Radiation damage to Hubble has been 4.3 years out of phase with the Solar cycle

# Radiation damage to the Hubble Space Telescopehas been several years out of phase with the Solar cycle
Source: [https://arxiv.org/html/2608.18214](https://arxiv.org/html/2608.18214)
a\]Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UKb\]Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK c\]Department of Earth Science and Engineering, Imperial College London, London SW7 2BP, UK d\]Centre for Astrophysics Research, Department of Physics, University of Hertfordshire, Hatfield AL10 9AB, UK e\]Physics Department, Newcastle University, Newcastle upon Tyne NE1 7RU, UK f\]Centre for Electronic Imaging, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK g\]Laboratoire d’Astrophysique, EPFL, Observatoire de Sauverny, 1290 Versoix, Switzerland

Juan Paolo Lorenzo Gerardo Barrios[https://orcid.org/0000-0002-5605-0029](https://orcid.org/0000-0002-5605-0029) Maximilian von Wietersheim\-Kramsta[https://orcid.org/0000-0003-4986-5091](https://orcid.org/0000-0003-4986-5091)Richard Massey[https://orcid.org/0000-0002-6085-3780](https://orcid.org/0000-0002-6085-3780) Richard G\. HayesJacob A\. Kegerreis[https://orcid.org/0000-0001-5383-236X](https://orcid.org/0000-0001-5383-236X)David Lagattuta[https://orcid.org/0000-0002-7633-2883](https://orcid.org/0000-0002-7633-2883)Zane D\. Lentz[https://orcid.org/0000-0003-4428-7843](https://orcid.org/0000-0003-4428-7843) James W\. Nightingale[https://orcid.org/0000-0002-8987-7401](https://orcid.org/0000-0002-8987-7401)Jesper Skottfelt[https://orcid.org/0000-0003-1310-8283](https://orcid.org/0000-0003-1310-8283)Felix Vecchi[https://orcid.org/0009-0004-7808-1979](https://orcid.org/0009-0004-7808-1979)Affiliation:\[Affiliation:\[Affiliation:\[Affiliation:\[Affiliation:\[Affiliation:\[Affiliation:\[

###### Abstract

As well as obtaining beautiful images of the Universe, theHubble Space Telescope’s CCD detectors are sensitive radiation dosimeters that have been monitored in Low Earth Orbit for more than 24 years\. The rate of radiation damage they received has varied over each Solar cycle, but several years out of phase with the appearance of sunspots or coronal mass ejections\. We investigate functional forms that successfully fit the time series of damage to telescopes elsewhere in the Solar system\. We obtain remarkably accurate fits toHubbledata but with physically absurd parameter values\. During image post\-processing, such fits can be used empirically, to correct more than 99\.5% of the radiation damage’s effect on image quality\. However, fits to the time series with physically reasonable parameters produce worse performance\. Our results highlight the diversity of radiation environments in different parts of our Solar system, and the complexity of Low Earth Orbit in particular\. Our results also motivate continued monitoring of radiation damage to currently operational spacecraft, to more reliably predict the rate of degradation in \(and useful lifespan of\) future missions\.

###### keywords

Radiation damage — Low Earth Orbit — Charge Transfer Inefficiency — Hubble Space Telescope — sunspots — arCTIc — instrumentation — detectors

††authorinfo:Send correspondence to J\.P\.L\.G\.B\. at[jplgmb2@cam\.ac\.uk](mailto:[email protected])\.## 1INTRODUCTION

