the planet can be expressed as-

$\mathrm{V}=\frac{4}{3}{\mathrm{Ï€¸é}}^3$

Then equation $\left(2\right)$becomes-

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{{\mathrm{gD}}^2}{4\mathrm{G}×\left({\displaystyle \frac{4}{3}}{\mathrm{Ï€¸é}}^3\right)}\\ & =& \frac{3{\mathrm{gD}}^2}{16{\mathrm{GÏ€¸é}}^3}\end{array}$

For $\mathrm{D}=2\mathrm{R}$,

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{3g{D}^2}{16G\mathrm{Ï€}{\left({\displaystyle \frac{D}{2}}\right)}^3}\\ & =& \frac{24g}{16G\mathrm{Ï€}D}........................................\left(3\right)\end{array}$

For the planet Mercury,

Substituting $\mathrm{D}=4879\mathrm{km}$,$\mathrm{G}=6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2$and$\mathrm{g}=3.7\mathrm{m}/{\mathrm{s}}^2$in the equation 3),

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{24×3.7\mathrm{m}/{\mathrm{s}}^2}{16×6.7×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×3.14×4879\mathrm{km}}\\ & =& \frac{24×3.7\mathrm{m}/{\mathrm{s}}^2}{3.35×{10}^{-09}{\mathrm{Nm}}^2/{\mathrm{kg}}^2×4879×{10}^3\mathrm{m}}\\ & =& \frac{24×3.7\mathrm{m}/{\mathrm{s}}^2}{0.0163{\mathrm{Nm}}^3/{\mathrm{kg}}^2}\\ & =& 5447.85{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\end{array}$

Hence, further as,

$\begin{array}{rcl}\mathrm{ÒÏ}& =& 5447.85{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\\ & =& 5447.85{\mathrm{kg}}^2.\mathrm{m}/{\mathrm{m}}^3.{\mathrm{s}}^2×\frac{1}{\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\\ & =& 5447.85\mathrm{kg}/{\mathrm{m}}^3\end{array}$

For the planet Venus,

Substituting$\mathrm{D}=12104\mathrm{km}$, $\mathrm{G}=6.67×{10}^{-11}{\mathrm{Nm}}^2/{\mathrm{kg}}^2$and$\mathrm{g}=8.9\mathrm{m}/{\mathrm{s}}^2$in the equation $\left(3\right)$,

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{24×8.9\mathrm{m}/{\mathrm{s}}^2}{16×6.017×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×3.14×12014\mathrm{km}}\\ & =& \frac{24×8.9\mathrm{m}/{\mathrm{s}}^2}{3.35×{10}^{-09}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×12014×{10}^3\mathrm{m}}\\ & =& \frac{24×3.7\mathrm{m}/{\mathrm{s}}^2}{0.0405\mathrm{N}.{\mathrm{m}}^3/{\mathrm{kg}}^2}\\ & =& 2192.59{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\end{array}$

Hence, further as,

$$\begin{gathered} \mathrm{ÒÏ}=2192.59{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2 \\ =2192.59{\mathrm{kg}}^2.\mathrm{m}/{\mathrm{m}}^3.{\mathrm{s}}^2×\frac{1}{\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2} \\ =2192.59\mathrm{kg}/{\mathrm{m}}^3 \end{gathered}$$

For the planet Earth,

Substituting $\mathrm{D}=12756\mathrm{km}$,$\mathrm{G}=6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2$and$\mathrm{g}=9.8\mathrm{m}/{\mathrm{s}}^2$in the equation

$\left(3\right)$

localid="1655123018356" $\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{24×9.8\mathrm{m}/{\mathrm{s}}^2}{16×6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×3.14×6792\mathrm{km}}\\ & =& \frac{24×9.8\mathrm{m}/{\mathrm{s}}^2}{3.35×{10}^{-09}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×6792×{10}^3\mathrm{m}}\\ & =& \frac{24×9.8\mathrm{m}/{\mathrm{s}}^2}{0.0427\mathrm{N}.{\mathrm{m}}^3/{\mathrm{kg}}^2}\\ & =& 2079.62.{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\end{array}$

