Chapter 8.2, Problem 56E

Solution

The equations for the orbits of Venus and Mars in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .

It has been determined that the equation for the orbit of Venus is $\frac{x^2}{\text{11707}\text{.24}}+\frac{y^2}{\text{11706}\text{.6986572224}}=1$ and the equation for the orbit of Mars is $\frac{x^2}{\text{51938}\text{.41}}+\frac{y^2}{\text{51485}\text{.3221640604}}=1$ .

Given:

The eccentricities and semi-major axis of the planets and Pluto.

Concept used:

The eccentricity of an elliptic orbit is given as $e=\frac{c}{a}$ , where $b^2=a^2-c^2$ .

Calculation:

It can be found from the given table that the eccentricity of Venus is $0.0068$ and its semi-major axis is $108.2$ Gm.

Then, for Venus, $e=0.0068$ and $a=108.2$ .

Put these values in $e=\frac{c}{a}$ to get,

  $0.0068=\frac{c}{108.2}$

Simplifying,

  $c=0.73576$

Put $a=108.2$ and $c=0.73576$ in $b^2=a^2-c^2$ to get,

  $b^2={\left(108.2\right)}^2-{\left(0.73576\right)}^2$

Simplifying,

  $b^2=\text{11707}\text{.24}-\text{0}\text{.5413427776}$

On further simplification,

  $b^2=\text{11706}\text{.6986572224}$

Put $a=108.2$ and $b^2=\text{11706}\text{.6986572224}$ in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to get,

  $\frac{x^2}{{\left(108.2\right)}^2}+\frac{y^2}{\text{11706}\text{.6986572224}}=1$

Simplifying,

  $\frac{x^2}{\text{11707}\text{.24}}+\frac{y^2}{\text{11706}\text{.6986572224}}=1$

This is the equation for the orbit of Venus.

Similarly, it can be found from the given table that the eccentricity of Mars is $0.0934$ and its semi-major axis is $227.9$ Gm.

Then, for Mars, $e=0.0934$ and $a=227.9$ .

Put these values in $e=\frac{c}{a}$ to get,

  $0.0934=\frac{c}{227.9}$

Simplifying,

  $c=\text{21}\text{.28586}$

Put $a=227.9$ and $c=\text{21}\text{.28586}$ in $b^2=a^2-c^2$ to get,

  $b^2={\left(227.9\right)}^2-{\left(\text{21}\text{.28586}\right)}^2$

Simplifying,

  $b^2=\text{51938}\text{.41}-\text{453}\text{.0878359396}$

On further simplification,

  $b^2=\text{51485}\text{.3221640604}$

Put $a=227.9$ and $b^2=\text{51485}\text{.3221640604}$ in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to get,

  $\frac{x^2}{{\left(227.9\right)}^2}+\frac{y^2}{\text{51485}\text{.3221640604}}=1$

Simplifying,

  $\frac{x^2}{\text{51938}\text{.41}}+\frac{y^2}{\text{51485}\text{.3221640604}}=1$

This is the equation for the orbit of Mars.

Conclusion:

It has been determined that the equation for the orbit of Venus is $\frac{x^2}{\text{11707}\text{.24}}+\frac{y^2}{\text{11706}\text{.6986572224}}=1$ and the equation for the orbit of Mars is $\frac{x^2}{\text{51938}\text{.41}}+\frac{y^2}{\text{51485}\text{.3221640604}}=1$ .