Did this satellite streak past the Hubble Space Telescope so close that it was out of focus? If so, how close was it?

Question

The new Nature Astronomy paper The impact of satellite trails on Hubble Space Telescope observations (Kruk et al. 2023) describes an AI-based object classification method used to find satellite trails in Hubble Space Telescope images.

Hubble images with satellite trails are now in the popular news. I checked several links and this one from Gizmodo's Satellite Swarms Like SpaceX’s Starlink Are Increasingly Spoiling Hubble Telescope Images caught my eye.

A particularly bad satellite streak appears in this Hubble image. Image: NASA/ESA/S. Kruk et al., 2023

Two linear features stand out; the vertical one associated with the overexposed object looks to me to be a diffraction spike of some type, but half of the photo is overwhelmingly dominated by a huge fuzzy, diffuse bright streak crossing at an angle.

The caption labels this only as a "particularly bad satellite streak" but I don't see any other way right now than to conclude that this satellite passed so close in front of Hubble that it was out of focus!

It's not broad due to overexposure - we can confirm this by checking gray levels and by noticing we can still see stars behind it.

Question: Did this satellite streak past Hubble so close that it was out of focus? If so, just how close was it?

note: the caption for Figure 1 does call this a "broad out-of-focus trail (ja4tg4lsq)" because that's what it looks like, but without an identification of the satellite (position (within a few kilometers I suppose) and direction during the exposure time should make that fairly easy) there could be other artifact mechanisms that might be responsible.

Since Hubble's aperture dimension is known, all we need is a plate scale for this image to estimate the distance due to DOF issues.

Related, potentially helpful:

According to @Roger Wood's answer to Has Hubble ever focused on something close enough that it had to move away from being focused at infinity?:

Hubble's resolution is 0.014 arc seconds (6.8e-8 radian) and its mirror diameter is 2.4 m. So objects closer than (2.4/2)/6.8e-8 m ~= 18,000 km will become detectably blurred and would benefit from refocussing.

Answer 1

As you note, that particular image has been identified as due an "out of focus" satellite (the other images presumably show more distant satellites in e.g. geosynchronous orbits). The Nature Astronomy paper references a paper by Tyson et al. (2020), which describes the approximate size of an out-of-focus object (full-width-half-maximum in radians) as

$\theta_{\mathrm{eff}}^{2} = \theta_{\mathrm{atm}}^{2} + (D_{\mathrm{satellite}}^{2} + D_{\mathrm{mirror}}^{2})/d^{2}$

where $\theta_{\mathrm{atm}}$ is the delivered seeing (in radians), $d$ is the range (distance) to the satellite, $D_{\mathrm{satellite}}$ is the satellite effective projected size, and $D_{\mathrm{mirror}}$ is the diameter of the telescope primary mirror (Bektešević et al. 2018). The mirror size enters because the telescope optics are focused for parallel rays, while satellites have a finite range. A simulation of a 2 m satellite at 550 km height seen at 40 deg zenith angle with the Rubin Observatory is shown in Figure 4. Because of the out-of-focus effect, the instantaneous image of the satellite has a donut shape, and the transverse profile of the trail has a double-peaked structure.

If you look carefully, you can see the double-peaked structure in that streak in the HST image.

Since HST is above the atmosphere, $\theta_{\mathrm{atm}}$ is zero, but otherwise that equation should be useful. Kruk et al. appear to use that in the following:

Assuming that an artificial satellite will pass at only 100 km from the pointing of HST, the 3 m Starlink satellite will produce a wide band of $6^{\prime\prime}$ or 120 pixels across the ACS detector, which might have an impact on the scientific exploitation of the HST data.

The image in question is identified in the paper as "observing ID ja4tg4lsq", which can be found in the HST archives (e.g., this search interface at ESA) if you want to try measuring its width and estimating the distance of the satellite from HST at the time of observations....

Edit: OK, I went ahead and downloaded the HST image. It looks like the FWHM of the trail is $\sim 27^{\prime\prime}$, which translates to a distance of $\sim 30$ km, assuming an object diameter of 3 m.