Chapter 8.2, Problem 54E

Solution

The closest distance between Mercury and the Sun’s surface.

It has been determined that the closest distance between Mercury and the Sun’s surface is $45.29976$ Gm.

Given:

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

The diameter of the Sun is $1.392$ Gm.

Concept used:

The eccentricity of an elliptic orbit is given as $e=\frac{c}{a}$ and the perihelion distance from the Sun is $a-c$ .

Calculation:

It can be found from the given table that the eccentricity of Mercury is $0.2056$ and its semi-major axis is $57.9$ Gm.

Then, for Mercury, $e=0.2056$ and $a=57.9$ .

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

  $0.2056=\frac{c}{57.9}$

Simplifying,

  $c=11.90424$

Then, the perihelion distance of Mercury from the Sun is given as,

  $a-c=57.9-11.90424=45.99576$

So, the perihelion distance of Mercury from the Sun is $45.99576$ Gm.

This is the closest distance between Mercury and the Sun, more specifically the center of the Sun.

Now, it is given that the diameter of the Sun is $1.392$ Gm.

Then, the radius of the Sun is $\frac{1.392}{2}=0.696$ Gm.

So, the closest distance between Mercury and the surface of the Sun is $45.99576-0.696=45.29976$ Gm.

Thus, the closest distance between Mercury and the Sun’s surface is $45.29976$ Gm.

Conclusion:

It has been determined that the closest distance between Mercury and the Sun’s surface is $45.29976$ Gm.