# Between Mercury and Venus, which planet produces a longer transit? Assuming both planets describe same path on the solar disk

Is it even possible to give a definite answer to this question? There are a lot of factors involved in this like Earth's position and motion relative to the planets and of course their own motion and apparent sizes in earth's sky.

Or is it that obvious that Venus transit will be longer because its orbital speed is less than Mercury?

• Question is ill-defined. What are you keeping the same? Same observation point on Earth, same transit latitude across the Sun? – ProfRob Mar 30 '20 at 15:04
• Yes, it is possible to give a definitive answer since all of the factors you mentioned (position, distance, motion) are all known or easily calculated. Depending on where the question is from, it may be sufficient to assume the mean orbit of each planet. – JohnHoltz Mar 30 '20 at 16:02
• same path on solar disk means same transit latitude across the Sun. – Lagaash Mar 30 '20 at 18:04

This is a cool question; I'll give it a go.

Or is it that obvious that Venus transit will be longer because its orbital speed is less than Mercury?

No, nothing here is obvious!

tl;dr: Times for the transit of Mercury can range from 5.37 to 8.30 hours, and for Venus they range from 7.94 to 8.12 hours under assumptions of coplanarity but allowing for eccentricity.

Let's assume all orbits are in one plane. We can get the orbital speed of body $$i$$ with semimajor axis $$a$$ and current distanc $$r$$ from the vis-viva equation;

$$v_i = \sqrt{GM\left( \frac{2}{r_i} - \frac{1}{_i} \right)}$$

where $$GM$$ is the standard gravitational parameter of the Sun which is about 1.327124E+20 m3/s2.

               a          ecc        pei           apo       v_peri   v_apo
Earth    1.49598E+11   0.016709  1.470984E+11  1.520976E+11  30286.6  29291.1
Venus    1.08208E+11   0.006772  1.074752E+11  1.089408E+11  35258.8  34784.4
Mercury  5.79091E+10   0.205630  4.600125E+10  6.981695E+10  58976.4  38858.6


Imagine Venus and Mercury crossing a line fixed at one end to the Sun and the other end to the center of the Earth. The linear speed of a planet $$i$$ relative to that line will be $$v_i$$ minus the speed of the line at the distance of planet $$i$$ or $$v_E r_i / r_E$$, and the angular velocity seen from Earth will be that divided by ($$r_E - r_i)$$ or

$$\dot{\theta}_i = \frac{v_i - (r_i/r_E) v_E}{r_E - r_i} = \frac{r_e v_i - r_i v_E}{r_E^2 - r_i r_E}$$

(seconds)     Venus          Mercury
Earth       peri   apo      peri   apo
peri       28570  29242    19335  29884
apo        28060  28626    19384  29646

(hours)       Venus          Mercury
Earth       peri   apo      peri   apo
peri       7.936  8.123    5.371  8.301
apo        7.794  7.952    5.384  8.235

• Sorry, but I got lost where you converted $v_p-v_E$ to the term with the square root GM etc. Are you sure that conversion is correct? – JohnHoltz Mar 31 '20 at 16:21
• @JohnHoltz yikes! Okay I'll fix that asap! – uhoh Mar 31 '20 at 19:52
• @JohnHoltz I've just substantially revised my answer. – uhoh Apr 11 '20 at 9:43