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What was the diameter of Explorer I's whip antennas, and what material were they made of? I am trying to write a simulation of the satellite's spin reorientation. The Smithsonian lists antenna length as 22 inches.

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I can't tell you exactly, but I can suggest a methodology that you can think about applying to try to deduce the answer.

From the NASA page A Moment in Time: Explorer 1 I cropped the following image and played tricks with cropping further and zooming and estimating. I didn't pull up OpenCV in Python and do edge detection but that would be the next step.

Wikipedia gives the rocket diameter as 15.2 centimeters (six inches). I took three spots and estimated the antenna diameter at the midpoint of the sloped edges, then took the ratio to the rocket width at a nearby location at presumably a similar distance from the camera (camera is not at infinity).

I got [0.61, 0.54, 0.43] centimeters for the three estimates, median and average are 0.54 and 0.53 cm respectively, and I won't report a standard deviation for only three numbers. (okay it's 0.07 cm, i.e. big).

Since oscillators can drift in frequency and metal can expand and contract thermally, it is desired to have a finite bandwidth over which the antenna is impedance matched. Thin wire antennas have a very narrow low-SWR range. This is one reason why monopole, dipole, even Yagis are made with larger diameter tubing than they might need to be structurally. So the 5mm diameter may be related to both desired mass and SWR performance.

As they are so flexible, they are not solid metal but instead probably some kind of coiled or braided multifilament, but I don't know how they did that.

enter image description here

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You could repeat the process with the following image from here but there's no guarantee it's the same antenna (unless you can track down that information).

enter image description here

Or this one from here

enter image description here

But don't use this one or this one.

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