Above the protection of the Earth’s atmosphere, Charge\-Coupled Device \(CCD\) imaging detectors are gradually damaged by the harsh radiation environment\. High\-energy charged particles displace atoms from the silicon wafer, creating lattice defects that disrupt the smooth transport of photoelectrons during detector readout\[[17](https://arxiv.org/html/2608.18214#bib.bib28)\]\. The defects temporarily capture electrons and release them after characteristic delays\[[34](https://arxiv.org/html/2608.18214#bib.bib10),[14](https://arxiv.org/html/2608.18214#bib.bib11)\], shifting charge away from its original location and producing spurious trails behind astronomical sources\[[18](https://arxiv.org/html/2608.18214#bib.bib12),[19](https://arxiv.org/html/2608.18214#bib.bib13),[1](https://arxiv.org/html/2608.18214#bib.bib17)\]as illustrated in figure[1](https://arxiv.org/html/2608.18214#S1.F1)\. Several species of defect can be created, corresponding to different topological configurations of dislocated atoms\. Each species of trap delays charge for a different time, superimposing trails of different length\. This spurious trailing, which depends non\-linearly on source brightness, morphology, and illumination history\[[31](https://arxiv.org/html/2608.18214#bib.bib30)\], can be the most serious obstacle to some scientific measurements\[[10](https://arxiv.org/html/2608.18214#bib.bib3),[37](https://arxiv.org/html/2608.18214#bib.bib27),[4](https://arxiv.org/html/2608.18214#bib.bib16),[38](https://arxiv.org/html/2608.18214#bib.bib18),[5](https://arxiv.org/html/2608.18214#bib.bib26),[28](https://arxiv.org/html/2608.18214#bib.bib33)\]\.

![Refer to caption](https://arxiv.org/html/2608.18214v1/CCD_diagram_PAPER.png)![Refer to caption](https://arxiv.org/html/2608.18214v1/CTI_trailing_1000DPI_squarezoom_ADU_compressed.png)Figure 1:Left:Photoelectrons created in a CCD pixel are counted at the end of an exposure by shifting them in the parallel then serial direction to an amplifier and ADC at the corner\.Right:Radiation damage to theHubble Space Telescopehas created defects that delay electron flow for a duration similar to the pixel\-to\-pixel transfer time in the parallel direction\[[2](https://arxiv.org/html/2608.18214#bib.bib20),[33](https://arxiv.org/html/2608.18214#bib.bib7)\]\. This creates spurious trailing behind all image features whose electrons have to move past many traps\.The physics of radiation damage to to silicon is understood well\. Models of electron flow through damaged CCDs can reproduce the observed trailing, and these models can be inverted to restore the true image\[[7](https://arxiv.org/html/2608.18214#bib.bib2)\]\. Following continuous study over its 24 years in space, gradually improving models of theHubble Space Telescope\(HST\)Advanced Camera for Surveys/Wide Field Channel\(ACS/WFC\),\[[24](https://arxiv.org/html/2608.18214#bib.bib5),[25](https://arxiv.org/html/2608.18214#bib.bib4),[3](https://arxiv.org/html/2608.18214#bib.bib31),[23](https://arxiv.org/html/2608.18214#bib.bib6),[8](https://arxiv.org/html/2608.18214#bib.bib19)\]now correct better than 99\.5% of image trailing\[[22](https://arxiv.org/html/2608.18214#bib.bib37)\]\(figure[2](https://arxiv.org/html/2608.18214#S1.F2)\)\.

![Refer to caption](https://arxiv.org/html/2608.18214v1/correction.png)

Figure 2:Left:An image acquired by theHubble Space Telescopein January 2025, before any processing\. Note the spurious trailing above every image feature, caused by radiation\-induced Charge Transfer Inefficiency\.Right:The same image, corrected following the procedure in Massey et al\. \(2026\)\.Hubble’s CCD detectors are thus \(amongst other uses\) phenomenally precise radiation dosimeters that have been continuously monitored in Low Earth Orbit for longer than two Solar cycles \(figure[3](https://arxiv.org/html/2608.18214#S2.F3)\)\. Measurements of the gradual degradation of CCD performance in various astronomical telescopes may inform models of the radiation environment at different locations within our Solar system, such as SPENVIS\[[16](https://arxiv.org/html/2608.18214#bib.bib35),[27](https://arxiv.org/html/2608.18214#bib.bib36)\]\.

In this paper, we assess a new explanation for the curious time series of CCD degradation inHubble\(Section[2](https://arxiv.org/html/2608.18214#S2)\)\. We also describe recent improvements to the model of electron flow, which have stabilised its performance in extreme regimes, improving the correction of bias and dark frames as well as science exposures \(Section[3](https://arxiv.org/html/2608.18214#S3)\)\. We conclude in Section[4](https://arxiv.org/html/2608.18214#S4)\.

