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Planet Labs recently tweeted ( https://twitter.com/planet/status/1481790983155044352 ) a video showing a Falcon 9 first stage landed on a pad.

Strangely, the image looks like it was taken by a camera which was dangling on a rope. What is causing these oscillations? I thought Earth-observing satellites need to be extremely stable to provide a clear picture, but the video seems to suggest that the satellite is making lots of small corrections of its orientation with respect to Earth. Is this possibly correct? Or are those corrections slow and the video was just sped up?

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Don't look at this from the perspective of why this was so lousy. Look at it from the perspective why this is so good.

I thought Earth-observing satellites need to be extremely stable to provide a clear picture.

This is extremely stable. Look at how clear each frame is.

but the video seems to suggest that the satellite is making lots of small corrections of its orientation with respect to Earth. Is this possibly correct?

This short video was taken from a satellite in low Earth orbit at 450 km altitude. That means that the satellite is moving at a speed of over 7.6 km/s. The satellite is small, essentially the size of a mini fridge.

The satellite and imaging equipment needs to be rotating / moving very precisely to keep the imagery focused on one spot and to avoid motion blur. To repeat an image, the satellite has to reposition itself / its imager for every video frame. This control not perfect; nothing ever is.

The video in question is one of PlanetLab's videos products. From PlanetLab's developer resource center on their video products,

The SkySat Video products include a video mpeg-4 file, with all captured frames used to produce the video as L1A panchromatic scenes.

So what are "L1A panchromatic scenes"? From PlanetLab's developer resource center on SkySat,

Basic L1A Panchromatic assets are non-orthorectified, uncalibrated, panchromatic-only imagery products with native sensor resolution (0.72-0.81m), that have been made available roughly two hours before all other SkySat asset types are available in the catalog. These products are designed for time-sensitive, low-latency monitoring applications, and can be geometrically corrected with associated rational polynomial coefficients (RPCs) assets (derived from satellite telemetry).

Level 1A (L1A) data from an imaging satellite are very close to raw data. In particular, the SkySat L1A data are "non-orthorectified", which means the imagery has not been processed (e.g., rubber-sheeting) so as to make the imagery line up with ground correction points. The imagery is what the satellite saw. Yes, it's a bit wobbly. That is to be expected with L1A data.

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This is very likely the satellite's attitude control. If the satellite is not controlled, it will not just remain in its current orientation, but instead will at least rotate slowly. Maybe the solar radiation pressure will lead to a stable orientation after some time (Kepler used something similar and the new JWST has a momentum flap to keep its attitude stable) or maybe it will continue spinning or even speed up due to complex interactions. To keep the satellite pointed exactly at a target precise attitude control is needed.

How strong is the wobble? The satellites are at an altitude of roughly $500\,\mathrm{km}$ [1]. The landing zone is $86\,\mathrm{m}$ in diameter [2]. From the video we can roughly see it oscillating by half the landing zone width. So what is the change in angle?

$$\Delta\varphi = \arctan\frac{0.5 \cdot 86\,\mathrm{m}}{500\,\mathrm{km}} \approx 0.00493°\,.$$

We assume the satellite is roughly above the landing pad for the small angle approximation. This shouldn't alter the result too much.

For comparison the angular size of the Moon from Earth is about $0.495°$. So the pointing precision the satellite is doing here is about as good as if you filmed the moon with your cellphone and managed to shake it so little that it moves less than $1\%$ of the moons diameter.

The satellites are primarily meant to take still photos where a bit of shaking is ok so this stability in the video is still quite astonishing. That being said I have no clue why they did not just stabilize the video in post; cropping the edges a bit.

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