Question

The Sun’s center is at one focus of Earth’s orbit. How far from this focus is the other focus,

(a) in meters and

(b) in terms of the solar radius,${\mathbf{\text{6.96×10}}}^{\mathbf{\text{8}}}\mathbf{\text{m}}$? The eccentricity is$\mathbf{\text{0.0167}}$, and the semimajor axis is${\mathbf{\text{1.50×10}}}^{\mathbf{\text{11}}}\mathbf{\text{m}}$.

Short answer

1. Distance between two foci of earth’s orbit in meters${\text{5.0×10}}^{\text{9}}\text{m}$.
2. Distance between two foci of earth’s orbit in terms of the solar radius ${\text{R}}_{\text{s}}$is $7.2\times {\text{R}}_{\text{s}}$.

Step-by-step solution

Step 1: Given

${\text{R}}_{\text{s}}=6.96\times {10}^{\text{8}}\text{m}$

Eccentricity $\left(\text{e}\right)=0.0167$

Semi major axis$\left(\text{a}\right)=1.50\times {10}^{11}\text{m}$

Determining the concept

As the distance between the focus of the ellipse and the center of the ellipse iseccentricity × semi-major axis, calculate the distance between the two foci in meters of earth’s orbit both in meters and in terms of the radius of the sun.

The formula is as follows:

Distance (focus and center of the ellipse) = eccentricity × semi-major axis

(a) Determining the distance between two foci of the earth’s orbit in meters

From the diagram,

$\begin{array}{c}{\mathrm{Distance}}_{{\mathrm{F}}_1\mathrm{and}{\mathrm{F}}_2}=((\mathrm{a}\times \mathrm{e})+(\mathrm{a}\times \mathrm{e}))\\ =((0.0167\times 1.50\times {10}^{11}\mathrm{m})+(0.0167\times 1.50\times {10}^{11}\mathrm{m}))\\ =(2.5\times {10}^9\mathrm{m})+(2.5\times {10}^9\mathrm{m})\\ =5.0\times {10}^9\mathrm{m}\end{array}$

Hence, the distance between two foci of the earth’s orbit is in meters ${\text{5.0×10}}^{\text{9}}\text{m}$.

(b) Determining the distance between two foci of the earth’s orbit in terms of the solar radius Rs

Now,

${\mathrm{Distance}}_{{\mathrm{F}}_1\mathrm{and}{\mathrm{F}}_2}=5.0\times {10}^9\text{m}$

Multiply and divide to the right-hand side by the radius of the sun${\text{R}}_{\text{s}}$,

$\begin{array}{c}{\mathrm{Distance}}_{{\mathrm{F}}_1\mathrm{and}{\mathrm{F}}_2}=5.0\times {10}^9\text{m}\times \frac{{\mathrm{R}}_{\mathrm{s}}}{{\mathrm{R}}_{\mathrm{s}}}\\ =5.0\times {10}^9\text{m}\times \frac{{\mathrm{R}}_{\mathrm{s}}}{6.96\times {10}^8\mathrm{m}}\\ =7.2\times {\mathrm{R}}_{\mathrm{s}}\end{array}$

Hence, the distance between two foci of the earth’s orbit in terms of the solar radius${\mathrm{R}}_{\mathrm{s}}$is$7.2\times {\mathrm{R}}_{\mathrm{s}}$.

Therefore, using the formula for the distance between the focus of the ellipse and the center of the ellipse, the distance between the two foci of the earth can be found.