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PHYS-1000
How Long is a Day on Mercury and Venus?
Fall 2012
Mercury
In this exercise, you will calculate the solar day for a person on Mercury. The period of revolution, the time
interval for Mercury to orbit Sun (relative to the stars), for Mercury is 88 days (and when I say “day” as a
unit, I am referring to Earth days). Mercury’s period of rotation, the time interval to rotate once about its
axis (relative to the stars), is 59 days–this is 2/3 of the orbital period of Mercury.
1. What is the definition of solar day? Include a diagram with your definition.
Solution: A solar day (or synodic day) is the time it takes the sun to successively pass the meridian
(astronomical noon). A mean solar day on Earth is 24 hours. The earth has to rotate more than 360◦
for the sun to come back to “noon.” See Figure 1.
2. What is the definition of sidereal day? Include a diagram with your definition.
Solution: One sidereal day is the time it takes for a star in the sky to come back to the same place
in the sky. Because, for all intents and purposes, the sky is “fixed,” a sidereal day is when the earth
rotates 360◦ . See Figure 1.
Figure 1: Sidereal versus solar (synodic) motion as referenced in questions 1 and 2.
A picture of Mercury’s orbit is shown in Figure 2. Assume that it is circular orbit, and Mercury’s axis
is not tilted. A person standing on Mercury at its equator is represented by the arrow. The dotted lines
divide the orbit into thirds. From this view, Mercury is revolving (orbiting the sun) counterclockwise
and is rotating (spinning on its axis) counterclockwise.
3. At the location shown, what time is it for the person on Mercury?
Solution: noon
4. After one complete rotation of Mercury, so that the arrow has the same orientation with respect to the
stars, where is Mercury? Sketch the planet with the arrow in the correct orientation on Figure 2 and
label how many Earth days have passed.
Solution: Noon. See position 2 on Figure 2.
5. What direction is the person facing when Mercury has only traveled one-third of an orbit? Sketch it on
Figure 2 as well and label how many Earth-days have passed.
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PHYS-1000
How Long is a Day on Mercury and Venus?
Fall 2012
Figure 2: Mercury’s orbit about the Sun.
Solution: See position 3 on Figure 2.
6. Where will the planet be when it is again noon for the person on Mercury (when the arrow points directly
toward the Sun)?
Solution: See position 1 on Figure 2.
7. How many Earth days have passed since it was last noon?
Solution: 176 Earth days.
8. How many times does Mercury orbit the Sun in one solar day (Mercurian day)?
Solution: It orbits the sun two times.
Venus
The rotational period of Venus is 243 Earth days. Its orbital period (period of revolution) is 225 Earth
days. Venus rotates “backwards” compared to the other planets (though Uranus and Pluto orbit on their
sides, meaning that their axis of rotation is nearly in the ecliptic plane). By “backwards” we mean that
if from a certain view, Venus orbits counterclockwise with respect to the Sun, then it rotates clockwise
about its own axis.
9. What is approximately a solar day on Venus? Use the same technique you used for Mercury above.
Start by sketching it and then determining where it is after a sidereal day, etc. Note that you won’t be
able to get an exact day using this method, but you can pin it down to a very small range of possible
answers.
Solution: See Figure 3.
Venus moves 360◦ in 225 Earth days. This means it moves 360◦ /225 Earth days = 1.6◦ per Earth day.
This means its sidereal day last (1.6◦ per Earth day) × (243 Earth days) = 388.8◦ per Venus solar day.
In other words, from sidereal noon to sidereal noon on Venus (where Venus is pointing to the same
far-off distant star), it has to pass through it’s original location (after 360◦ ) and then continue sliding
counterclockwise for another 28.8◦ . See position 2 on Figure 3.
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PHYS-1000
How Long is a Day on Mercury and Venus?
Fall 2012
To go half a sidereal day so that the arrow points in the opposite direction compared to where it starts,
243 Earth days / 2 = 121.5 Earth days have to pass. (1.6◦ per Earth day)×(121.5 Earth days) = 194.4◦
is where Venus will be located when this happens. This is marked at position 3 in Figure 3.
We can then reason that since Venus rotates clockwise, it’ll point at the sun again before it reaches
position 3 but sometime after it has moved half it’s orbit (directly opposite position 1).
Figure 3: Venus’s orbit about the Sun.
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