Hence, further as,

localid="1655122399974" $\begin{array}{rcl}\mathrm{ÒÏ}& =& 2079.62{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\\ & =& 2079.62{\mathrm{kg}}^2.\mathrm{m}/{\mathrm{m}}^3.{\mathrm{s}}^2×\frac{1}{\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\\ & =& 2079.62\mathrm{kg}/{\mathrm{m}}^3\end{array}$

For the planet Mars,

Substituting$D=6792km$,$\mathrm{G}=6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2$and$\mathrm{g}=3.7\mathrm{m}/{\mathrm{s}}^2$in the equation$\left(3\right)$

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{24×3.7\mathrm{m}/{\mathrm{s}}^2}{16×6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×3.14×6792\mathrm{km}}\\ & =& \frac{24×3.7\mathrm{m}/{\mathrm{s}}^2}{3.35×{10}^{-09}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×6792×{10}^3\mathrm{m}}\\ & =& \frac{24×3.7\mathrm{m}/{\mathrm{s}}^2}{0.0227\mathrm{N}.{\mathrm{m}}^3/{\mathrm{kg}}^2}\\ & =& 3911.89k{g}^2.m/N.m^3.s^2\end{array}$

Hence, further as,

$\begin{array}{rcl}\mathrm{ÒÏ}& =& 3911.89{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\\ & =& 3911.89{\mathrm{kg}}^2.\mathrm{m}/{\mathrm{m}}^3.{\mathrm{s}}^2×\frac{1}{\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\\ & =& 3911.89\mathrm{kg}/{\mathrm{m}}^3\end{array}$

For the planet Jupiter,

Substituting$\mathrm{D}=142,984\mathrm{km}$,$\mathrm{G}=6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2$and$\mathrm{g}=23.1\mathrm{m}/{\mathrm{s}}^2$in the equation (3),

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{24×23.1\mathrm{m}/{\mathrm{s}}^2}{16×6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×3.14×142,984\mathrm{km}}\\ & =& \frac{24×23.1\mathrm{m}/{\mathrm{s}}^2}{3.35×{10}^{-09}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×142,984×{10}^3\mathrm{m}}\\ & =& \frac{24×23.1\mathrm{m}/{\mathrm{s}}^2}{0.4789\mathrm{N}.{\mathrm{m}}^3/{\mathrm{kg}}^2}\\ & =& 1157.65{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{kg}}^2\end{array}$

Hence, further as,

$\begin{array}{rcl}\mathrm{ÒÏ}& =& 1157.65{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\\ & =& 1157.65{\mathrm{kg}}^2.\mathrm{m}/{\mathrm{m}}^3.{\mathrm{s}}^2×\frac{1}{\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\\ & =& 1157.65\mathrm{kg}/{\mathrm{m}}^3\end{array}$

For the planet Saturn,

Substituting$\mathrm{D}=120536\mathrm{km}$,$\mathrm{G}=6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2$and$\mathrm{g}=9.0\mathrm{m}/{\mathrm{s}}^2$in the equation (3),

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{24×9.0\mathrm{m}/{\mathrm{s}}^2}{16×6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×3.14×120536\mathrm{km}}\\ & =& \frac{24×9.0\mathrm{m}/{\mathrm{s}}^2}{3.35×{10}^{-09}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×120536×{10}^3\mathrm{m}}\\ & =& \frac{24×9.0\mathrm{m}/{\mathrm{s}}^2}{0.4037\mathrm{N}.{\mathrm{m}}^3/{\mathrm{kg}}^2}\\ & =& 535.05{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\end{array}$

Hence, further as,

$\begin{array}{rcl}\mathrm{ÒÏ}& =& 535.05{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\\ & =& 535.05{\mathrm{kg}}^2.\mathrm{m}/{\mathrm{m}}^3.{\mathrm{s}}^2×\frac{1}{\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\\ & =& 535.05\mathrm{kg}/{\mathrm{m}}^3\end{array}$