## 2TIME EVOLUTION OF RADIATION DAMAGE

We study a time series of the mean density of charge traps perACS/WFCpixel,ρtrap​\(t\)\\rho\_\{\\mathrm\{trap\}\}\(t\)that cause CTI, as measured by Massey et al\. \(2026\)\[[22](https://arxiv.org/html/2608.18214#bib.bib37)\]\. The timing and the relative amplitude of damage measured in this way roughly matches measurements of damage from the growth rate of sink pixels in the same CCDs\[[13](https://arxiv.org/html/2608.18214#bib.bib8)\]\.

Curiously, the rate of degradation ofHubble’s performance has been out of phase with the Solar cycle\. The rate of damage varies by∼\\sim18\.5% over each 11 year period, but the maximum rate of damage occurs approximately 4\.3 years before Solar maximum\. Massey et al\. \(2026\) fit a model using daily numbers of sunspots,nsunspot​\(t\)n\_\{\\mathrm\{sunspot\}\}\(t\)from the SILSO World Data Center\[[35](https://arxiv.org/html/2608.18214#bib.bib1)\],

ρtrap​\(t\)=ρ0\+AGCR×t\+Asunspot​∫t0t\(nsunspot​\(t′−tlag\)\)η​d​t′,\\rho\_\{\\mathrm\{trap\}\}\(t\)=\\rho\_\{0\}\+A\_\{\\mathrm\{GCR\}\}\\times t\+A\_\{\\mathrm\{sunspot\}\}\\int\_\{t\_\{\\mathrm\{0\}\}\}^\{t\}\\Big\(n\_\{\\mathrm\{sunspot\}\}\(t^\{\\prime\}\-t\_\{\\mathrm\{lag\}\}\)\\Big\)^\{\\eta\}~dt^\{\\prime\},\(1\)wherettis the time in days since launch att0t\_\{0\}\. Best\-fit values of parametersρ0\\rho\_\{0\},AGCRA\_\{\\mathrm\{GCR\}\},AsunspotA\_\{\\mathrm\{sunspot\}\},η\\etaandtlagt\_\{\\mathrm\{lag\}\}are listed in table[1](https://arxiv.org/html/2608.18214#S2.T1)and the fit is shown as a grey curve in figure[3](https://arxiv.org/html/2608.18214#S2.F3): it is almost indistinguishable from the best\-fit sinusoid\. The negative sign of best\-fit parameterAsunspotA\_\{\\mathrm\{sunspot\}\}implies that the appearance of sunspotsreducesthe rate of CCD degradation in Low Earth Orbit, as if the increased particle flux or Solar wind suppresses Galactic Cosmic Rays\[[20](https://arxiv.org/html/2608.18214#bib.bib29),[21](https://arxiv.org/html/2608.18214#bib.bib23),[39](https://arxiv.org/html/2608.18214#bib.bib24)\]\. Those authors were unable to distinguish between thetlag=430−5\+11t\_\{\\mathrm\{lag\}\}=430^\{\+11\}\_\{\-5\}day time lag being a phase difference in Solar physics between the production of energetic particles and sunspots, or a local delay caused by the Earth’s magnetic field\[[15](https://arxiv.org/html/2608.18214#bib.bib22),[26](https://arxiv.org/html/2608.18214#bib.bib25)\]\.

![Refer to caption](https://arxiv.org/html/2608.18214v1/timeseries_annotated_v6.png)

Figure 3:Top:Measurements of the accumulated damage to CCD detectors onboard the Hubble Space Telescope \(black points\), from the amplitude of trails behind warm pixels\[[22](https://arxiv.org/html/2608.18214#bib.bib37)\]\. The grey band shows the mean number of sunspots in a 3 month moving window \(centered on the date shown\), and the standard deviation of the number of suspots per day within that window\. Vertical grey lines indicate coronal mass ejection \(CME\) events with a flux of high energy particles\>\>1010protons cm\-2sr\-1s\-1\.Bottom:The deviation of the damage from linear growth \(black points\), plus three models attempting to predict it\. These include a model based on sunspots, with a 430 day lag \(grey\), a better\-fitting model based on CMEs but with an unphysical 8 year lag \(blue\), and a pragmatic piecewise\-linear empirical model \(red\) that has no physical motivation but enables 99% of the imaging trailing to be corrected\.Here we try fitting a model in which the rate of growth of charge traps,d​ρtrap​\(t\)/d​td\\rho\_\{\\mathrm\{trap\}\}\(t\)/dt, is assumed to correlate with Coronal Mass Ejection \(CME\) events\. We use measurements of the fluence of particles with energy\>\>10 MeV during CMEs, obtained by the NOAAGeostationary Operational Environmental Satellites\(GOES\)111GOES data is from[www\.ngdc\.noaa\.gov/stp/space\-weather/interplanetary\-data/solar\-proton\-events](https://www.ngdc.noaa.gov/stp/space-weather/interplanetary-data/solar-proton-events/SEP%20page%20code.html)\.in Earth Geosynchronous orbit\. Specifically, we fit

