### Armchair science

I noted with interest this update from the DSCOVR mission:

Cool! The DSCOVR mission is an Earth-imaging mission that takes a continuous stream of Earth images to monitor it for climate and other changes over time. It happened to catch the Moon crossing the face of the Earth (go to the link for the animation - you'll be glad you did). This will happen twice a year, when the plane of DSCOVR's orbit intersects the plane of the Moon's orbit in line with the Earth.

The DSCOVR web page says that the satellite orbits "a million miles" from Earth. It turns out, you can calculate that distance just from this image and from the known sizes of the Earth and the Moon. The Earth has a mean radius of 3,959 miles, and the Moon has a mean radius of 1,079 miles. The ratio between the two is 3.67 - meaning the Earth is 3.67 times as big as the Moon. But if you open up that image in Photoshop and measure the Earth and Moon in pixels, you will find this:

The Earth is 1595 pixels big, and the Moon is 596 pixels big. This is a ratio of 2.72 - what's up with that? The answer is, perspective. The Earth is farther away in this picture than the Moon, so it looks smaller. We know that the Earth and the Moon are roughly 250,000 miles apart. So, how far is DSCOVR from the Earth? Based on these measurements, this diagram, and a little trigonometry, we can calculate it:

Point "S" is the spacecraft. Point "E" is the north pole of the Earth. Point "M" is the north pole of the Moon. Point "X" is the center of the Moon, and point "Y" is the center of the Earth. We know the following distances:

The dashed line shows the projection of the edge of the moon from the spacecraft's point-of-view. The pixel measurements are taken at the same distance - since they are both projected against the same plane (in reality, the plane of the imaging system). The point labeled "Mp" is that projected point in the image. We have similar triangles which can help us figure out the distance D:

The image shows the calcuations, which gets a distance of 965,000 miles. Close enough to "one million miles" for government work! Especially given the precision with which I specified the Earth-Moon distance (the actual distance varies between 221,457 miles and 252,712 miles).

Finally, here is a picture showing how big the Earth would have looked if it had been at the same distance as the Moon in the DSCOVR photo (I show it as a dim ghost behind the original image):

Of course, we would all be having a very bad day if the Earth and Moon had been at the same distance at that time...

Given that complex orbit, this method is probably the most accurate way I have of conveniently determining DSCOVR's distance from the Earth. At least on the days that the Moon occults or is occulted by the Earth.

Phil also pointed out that the moon will look smaller due to perspective when it is on the other side of its orbit (the back of my envelope tells me it will be 59% of the apparent size) - so it might look something like this by comparison:

And yes, I'm showing the other face of the moon - that's the one DSCOVR will see in that geometry. We'll see...

Cool! The DSCOVR mission is an Earth-imaging mission that takes a continuous stream of Earth images to monitor it for climate and other changes over time. It happened to catch the Moon crossing the face of the Earth (go to the link for the animation - you'll be glad you did). This will happen twice a year, when the plane of DSCOVR's orbit intersects the plane of the Moon's orbit in line with the Earth.

The DSCOVR web page says that the satellite orbits "a million miles" from Earth. It turns out, you can calculate that distance just from this image and from the known sizes of the Earth and the Moon. The Earth has a mean radius of 3,959 miles, and the Moon has a mean radius of 1,079 miles. The ratio between the two is 3.67 - meaning the Earth is 3.67 times as big as the Moon. But if you open up that image in Photoshop and measure the Earth and Moon in pixels, you will find this:

The Earth is 1595 pixels big, and the Moon is 596 pixels big. This is a ratio of 2.72 - what's up with that? The answer is, perspective. The Earth is farther away in this picture than the Moon, so it looks smaller. We know that the Earth and the Moon are roughly 250,000 miles apart. So, how far is DSCOVR from the Earth? Based on these measurements, this diagram, and a little trigonometry, we can calculate it:

Point "S" is the spacecraft. Point "E" is the north pole of the Earth. Point "M" is the north pole of the Moon. Point "X" is the center of the Moon, and point "Y" is the center of the Earth. We know the following distances:

- Point X to point M: 1,079 miles (and 596 pixels)
- Point Y to point E: 3,959 miles, (and 1,595 pixels)
- Point X to point Y: 250,000 miles

The dashed line shows the projection of the edge of the moon from the spacecraft's point-of-view. The pixel measurements are taken at the same distance - since they are both projected against the same plane (in reality, the plane of the imaging system). The point labeled "Mp" is that projected point in the image. We have similar triangles which can help us figure out the distance D:

- Triangle Y-S-Mp
- Triangle D-S-M

The image shows the calcuations, which gets a distance of 965,000 miles. Close enough to "one million miles" for government work! Especially given the precision with which I specified the Earth-Moon distance (the actual distance varies between 221,457 miles and 252,712 miles).

Finally, here is a picture showing how big the Earth would have looked if it had been at the same distance as the Moon in the DSCOVR photo (I show it as a dim ghost behind the original image):

Of course, we would all be having a very bad day if the Earth and Moon had been at the same distance at that time...

**Edit (Aug 5 7:15PM MDT)**: I had an email exchange with Phil Plait, the Bad Astronomer, and he pointed out I can get the exact Earth-Moon distance on the day in question (July 16) from the United States Naval Observatory website. On that date, it was 243,388 miles. So, plugging that into the formula I get that DSCOVR was 940,000 miles from Earth on that date. L1 is about 929,000 miles from the Earth-Moon barycenter, which in this photo is about 2900 miles closer to the spacecraft than Earth is, making the L1 point 932,000 miles from the center of the Earth. Which is very close to what I calculated - less than 1% difference. Part of this is difference is attributable to the inaccuracy of measuring the pixel sizes of the Earth and Moon, and part is attributable to the fact that DSCOVR is not actually *at* the L1 point; rather, it is orbiting about it in a complex pattern known as a "Lissajous Orbit".Given that complex orbit, this method is probably the most accurate way I have of conveniently determining DSCOVR's distance from the Earth. At least on the days that the Moon occults or is occulted by the Earth.

Phil also pointed out that the moon will look smaller due to perspective when it is on the other side of its orbit (the back of my envelope tells me it will be 59% of the apparent size) - so it might look something like this by comparison:

And yes, I'm showing the other face of the moon - that's the one DSCOVR will see in that geometry. We'll see...

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