Thursday, February 21, 2008

Shooting the moon

With this past lunar eclipse flickr was saturated with moon photos. Some good and others ... well not so good. For the most part, the "not so good" photos suffered from motion blur. "I used a tripod and mirror lockup", a tripod and mirror lockup will only prevent motion due to camera shake and does not account for the relative motion of the moon with respect to an earth observer.

Shooting the moon

Back of the envelope calculation:

For the purposes of this calculation, to keep things simple and two dimensional, let's forget about facts like the orbits are elliptical, the moon's orbital plane rotates, the earth is tilted, the moon's center of rotation is not the center of the earth and so on.

Question: What shutter speed will ensure that the motion of the moon does not appear in a photo?

First lets list some facts about the earth's rotation and orbit of the moon:*

  1. The earth makes a full counter clockwise rotation in 24 hours.

      The angular velocity of the earth is w_earth = (2Pi radians / 24 hours)*(1 hour / 60 mins)*(1 min / 60 seconds) =7.27x10^(-5) rads/s.

  2. The moon has a linear velocity of 1.023 km/s relative to the stars (sun) and is about 385000 km from the earth.

      Linear velocity is related to angular velocity by the radius, v=rw, which implies that the angular velocity of the moon around the earth is, w_moon= 2.6 x10^(-6) rads/s. This means relative to an earth observer, w_moon_rel = 7.5x10^(-5) rads/s (assuming the moon's orbit is ccw). In other words the moon has a linear velocity relative to an earth observer of v_moon_rel= 28.9 km/s

  3. The moon's radius is 1737 km.

      Using (2), for the moon to move 1 diameter in the nights sky it takes 3474 km/28.9 km/s = 120 secs or 2 minutes.
*-numbers are not exact, it's a back of the envelope calculation after all.

Ok so what does this have to do with photography? Well suppose you are using a 400mm lens so that the moon appears to be roughly 600 pixels across on the camera's sensor. By the above calculations the moon has moved 600 pixels in 2 mins or 1 pixel in 1/5th of a second. So to freeze the motion of the moon you should use a shutter speed faster than 1/5th like 1/10 or better.