d​ρtrapd​t​\(t\)=AGCR\+δ⁡\(t−tCME,i\)×ACME×\(PCME,i\)η\\displaystyle\\frac\{d\\rho\_\{\\mathrm\{trap\}\}\}\{dt\}\(t\)=A\_\{\\mathrm\{GCR\}\}\+\\delta\(t\-t\_\{\\mathrm\{CME\},i\}\)\\times A\_\{\\mathrm\{CME\}\}\\times\(P\_\{\\mathrm\{CME\},i\}\)^\{\\eta\}\(2\)i\.e\.ρtrap​\(t\)=ρ0\+AGCR×t\+ACME​∑i\(PCME,i\)η,\\displaystyle\\text\{i\.e\.\}~~~~\\rho\_\{\\mathrm\{trap\}\}\(t\)=\\rho\_\{0\}\+A\_\{\\mathrm\{GCR\}\}\\times t\+A\_\{\\mathrm\{CME\}\}\\sum\_\{i\}\(P\_\{\\mathrm\{CME\},i\}\)^\{\\eta\}\\,,\\,~~~~~~\(3\)wherePCMEP\_\{\\mathrm\{CME\}\}is the integral 5\-minute average proton flux, in units of protons cm\-2sr\-1s\-1, during an event at timetCME,it\_\{\\mathrm\{CME\},i\}and the indexiiruns over all such events since launch\. This model fits the degradation of CCDs inEuclid\[[36](https://arxiv.org/html/2608.18214#bib.bib14)\]andGaia\[[29](https://arxiv.org/html/2608.18214#bib.bib39)\]222Gaiadata prefer small modulation ofAGCRA\_\{\\mathrm\{GCR\}\}over an period longer than the 10 year mission during \(Claudio Pagani, priv\. comm\.\); it would be very interesting to measure the period and phase of that with respect to the Solar cycle\.both of which are at Lagrange point L2, but gives aterriblefit to the measured values ofρtrap​\(t\)\\rho\_\{\\mathrm\{trap\}\}\(t\)inHubblefor any values of parametersρ0\\rho\_\{0\},AGCRA\_\{\\mathrm\{GCR\}\},ACMEA\_\{\\mathrm\{CME\}\}andη\\eta\.

Qualitatively, a remarkably accurate fit can be obtained by allowing a delaytlagt\_\{\\mathrm\{lag\}\}between the CME and the damage \(and softening the step functions to keepρtrap\\rho\_\{\\mathrm\{trap\}\}differentiable\), via