For the planet Uranus,

Substituting$\mathrm{D}=51118\mathrm{km}$,$G=6.67×{10}^{-11}N.m^2/k{g}^2$and $\mathrm{g}=8.7\mathrm{m}/{\mathrm{s}}^2$in the equation (3),

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{24×8.7\mathrm{m}/{\mathrm{s}}^2}{16×6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×3.14×51118\mathrm{km}}\\ & =& \frac{24×8.7\mathrm{m}/{\mathrm{s}}^2}{3.35×{10}^{-09}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×51118×{10}^3\mathrm{km}}\\ & =& \frac{24×8.7\mathrm{m}/{\mathrm{s}}^2}{0.1712\mathrm{N}.{\mathrm{m}}^3/{\mathrm{kg}}^2}\\ & =& 1219.62{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\end{array}$

Hence, further as,

$\begin{array}{rcl}\mathrm{ÒÏ}& =& 1219.62{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\\ & =& 1219.62{\mathrm{kg}}^2.\mathrm{m}/{\mathrm{m}}^3.{\mathrm{s}}^2×\frac{1}{\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\\ & =& 1219.62\mathrm{kg}/{\mathrm{m}}^3\end{array}$

For the planet Neptune,

Substituting $\mathrm{D}=49528\mathrm{km}$, $\mathrm{G}=6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2$and $\mathrm{g}=11.0\mathrm{m}/{\mathrm{s}}^2$in the equation (3),

$\begin{array}{rcl}\mathrm{ÒÏ}& =& \frac{24×11.0\mathrm{m}/{\mathrm{s}}^2}{16×6.67×{10}^{-11}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×3.14×49525\mathrm{km}}\\ & =& \frac{24×11.0\mathrm{m}/{\mathrm{s}}^2}{3.35×{10}^{-09}\mathrm{N}.{\mathrm{m}}^2/{\mathrm{kg}}^2×49525×{10}^3\mathrm{m}}\\ & =& \frac{24×11.0\mathrm{m}/{\mathrm{s}}^2}{0.1659\mathrm{N}.{\mathrm{m}}^3/{\mathrm{kg}}^2}\\ & =& 1591.32{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\end{array}$

Hence, further as,

$\begin{array}{rcl}\mathrm{ÒÏ}& =& 1591.32{\mathrm{kg}}^2.\mathrm{m}/\mathrm{N}.{\mathrm{m}}^3.{\mathrm{s}}^2\\ & =& 1591.32{\mathrm{kg}}^2.\mathrm{m}/{\mathrm{m}}^3.{\mathrm{s}}^2×\frac{1}{\mathrm{kg}.{\mathrm{m}}^2/{\mathrm{s}}^2}\\ & =& 1591.32\mathrm{kg}/{\mathrm{m}}^3\end{array}$

Thus, the average density of the planets in the decreasing order are of the planets Mercury, Mars, Venus, Earth, Neptune, Uranus, Jupiter and Saturn which are $5447.85\mathrm{kg}/{\mathrm{m}}^3$, $3911.89\mathrm{kg}/{\mathrm{m}}^3$, $2192.59\mathrm{kg}/{\mathrm{m}}^3$, $2079.62\mathrm{kg}/{\mathrm{m}}^3$, $1591.32\mathrm{kg}/{\mathrm{m}}^3$, $1219.62\mathrm{kg}/{\mathrm{m}}^3$, $1157.65\mathrm{kg}/{\mathrm{m}}^3$, $535.05\mathrm{kg}/{\mathrm{m}}^3$respectively.

Along with the planet Earth, the other eight major planets are also not uniform as they spin around their axis and around the sun. However, the force of spin mainly acts against the gravity and it eventually causes the planets to more bulge out around their respective equator.

Thus, due to the spinning force, the eight major planets have also variable average densities.

If the average density of Saturn is same as of the Earth, then from the equation 3), the equation of the acceleration due to gravity of Saturn can be expressed as,

For,and,

Hence, further as,

Thus, the value of the at the Saturn’s surface is.