2.1

CubeSat concept

To analyse the concept mentioned above, we consider a 6U CubeSat inside which 2.5U are allocated to the optical relay, 1U to the dichroic splitting the visible and infrared channels and the two cameras, and the remaining to electronics, ADCS and motors. To collect the stellar radiation we consider a compact 3-mirrors optical relay, similar to those proposed already in a number of CubeSat design ([4] - [8]) providing roughly an f/6 system with a FoV of 0.16 degrees – key specifications in Table 1.

Table 1:

main optical parameters of the CubeSat design under consideration.

The detection system consists in two cameras, a broadband visible CMOS (400-900 nm) and a mid-IR microbolometer operating in the thermal infrared (around the 8-15μm range). Also in this case, we take reference data from detector systems available in the CubeSat market to define the specific detector parameter basing on available designs (e.g. [4]) and off-the-shelf items. Particularly, we consider a panchromatic camera in the visible range similar to that specified in [1] as well as an uncooled microbolometer with performance similar to [9] [10] – summary of key parameters in Table 2.

Table 2:

main parameters of the detection system under consideration.

The CubeSat is foreseen to be launched on a drifting low-Earth orbit, at 700km height. In this configuration, the satellite automatically acquires images in the two channels, in short exposures to allow for frequent mapping of specific portions of the sky – in the search for bright transient events. At this height, the satellite speed would be around 7.5 km/s, with an orbital period of around 100min, providing 14.6 revolutions per day. We consider a mission lifetime of 3 years and assume that the CubeSat attitude is controlled via three reaction wheels delivering and absolute pointing accuracy of 0.1°.

Key to the monitoring activity required to detect transient events, is the radiometric stability of the system, which is achieved by regularly calibrating both channels, with background and flat-field acquisitions. As introduced in the section before, the calibration takes advantage of a moveable aluminium cover mounted on the outer surface of the CubeSat, one side of which is covered in black coating (e.g. Aeroglaze) and the other side protected with MLI, with four pinholes mounted on it. Conceptually, the calibration sequence consist on a two-point (gain/offset) calibration, done by acquiring regularly background and bright flat-field images. It is assumed that the on-board system is used only for relative radiometric calibration and monitoring of effects such as ageing of the optics and detector degradation, since a more precise absolute calibration can be carried out observing stellar standards.

Figure 1:

conceptual representation of the 6U CubeSat design under study. The colours indicate the different function of each unit, blue for CubeSat units housing the optical system, light orange for the detector and beam splitting unit and green for the units housing control electronics for the mechanisms, mass memory, batteries, communication and ADCS (attitude determination and control system). The orange lines surrounding the CubeSat represent a thin MLI substrate that covers most of its outer shell; not represented are the deployable solar panels assumed to be mounted on all the four long sides of the CubeSat.

2.2

Calibration concept

Since a relatively good stability of the detectors is foreseen (which is also well characterizable on-ground), a gain calibration is assumed to be sufficient every 7 days – period after which the orbit has sufficiently drifted and the thermal environment’s changes becomes significant (especially in IR).

The sequence foreseen begins by calibrating the mid-IR channel (Figure 2), they key part of which consist in pointing the black side of the half-open cover to the sun for about 5-10 minutes to heat it up to about 370K. This is then closed to allow flat-fields to be measured. Temperature sensors mounted on the cover allows to compensate the difference in radiance emitted by the cover due to the change of the incident solar radiation throughout the year. In this scenario, we consider that background acquisitions are done before and after the flat-field ones and subsequently weighted to determine the calibration background. This is done through a series of chop/nod acquisitions centered around a relatively dark part of the sky (which will vary depending on the position of the cubesat around the orbit). We assume that in this case the image take is carried out in a short amount of time, therefore no considerable variation in the temperature of the optics is expected (generally the timescale in which optics and the structure respond to temperature changes is longer compared to the calibration timescale).

Figure 2:

sequence foreseen for the infrared gain calibration, the colours indicate the type of action of the sequence: blue for initialization movements of the CubeSat, light-orange for calibration maneuvers and gray for calibration acquisitions. Gray cells contain also additional information on the specific step.