ρtrap​\(t\)=ρ0\+AGCR×t\+ACME​∑tlauncht\(PCME,i\)η1\+exp⁡\(\(t−tCME,i−tlag\)/λ\)\.\\rho\_\{\\mathrm\{trap\}\}\(t\)=\\rho\_\{0\}\+A\_\{\\mathrm\{GCR\}\}\\times t\+A\_\{\\mathrm\{CME\}\}\\sum\_\{t\_\{\\mathrm\{launch\}\}\}^\{t\}\\frac\{\(P\_\{\\mathrm\{CME\},i\}\)^\{\\eta\}\}\{1\+\\exp\{\(\(t\-t\_\{\\mathrm\{CME\},i\}\-t\_\{\\mathrm\{lag\}\}\)/\\lambda\)\}\}~\.\(4\)This function \(solid blue curve in figure[3](https://arxiv.org/html/2608.18214#S2.F3)\) even incorporates a previously\-unexplained reduction in the rate of damage in 2025\. However, the best\-fit parameters \(table[1](https://arxiv.org/html/2608.18214#S2.T1)\) include an 8 year lag between CME protons reaching Geosynchronous orbit and damage happening to detectors in Low Earth Orbit that is so long it must clearly be unphysical\. This model’s fit is nonetheless even more remarkable when compared to a fit using the pattern of sunspots in the subsequent Solar cycle\. Forcing the fit to a local minimum neartlag≈8−11=−3t\_\{\\mathrm\{lag\}\}\\approx 8\-11=\-3years fails to reproduce features inρtrap​\(t\)\\rho\_\{\\mathrm\{trap\}\}\(t\)\(and is even more unphysical: the as\-yet uncounted number of sunspots in 2027 are needed to model the accumulation of damage in 2024\)\.

The best\-fit value ofACMEA\_\{\\mathrm\{CME\}\}is positive\. If it is instead forced to be negative, as if the particle flux from the Sun during CME events suppresses Galactic Cosmic Rays reaching Low Earth Orbit, the phase of the damage is recovered with atlag=424\.7±2\.1t\_\{\\mathrm\{lag\}\}=424\.7\\pm 2\.1day lag almost identical to that of the sunspot model \(table[1](https://arxiv.org/html/2608.18214#S2.T1)\)\. However, the poor overall fit \(dotted blue curve in figure[3](https://arxiv.org/html/2608.18214#S2.F3)\) fails to reproduce the dynamic range in the rate of damage\. Even with a best\-fit value ofη≈0\\eta\\approx 0\(counting all CMEs equally effectively reproduces the sunspot numbers with coarser time resolution\), this model cannot simultaneously match the steep excess rate of damage during∼\\sim20102010and the shallow reduction in the rate of damage from∼\\sim20142014–20202020\.

Table 1:Best\-fit parameters for models of the time series of damage toHubble’s CCD detectors,ρtrap​\(t\)\\rho\_\{\\mathrm\{trap\}\}\(t\)\. The sunspot model \([1](https://arxiv.org/html/2608.18214#S2.E1)\) produces the grey curve in figure[3](https://arxiv.org/html/2608.18214#S2.F3), with parameter values fitted by Massey et al\. \(2026\)\. The Coronal Mass Ejection model \([4](https://arxiv.org/html/2608.18214#S2.E4)\) with a postive/negative value ofACMEA\_\{\\mathrm\{CME\}\}produces the solid/dotted blue curve in figure[3](https://arxiv.org/html/2608.18214#S2.F3)\.
This analysis leaves us with more questions than answers\. Good fits are possible with apparently unphysical parameters; reasonable parameters and reasonable functional forms lead to qualitatively poor fits\. Pragmatically, accurate correction for CTI is possible with any well\-fitting model, including even a simple piecewise\-linear fit \(red curve and corrected data points in figure[3](https://arxiv.org/html/2608.18214#S2.F3)\), but this approach is not useful to predict future performance\.

## 3IMPROVEMENTS TO A MODEL OF ELECTRON TRANSPORT THROUGH A DAMAGED CCD

Simulating the passage of a cloud ofnen\_\{\\mathrm\{e\}\}electrons through a CCD substrate to readout electronics requires three ingredients\. These are a model of the volume \(or cross\-sectional area\) of that cloud,V⁡\(ne\)V\(n\_\{\\mathrm\{e\}\}\), which determines how many traps it encounters, plus probabilities of capture by and release from a charge trap, which may be functions of time\. Here we report recent improvements in two of those ingredients\.

