# What are the perihelion and aphelion speeds of Mercury? What are the perihelion and aphelion distances of Earth? How would these speeds be calculated?

Apr 22, 2016

I suppose you wanted the speeds for both Earth AND Mercury.

#### Explanation:

I would use the radial distances $r$ both at perihelion $P$ and aphelion $A$ into Newton's Second Law $\Sigma \vec{F} = m \vec{a}$ using also centripetal acceleration: Use the values from the literature for the mass of the Sun, ${M}_{S}$, $G$, and the radial distances (you may find them in books, I think).

Apr 22, 2016

Mercury;s perihelion speed = 69.8 km's and aphelion speed is 46.0 km/s. Earth's perihelion = 147.1 million km and aphelion = 152.1 million km.

#### Explanation:

Let ${v}_{a}$ = speed , when the distance of a planet from the Sun is the semi-major axis a of the planet's orbit., ${v}_{p e}$ = perihelion speed and${v}_{a p}$ = speed at aphelion. Let e be the eccentricity of the orbit.

Then, approximating ${v}_{a}$ = average speed v,

${v}_{p e} = {v}_{a} \left(1 + e\right) = v \left(1 + e\right)$, nearly and ${v}_{a p} = {v}_{a} \left(1 - e\right) = v \left(1 - e\right)$, nearly.

I can assure you that these approximations give 3-sd values without error, if there are no bugs in the data used..

For Mercury v = 47.4 km/s, a = 57.9 million (M) km and e = 0.2055. .

So, ${v}_{p e} = 47.4 \left(1 + 0.2055\right) = 69.8 k \frac{m}{s}$ and
${v}_{a p} = 47.4 \left(1 - 0.2055\right) = 46.0$ km/s#, nearly.

This way, the perihelion and aphelion speeds of the Earth can be approximated using relevant data from sources like
NASA Planetary Fact Sheet.