After that, to calibrate the visible channel the closed cover is still used, this time by observing the sun through its pinholes (Figure 3). In this case, the flat-field is created by averaging over various frames acquired while scanning over the sun. To allow sufficient signal every calibration frame, a relatively slow scan speed is considered (around 0.6°/s). With this speed, scanning the sun disc over the focal plane while acquiring images at the exposure time of Table 3, leads to a total calibration time of about 1.2 seconds, assuming 10 frames averaging every 20 pixels. Subsequently, dark images are acquired again by pointing with the cover closed to a relatively dark part of the sky, since we can assume that the radiance transmitted through the pinholes in this configuration is negligible. Similarly to the IR case, also when performing the sun calibration it is assumed that the radiance entering the system is scaled during post-processing for the Earth-Sun distance variation over the year.

Figure 3:

sequence foreseen for the infrared gain calibration, the colours indicate the type of action of the sequence: blue for initialization movements of the CubeSat, green for nominal operation maneuvers and gray for calibration acquisitions. Gray cells contain also additional information on the specific step.

Table 3:

principal cover and calibration parameters considered in the study.

In addition to the calibration sequences described above, since IR observations are strongly affected by the instrument temperatures, a regular daily calibration is considered as well. Every orbit, at different positions around it, a few chop/nod background acquisitions allow minimizing the difference between the calibration background and the IR scene, by following the temperature variations of the satellite while minimizing the impact on the time available for science acquisitions. Assuming a 3ms exposure and a target of acquiring 100 frames (chop/nod differences) per day, this leads to a reserved calibration time of less than 0.002% (of a day).

Moreover, one can take advantage of the larger detector area compared to the active FoV (i.e. exposed to external radiation) to further improve the accuracy of background subtraction in both channels. In the visible, unexposed pixels allow compensating for the temperature-dependent level of the detector dark, using the calibrated background to compensate for non-uniformities in the response of each individual pixel. In the infrared, monitoring the drift of the detector’s surrounding structure allows the scaling of the background level generated by the optics in-between the calibration events.

3.1

Cover heatup

Basing on the calibration scenarios of section 2.2, the heatup of the cover can be analyzed by considering two extreme cases, defined by the minimum and maximum solar radiance (1322/1414 W/m²), as well as albedo (24%/40%) to which the satellite is exposed to. By considering these boundary conditions at an orbit of 700km, one can calculate that at equilibrium, the black side of the cover reaches ~270K.

In this scenario, when exposing a small 8cm in diameter-wide cover to the sun, one can estimate that its surface temperature would heat up to 370K (~100 °C) in a period of 5-10 minutes. With a fixed duration, yearly differences in the effective surface temperature reachable with a fixed heatup time can be monitored using calibrated temperature sensors (four foreseen) embedded within the cover structure. Preliminary simulations show that such a device exhibits a maximum temperature gradient of 3K across its surface after being heated up and closed to irradiate towards the inside of the CubeSat.

3.2

Sun imaging

To calibrate the visible channel, pinholes are used to create a blurred image of the sun onto the focal plane. Their dimension should be sized so that enough power is transmitted without saturating, while allowing for a sufficiently high well fill (ideally around 50%). With the CubeSat configuration under study, one can see that the solar radiance integrated over the detector band (~205 W/m²), transmitted through the four, 500μm pinholes, produces a well fill of around 60% in 0.15ms (with the CCD parameters of Table 2).

4.1

Radiometric stability, visible channel

The contributors to the radiometric instability are differentiated between those present during calibration and those in-between (mid-term), based on the calibration concept of building a flat-field by observing the sun through the pinholes, and acquiring dark images with the cover closed while pointing to a ~dark patch of the sky.

The main contributors present during calibration are the following:

In between calibrations and on the long term the following contributors are taken into account:

Scanning of Sun’s image

To analyse the principal effects of scanning the sun’s image over the detector, we implement a model of a simple imaging system, where the sun’s image is convolved with the PSF generated by a 4-aperture pupil mask. Due to the broadband nature of the detection system, no interference pattern is expected to form.

Firstly, we create a 1D model of the solar disk with a limb-darkened profile, to which we include potential sunspots and flares in the form of narrow peaks across the profile, of varying intensity and width. The profile is then convolved with a simplified 1D PSF of the pupil, created by assuming that the pinholes are distributed symmetrically across 80% of its diameter, the resulting focal plane image is shown in Figure 4 - left.

Figure 4:

(left) example of the 1D profile of the sun as observed through the closed cover, the blue line represents the ideal limb-darkened only normalized profile, while the red one includes sunspots at selected locations, blurred in intensity and edge sharpness from the convolution with the cover+optics PSF. (right) normalized flat-field profile resulting from stacking the moving image of the sun over the focal plane.

