Astrophysics: orbital dynamics - duration of a transit of Venus

[joriarty]

Edit: perhaps this might belong in one of the Physics help forums? Oops. Mods, please move if so!

Homework Statement

Show that a transit of Venus across the Sun’s disk lasts at most about 8 hours. The synodic period of Venus is 584 days and its orbital radius is 0.723 A.U. The Sun’s angular diameter is 32′. Assume that the orbits of Venus and the Earth are coplanar and circular. Do not use any other numerical data. (Hint: find the angular velocity of Venus relative to the Earth-Sun line and as seen from the Earth. The greatest duration of transit is when Venus passes through a diameter of the solar disk.)

Homework Equations

ω_Venus - ω_Earth = ω_synodic

The Attempt at a Solution

I understand the concept of synodic period, I'm just unsure how to tackle this problem. The transit must occur at inferior conjunction. I know the synodic period, which I can use to calculate the synodic angular velocity as ~ 0.02568 degrees per hour. This gives a maximum transit duration of nearly 21 hours! That's not right.

How can I convert this synodic angular velocity (which is with respect to a reference frame that is co-rotating with the Earth-Sun line) to an angular velocity seen by an observer on Earth?

Please don't be too specific with your hints as this is coursework, and I do want to figure things out for myself. I just need a nudge in the right direction :)

Thanks!

 

[Filip Larsen]

Try think of this as a purely circular geometric problem. Near inferior conjunction the Earth moves as if it rotates 360 degrees around Venus in 584 days at a distance of 1-0.723 AU. Now think about how much angle the 32' seen from Earth over 1 AU corresponds to when seen from Venus at 0.723 AU and then think about how fast this angle is covered when doing 360/584 deg/day.

 

[joriarty]

Got it now! Thank you :). Just needed to think in terms of Earth's frame of reference - I was trying to solve it from a "top down" solar system model.