Nikon D800 + Nikon 80-200mm f/2.8, 1/640 sec, f10, ISO 100 

INTRODUCTION

The International Space Station (or ISS) is around 248 miles above the earth, traveling at a speed of almost five miles per second, circling the earth every 92 minutes.  On occasion, the ISS will cross over the moon or sun.  It goes too fast to see it with the naked eye, but with the right equipment, preparation, and some precise timing, it's possible to photograph the ISS crossing the sun or moon.
EQUIPMENT
SLR
Nearly any SLR would work.  Key things to keep in mind are:
Sensor resolution - Even with a long telephoto lens, the ISS will only be a handful of pixels across.  Above 20 megapixels is ideal.
Frames per second - The ISS typically passes over the moon in under one second.  When the entire event is over in half a second, speed counts.  This means that for a slower SLR that shoots four frames per second, you might only capture two or three shots of the transit.  A faster SLR shooting at 8 FPS can get twice the shots.
 Telephoto lens
The longer the telephoto lens, the larger the moon will appear in your frame.  A 200 or 300mm lens would be the minimum to have a chance to capture the ISS in your photo.  A super-telephoto like a 150-600mm lens, along with a teleconverter, could fill up even more of the frame.
Tripod
A large, sturdy tripod is necessary to capture the shot.  Something on a swivel to easily follow the moon helps make sure everything stays in frame.
Remote shutter 
This isn't a requirement, but having a remote shutter can make it easier to get the timing and get the shot.  Keeping your smartphone in one hand to track the time, and the remote shutter in the other is easier than firing the shutter on the camera.  
Smartphone
You need to know the time down to the second, which means using a smartphone and visiting time.gov to ensure you know exactly when the ISS will be transiting the moon.
Solar filter
Unnecessary when shooting the moon, but if you want to photograph the ISS transiting the sun, you need to have one on your lens.  I use the Full Aperture Solarlite filter from Thousand Oaks Optical.  
PROCESS
1. Preparation work (One month beforehand)
The very first step to photographing the ISS in transit is to identify when the next lunar transit will be happening nearby.  transit-finder.com is a straightforward website where you can look up your house on a map, then determine the closest transits near your house.  You can forecast up to a month in advance.  
It's unlikely that a transit will occur directly over your house (but it can happen!).  You can view solar or lunar close passes on the map to see where they are in relation to you, and then determine where you can set up for the transit.  
For a given transit, note the exact time (down to the second and milisecond) that it occurs.  In this example, the ISS passes at 6:39am and 20.89 seconds.  
2. Before the transit occurs  (30 minutes beforehand)
Pack your bag with the equipment, and head to the location with time to spare.  Now I can set up everything in less than ten minutes, but giving at least a 30 minute buffer would be safe.  
Test your shutter speed and camera buffer by firing as many shots in a row as possible, and keep track of the number of seconds your camera will fire continuously.  My Nikon Z7 shoots for around eight or nine seconds before it slows down.  This gives a decent buffer of ±4 seconds before and after the transit.
Set up your tripod, and aim your camera at the moon.  If you are shooting the sun, *place your filter over the lens before aiming it at the sun*.  Lens Rentals has several excellent examples of what happens if you don't use a proper solar filter.  
Connect your remote shutter to your camera if you have one.  
Load time.gov on your smartphone to know the exact time down to the second.
3. Right before the transit occurs  (<60 seconds beforehand)
Now you wait until the ISS is scheduled to transit the moon, keeping a close eye on the time.  Make sure that the moon is kept in your viewfinder, as it will continue to move.  
I like to start firing shots two seconds before it's scheduled to pass.  For the example above, I would start firing at 6:39 and 18 seconds.  Once I start shooting, I keep taking photos until the camera's buffer is full.
4.  After the transit
Check your camera!  It's difficult, but zoom into the shots to see if you captured the ISS.
When you get home, load your photos and confirm if you were able to capture it.  If you see a small dot moving across the moon, you caught it!  
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