The orbit of the planet Mercury around the Sun is in elliptical shape with Sun at a focus. The semi-major axis

Question 76 of 126

Q. The orbit of the planet Mercury around the Sun is in elliptical shape with Sun at a focus. The semi-major axis is of length 36 million miles and the eccentricity of the orbit is 0.206. Find :

(i) How close the Mercury gets to Sun ?

(ii) The greatest possible distance between Mercury and Sun.

Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2016Subjective· 10mImportance★★★★★

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Perihelion $=a(1-e)$ and aphelion $=a(1+e)$ give Mercury's closest and farthest distances from the Sun.

Set-up. Place the ellipse with its centre at the origin and the Sun at one focus $S$ . For an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ with $a>b$ , the foci are at $(\pm c,0)$ where $c=ae$ (since $e=c/a$ ). The two vertices on the major axis are at $(\pm a,0)$ .
Distance from a focus to the nearer vertex. Take the focus at $S=(c,0)=(ae,0)$ and the nearer vertex at $(a,0)$ . Distance $=a-ae=a(1-e)$ . This is the least distance between a point on the ellipse and that focus (the perihelion, i.e. the point of closest approach).
Distance from a focus to the farther vertex. The farther vertex is at $(-a,0)$ . Distance $=a-(-ae)=a+ae=a(1+e)$ . This is the greatest distance between a point on the ellipse and the focus (the aphelion).

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Q.The orbit of the planet Mercury around the Sun is in elliptical shape with Sun at a focus. The semi-major axis is of length 36 million miles and the eccentricity of the orbit is 0.206. Find :