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This computer-generated view depicts how the wide angle camera on NASA's Cassini spacecraft will take the images of Saturn and its rings on July 19, 2013.
This computer-generated view depicts how the wide angle camera on NASA's Cassini spacecraft will take the images of Saturn and its rings on July 19, 2013. Image credit: NASA/JPL-Caltech

Timing is Everything – So Let's Do the Math

By Scott Edgington
July 18, 2013

Scott Edgington
Scott Edgington
For any typical observation Cassini makes of the Saturn system, the navigators spend weeks making sure that the path of the spacecraft remains within a few thousand miles (kilometers) of the path assumed during design of the observations. As the spacecraft approaches the observation, knowledge of the spacecraft’s path is improved from hundreds of miles (kilometers) to hundreds of yards (meters). For Cassini to take its picture of Earth as part of its imaging of the full Saturn system on July 19, we have to put it in the right place in Saturn's shadow, make sure we are pointed towards Earth and our other targets and snap the images at the right time. On the spacecraft, the Earth imaging takes place between 22:47:13 UTC (03:47:13 pm PDT) and 23:01:56 UTC (04:01:56 pm PDT). The end-to-end mosaic images of Saturn and rings all fall between 22:24:00 UTC on July 19 and 02:43:00 UTC on July 20 (03:24 to 07:43 pm PDT). (For spacecraft event time, we use UTC, which is Coordinated Universal Time.)

But, wait, you say, isn't that different from the times we've asked people to wave at Saturn? Yes, it is because we have to account for the time it takes for photons to leave Earth and arrive at the detectors on Cassini's cameras. To determine the time we've asked Earthlings to wave, we needed to know the exact distance these photons will travel from the Earth to Saturn and Cassini at the time that we wish to capture the image. We get this information from data called ephemerides that are produced by the Cassini navigation team here at NASA's Jet Propulsion Laboratory in Pasadena, Calif. The ephemerides give us the position of the Earth and other solar system objects (including Cassini and Saturn) relative to the center of our solar system. With this information, we calculated the distance that light has to travel.

The distance is calculated multiple times because Cassini and Saturn are one place when the photons depart Earth, but another place when the photons arrive. For the first iteration, the geometric distance is computed from the ephemerides and divided by the speed of light to determine the time it takes for photons to travel that distance. For the next iteration, the distance is computed from Earth at the desired time to Cassini and Saturn at the desired time plus the time computed in the previous iteration. Repeating this process eventually leads to an estimate of the one-way light travel time that’s accurate to about a microsecond.

First frame of How Does A Spacecraft Take a Movie? video
Filters, pixels and math star in this narrated cartoon. It shows how a spacecraft records a color image as data and transmits it to Earth. and how the image is reconstructed from the data.
From the ephemerides, and multiple light travel time calculations, we know that photons will have had to travel about 898.5 million miles (1.446 billion kilometers) from Earth on July 19 to reach Cassini at 22:47:13 UTC, which is the beginning of the Earth imaging. Knowing that the speed of light is 670,616,629 mph (299,792,458 meters per second) and dividing the distance by this speed of light, this translates into a time delay of about 80.4 minutes (1 hour, 20 minutes and 24 seconds). We call this the one-way light time (OWLT). If we are to capture your photons, we would need you to be waving 80.4 minutes prior to your photons reaching the spacecraft cameras, i.e. at 21:26:49 UTC (02:26:49 PM PDT5:26:49 PM EDT). That's how we came up with the Earth-waving window, to the nearest minute, of 21:27 UTC to 21:42 UTC (2:27 to 2:42 p.m. PDT).

Similarly, we use the one-way light time to tell us when we should be listening for Cassini when it is communicating with us. Cassini sends data back as radio waves, which travel at the speed of light. If Cassini sends a signal at any particular time, it would take exactly the one-way light time for the signal to reach us. Our computers here at JPL are set up to calculate the one-way light time at any instant during Cassini’s mission to a high degree of accuracy and in practice, we are often talking amongst the team about it down to the second. This way we can ensure that we are transmitting or receiving data (or even waving) at the correct times.

Lest you think the math ends there, we use similar calculations to estimate how big objects will be in our field of view. For instance, how do we know how big Earth will be?

The highest resolution camera (narrow-angle camera or NAC) on Cassini’s imaging science subsystem is an array of 1,024 by 1,024 pixels (a 1 mega-pixel camera!) with an angular field of view of 6.134 milli-radians. This means that each pixel has an angular view of 5.9907 micro-radians of the sky. The Earth, with a diameter of 7,926 miles (12,756 kilometers), is at a distance of 898.5 million miles (1.446 billion kilometers) from Cassini at this time. Dividing the size of the earth by the Earth-Cassini distance, this translates to an angle of about 8.83 micro-radians. Ideally, this means that the Earth will be roughly 1.47 pixels across in the image. Due to the optical properties of the NAC telescope, the Earth will span a bit more than this value. We call this factor the point spread function and it is equal to 1.3 for the NAC. The end result is that the Earth’s photogenic image will span approximately 1.9 pixels across (thus a 2 by 2 image). However, since not all of Earth is illuminated at this time -- less than half -- the pale blue shine that we'll see of Earth will be less than a pixel.

Still, we'll all be there in that less-than-pixel! On July 19, I'll be thinking of how far Cassini has traveled in the nearly 16 years since launch to see Earth from such great a distance and all the great collaborations that have made this view of ourselves possible.

Scott G. Edgington is the Cassini deputy project scientist, based at NASA's Jet Propulsion Laboratory, Pasadena, Calif.

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