This is Question 23 from USAAAO First Round 2015. The answer is D. Can someone show how to get the answer?

Calculate the planetary phase angle (counterclockwise from Earth, a = 1.0 AU) that a probe may correctly complete a Hohmann transfer orbit to Venus (a = 0.7 AU)

  • a. 141 degrees
  • b. 17.5 degrees
  • c. 121 degrees
  • d. 241 degrees
  • e. 343 degrees

What I tried:

  • Semimajor axis of Hohmann orbit: 0.85 AU
  • Period of Hohmann orbit: 0.7836 yr
  • Time of flight: 0.3918 yr
  • Period of Venus: 0.5856 yr
  • Angular distance travelled by Venus during flight: 241°

The last one looks suspiciously similar to the answer, but using that angular distance the phase angle should be 61°. Feel free to correct me anywhere that I’m wrong

Phase angle=61°Phase angle=48.99°

  • $\begingroup$ I'm not sure what you are asking. It seems to be "Here is a test question. The answer is D. I did some working and I got D. Please tell me what I did right." That is sort of an odd question to ask. Could you clarify. $\endgroup$ – James K Mar 23 '20 at 20:26
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    $\begingroup$ I think the question here is why is the angular distance Venus travelled in that time the answer $\endgroup$ – Tosic Mar 23 '20 at 20:29
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    $\begingroup$ @DavidHammen Thanks for replying. It would be great if you can show us the correct diagram. $\endgroup$ – Je Qin Chooi Mar 24 '20 at 8:01
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    $\begingroup$ @DavidHammen - I think the fourth object is supposed to be Venus at the initial time (given it's the same colour as the labelled Venus) $\endgroup$ – user24157 Mar 24 '20 at 11:32
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    $\begingroup$ I edited the images just for readability (I couldn't read them easily as they were so huge) - if you think it causes issues please revert. $\endgroup$ – Rory Alsop Mar 24 '20 at 17:47

Let's assume that Earth and Venus have circular, coplanar orbits. The spacecraft advances 180° in heliocentric ecliptic longitude in 143 days; meanwhile, Venus advances 241° and Earth advances 141°. When the spacecraft departs from Earth (left), Venus should be 61° behind Earth as seen from the Sun. When the spacecraft arrives at Venus (right), Venus should be 39° ahead of Earth.

Earth-Venus departure and arrival

The Sun-Earth-Venus angle marked in green in your second figure is Venus's elongation, 43° east of the Sun as seen from Earth at departure time.

In another context, "phase angle" could refer to the Sun-Venus-Earth angle. At departure time, an Earth-based observer would see Venus in a 76° phase angle or 62% illuminated.

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    $\begingroup$ Which means the USAAAO got it wrong when setting the question. $\endgroup$ – user24157 Mar 25 '20 at 8:44

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