### 3\.1Cross\-sectional volume of a cloud of electrons

A simple parameterisation of the fraction of a pixel filled by a cloud ofnen\_\{\\rm e\}electrons has often\[[32](https://arxiv.org/html/2608.18214#bib.bib9),[23](https://arxiv.org/html/2608.18214#bib.bib6),[22](https://arxiv.org/html/2608.18214#bib.bib37)\]been

V⁡\(ne\)=\{\(ne−dw−d\)βfor​ne\>d0for​ne<d,V\(n\_\{\\rm e\}\)=\\begin\{cases\}\\left\(\\dfrac\{n\_\{\\rm e\}\-d\}\{w\-d\}\\right\)^\{\\beta\}&\\text\{for \}n\_\{\\rm e\}\>d\\\\\[8\.0pt\] 0&\\text\{for \}n\_\{\\rm e\}<d,\\end\{cases\}\(5\)wherewwis the full depth,d⩾0d\\geqslant 0is the depth of a supplementary buried channel or ‘notch’ built in to some detectors, and the powerβ\\betais typically∼\\sim0\.50\.5\. However, this function is not differentiable atne=dn\_\{\\rm e\}=d, which creates a very large cross\-section for trailing of the next electron\. The sudden change in gradient also creates asymmetry in trailing of e\.g\. bias or dark exposures whenne≈dn\_\{\\rm e\}\\approx d: positive noise fluctuations are exposed to traps, and are trailed away, but negative noise fluctuations are untrailed\.

To rectify these asymmetries, we now allowd<0d<0and use

V⁡\(ne\)=\{\(ne−dw−d\)βfor​ne\>0​and​d<0\(−dw−d\)β​exp⁡\(−βd​ne\)for​ne<0​and​d<0,V\(n\_\{\\rm e\}\)=\\begin\{cases\}\\left\(\\dfrac\{n\_\{\\rm e\}\-d\}\{w\-d\}\\right\)^\{\\beta\}&\\text\{for \}n\_\{\\rm e\}\>0\\text\{ and \}d<0\\\\ \\left\(\\dfrac\{\-\\,d\}\{w\-d\}\\right\)^\{\\beta\}~\\exp\{\\left\(\\dfrac\{\-\\beta\}\{d\}n\_\{\\rm e\}\\right\)\}&\\text\{for \}n\_\{\\rm e\}<0\\text\{ and \}d<0,\\end\{cases\}\(6\)where the coefficient in the exponent ensures smoothness ind​V/d​nedV/dn\_\{\\rm e\}atne=0n\_\{\\rm e\}=0\. Thene=0n\_\{\\rm e\}=0location of the transition between these two curves is arbitrary, and could be another free parameter — but empirically fitting it in the∼\\simzero signal regime would likely be impossible\. Nonetheless, the now\-finite gradient atne=0n\_\{\\rm e\}=0and the extension of the function to negativenen\_\{\\rm e\}\(which is needed in the presence of measurement noise\) suitably stabilises trailing in bias and dark exposures\.

Figure 4:The effective volume of a cloud ofnen\_\{\\rm e\}electrons, from more detailed simulations\[[9](https://arxiv.org/html/2608.18214#bib.bib34)\]of electron density within solid\-state structure using Silvaco TCAD\. Over the full range of signal levels, our new model \(red curves, equation[6](https://arxiv.org/html/2608.18214#S3.E6)with parameters listed in table[2](https://arxiv.org/html/2608.18214#S3.T2)\)\. is as good a fit as that proposed by Clarke et al\. \(dotted black curves\)\. However, our new model usefully extends to negativenen\_\{\\rm e\}, which is needed in the presence of measurement noise, and its finite gradient atne=0n\_\{\\rm e\}=0stabilises correction of CTI in bias or dark exposures where pixel values are close to zero\.In practice, we find that parametersβ\\betaandddare highly degenerate when constrained using the trailing of warm pixels\. Instead, we fitβ\\betaandυ≡log10\(α\)≡βlog10\(−d/\(w−d\)\)\\upsilon\\equiv\\log\_\{10\}\{\(\\alpha\)\}\\equiv\\beta\\log\_\{10\}\{\(\-d/\(w\-d\)\)\}, which are approximately orthogonal\. The inspiration for this can be seen by rewriting the second case of equation \([6](https://arxiv.org/html/2608.18214#S3.E6)\) as333Using logarithms to base 10 rather thaneein the definition ofυ\\upsilonis inelegant, but this choice is now locked in to the code\.V⁡\(ne\)=exp⁡\(υ​ln⁡\(10\)−β​ne/d\)V\(n\_\{\\rm e\}\)=\\exp\{\(\\upsilon\\ln\{\(10\)\}\-\\beta n\_\{\\rm e\}/d\)\}\.

Table 2:Best\-fit parameters of equation \([6](https://arxiv.org/html/2608.18214#S3.E6)\), which models the effective volume of a cloud ofnen\_\{\\rm e\}electrons\. These produce the red curves in figure[4](https://arxiv.org/html/2608.18214#S3.F4)\.The form of volume\-filling function \([6](https://arxiv.org/html/2608.18214#S3.E6)\) is inspired by more detailed solid\-state Silvaco TCAD\[[12](https://arxiv.org/html/2608.18214#bib.bib38)\]simulations of electron density within an \(albeit different\) CCD\[[9](https://arxiv.org/html/2608.18214#bib.bib34)\], reproduced here in figure[4](https://arxiv.org/html/2608.18214#S3.F4)\. Whereas we achieve algorithmic stability by shifting the base power law horizontally, they chose to shift it vertically, asV=γc​\(ne\)βc\+αcV=\\gamma\_\{c\}\(n\_\{\\rm e\}\)^\{\\beta\_\{c\}\}\+\\alpha\_\{c\}\. They obtain best\-fit values\{αc=0\.89\\\{\\alpha\_\{c\}=0\.89,βc=0\.51\\beta\_\{c\}=0\.51,γc=0\.09\}\\gamma\_\{c\}=0\.09\\\}for parallel transport and\{αc=3\.195\\\{\\alpha\_\{c\}=3\.195,βc=0\.59\\beta\_\{c\}=0\.59,γc=0\.04\}\\gamma\_\{c\}=0\.04\\\}for the serial register\. Approximating

β≈βc​and​d=w1−α\(−1/β\)≈w1−\(1\+γc​wβc/αc\)1/βc≈−\(αc/γc\)\(1/βc\)\\beta\\approx\\beta\_\{c\}\\text\{~~~and~~~\}d=\\frac\{w\}\{1\-\\alpha^\{\(\-1/\\beta\)\}\}\\approx\\frac\{w\}\{1\-\(1\+\\gamma\_\{c\}w^\{\\beta\_\{c\}\}/\\alpha\_\{c\}\)^\{1/\\beta\_\{c\}\}\}\\approx\-\(\\alpha\_\{c\}/\\gamma\_\{c\}\)^\{\(1/\\beta\_\{c\}\)\}\(7\)suggests we could used≈−86d\\approx\-86for parallel, andd≈−1530d\\approx\-1530for serial\. However, explicitly refitting their data \(approximately recovered via[plotdigitizer\.com](https://plotdigitizer.com/)\) yields the parameters in table[2](https://arxiv.org/html/2608.18214#S3.T2)and red curves in figure[4](https://arxiv.org/html/2608.18214#S3.F4)\. Both functional forms fit the data well, although the logarithmic axes emphasise their differences at the low end\.

### 3\.2Probability of electrons being released from charge traps

Charge traps typically release captured electrons with a characteristic half\-life, producing an \(approximately\) exponential trail profile\. The characteristic release time,τ\\tautypically takes one of several values, depending on the topological configuration of displaced silicon atoms \(such as a vacancy, divacancy, or interstitial defect\), or the atomic number of an impurity\. However, if the CCD is kept cryogenically cold, trap pumping measurements in laboratories\[[11](https://arxiv.org/html/2608.18214#bib.bib32),[6](https://arxiv.org/html/2608.18214#bib.bib15),[30](https://arxiv.org/html/2608.18214#bib.bib21)\]and in space\[[36](https://arxiv.org/html/2608.18214#bib.bib14),[29](https://arxiv.org/html/2608.18214#bib.bib39)\]suggest newly\-generated traps develop a broadened, roughly lognormal distribution oft​a​utauaround the preferred values\. This is probably because the silicon atoms anneal only slowly towards the common, lowest\-energy configurations\.

The lognormal distributionN⁡\(τ\)N\(\\tau\)can be explained as a normal distribution of trap energy band gapsEE,

N⁡\(E\)∝exp⁡−\(E−Ei\)22​σi2,N\(E\)\\propto\\exp\{\\dfrac\{\-\(E\-E\_\{i\}\)^\{2\}\}\{2\\,\\sigma\_\{i\}^\{2\}\}\},\(8\)whereEiE\_\{i\}is the lowest energy state andσi\\sigma\_\{i\}is some scatter reflecting the lack of annealing\. Following solid\-state theory\[[34](https://arxiv.org/html/2608.18214#bib.bib10),[14](https://arxiv.org/html/2608.18214#bib.bib11),[18](https://arxiv.org/html/2608.18214#bib.bib12)\], a trap’s characteristic release time isτ∝eE/k​T\\tau\\propto e^\{E/kT\}at temperatureTT, so

N⁡\(log⁡τ\)​d​log⁡τ\\displaystyle N\(\\log\\tau\)~d\\log\\tau=\\displaystyle=N⁡\(E\)​d​E\\displaystyle N\(E\)~dE\(9\)N⁡\(log⁡τ\)\\displaystyle N\(\\log\\tau\)=\\displaystyle=d​Ed​log⁡τ​N​\(E\)=k​Tlog⁡e​exp⁡−\(E−Ei\)22​σi2\\displaystyle\\dfrac\{dE\}\{d\\log\\tau\}N\(E\)~~=~~\\dfrac\{kT\}\{\\log\{e\}\}\\exp\{\\dfrac\{\-\(E\-E\_\{i\}\)^\{2\}\}\{2\\,\\sigma\_\{i\}^\{2\}\}\}\(10\)=\\displaystyle=k​Tlog⁡e​exp⁡\(−\(log⁡τ−log⁡τi\)22​\(log⁡ek​T​σi\)2\)\.\\displaystyle\\frac\{kT\}\{\\log\{e\}\}\\exp\{\\left\(\\dfrac\{\-\(\\log\\tau\-\\log\\tau\_\{i\}\)^\{2\}\}\{2\\left\(\\frac\{\\log\{e\}\}\{kT\}~\\sigma\_\{i\}\\right\)^\{2\}\}\\right\)\}~\.i\.e\. a lognormal distribution inτ\\tau\. A distribution of release times perturbs the profile of charge trails\. Nonetheless, we continue to find no statistically significant evidence forσi\>0\\sigma\_\{i\}\>0in measurements of the shape of trails \(in eitherHubbleorEuclid\)\.

## 4Conclusions

Models of Charge Transfer Inefficiency in radiation\-damaged CCDs continue to improve\. Once calibrated using in\-orbit data, they enable increasingly comprehensive correction during data postprocessing\[[22](https://arxiv.org/html/2608.18214#bib.bib37)\]\. However, the rate of degradation over time remains unsatisfactorily unpredictable\. Measurements from several spacecraft have now shown that the rate of degradation depends upon Solar activity — but that it is not necessarily in phase with the appearance of sunspots or coronal mass ejections\. The qualitatively different functions of time that are required to model the degradation of CCDs in Low Earth Orbit or at Lagrange point L2 also highlights the diversity of radiation environments in different parts of our Solar system\.

To predict the future performance of missions likePLATO\[[28](https://arxiv.org/html/2608.18214#bib.bib33)\], and as STScI prepare forHubble’s continuing operation in to the 2030s444[https://hst\-docs\.stsci\.edu/hstos](https://docs.google.com/forms/d/e/1FAIpQLSe0ICn0kcRYjDE141dY0y1-O_9gqxewsLSAORMPCYFVStGfrQ/viewform), we recommend that telescope operation teams continue in\-situ measurements of CTI, using all the capabilities of modern detectors such as electronic injection of charge into pre\-defined patterns\. As well as being useful for all that astronomy and earth\-imaging stuff, CCDs are highly sensitive radiation dosimeters that can teach us about the higher\-energy radiation environment throughout our Solar system\.

###### Acknowledgements\.

Authors were supported in Durham by the UK STFC \(grant ST/X001075/1\) and the UK Space Agency \(grant ST/X001997/1\)\. JAK and JWN are supported by STFC Ernest Rutherford Fellowships, and ZDL is supported by STFC studentship ST/Y509346/1\. Our statistical analysis software was partially developed through STFC grant ST/T002565/1, then InnovateUK grants TS/V002856/1 and TS/Y014693/1\.

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