Now, if one assumes that sun image is moved across the detector with a speed of 0.6°/s, it follows that with a frame duration of 0.15ms its image is averaged across ~20px during one exposure. This results in a further blurring of the sun profile that, after stacking all frames together, leads to an overall flat-field such as that one shown in Figure 4 (right), which presents a peak-to-valley non-uniformity of roughly 20%, which we assume can be accurately fit with a polynomial down to 1% error. Since this is characterizable with high precision, we assume it does not enter the correction as its shape is constant with calibrations (apart from the pointing errors already included).

Expected radiometric stability

Overall, by considering the calibration and mid-term contributors listed above, one can summarize the main contributors to radiometric instability to those of Table 5, which for the detector configuration listed in section 2 leads to a stability ranging between 0.01% at low signals and ~10^-5 at high signals, as shown in Figure 5.

Table 5:

list of principal contributors to the radiometric stability of the VIS channel, after calibration.

Figure 5:

expected radiometric stability of the signal measured in the visible channel, after gain and offset calibration. Left: stability with respect to the detector well fill, with highlighted in red the signal levels expected with dark and sun imaging. Right: same plot, compared with the input power per pixel.

4.2

Radiometric stability, mid-IR channel

Similarly to the visible channel, contributors to the radiometric instability are differentiated between those present during calibration and those in-between calibration events. In this case, the flat-field is measured by observing the hot cover, while the background is corrected for using a series of chop/nod acquisitions of the dark sky. The main contributors are categorized below, it is considered that for the duration of the calibration the detector temperature remains sufficiently stable not to play a significant role in the calibration instabilities – considering also that it is monitored anyway using the parts of the detector unexposed to radiation.

The main contributors present during calibration are the following:

In between calibrations and on the long term the following contributors are taken into account:

Modelling cover instabilities

Temperature gradients on the surface of the cover and the limited number of sensors that can be fitted on it, leads to an error in the determination of its true temperature (and consequently radiance). Assuming a maximum peak-to-valley temperature difference of 3K after heatup and cover closure, one can estimate the effective temperature error by considering how the sensors are placed over the surface and their measurement accuracy.

To quantify this parameter we follow a statistical (Monte-Carlo) approach, by generating a number of different temperature distributions (temperature nodes) accounting for a certain smoothness in the gradient over the surface (no sharp variations between consecutive nodes). For each distribution, we assume that sensors are placed such that the temperature they measure differs from that one of any neighboring point by no more than a given value (0.5K in our case). The readings from all four sensors are used to determine a mean temperature and from it the mean radiance (⟨I_cal⟩) emitted by the cover. After that, the error is determined from the standard deviation between ⟨I_cal⟩ and the true radiance calculated by considering each temperature node. Finally, the 85% percentile of all possible distributions determines the error accounted for in the computation.

Figure 6:

(left) example of a flattened cover temperature profile generated out of a MC run, where each blue dot represents the temperature of a point (node) in the surface of the cover, the red line the simplified profile around which each temperature node is defined and the black crosses the values measured by each sensor, all with respect to the mean calibration temperature of 370K. (right) histogram of the radiance error in % of the calibration radiance obtained from all MC runs with highlighted the 85% percentile value on the top-right.

In addition to the uncertainty in determining the cover’s true temperature, we assume that over time the emissivity of the cover degrades unequally due the uneven exposure to ageing events (e.g. radiation), thus reducing further the ability of scaling for the correct calibration radiances. In this scenario, we consider that a given area of the cover can degrade its emissivity by maximum 5% over the course of the mission, and that patches large up to 10% of the pupil diameter can degrade evenly at the same time. Also in this case a Monte-Carlo approach is well-suited for the analysis, allowing one to generate several degraded intensity profiles of the cover to determine the residual variation between the BOL radiance slope (assumed to be without local degradations) and a degraded (EOL) one. The 85% percentile of all residuals determine the radiometric error to be included when computing the overall radiometric stability.

Figure 7:

result of Monte-Carlo analysis of the global emissivity degradation across the cover. The values are expressed in % of the effective emissivity taken as reference from the BOL value.

Modelling IR background calibration

As the CubeSat circles around Earth, its temperature will change principally because of the different exposure to the sun and Earth’s albedo, causing the infrared background to fluctuate with it. This combines with the variation in intensity of the diffuse IR background, as a function of the sky location being observed. A regular and continuous background calibration allows to automatically follow the combination of these two effects, reducing the radiometric instability to the local differences between subsequent chop/nod pairs, with a cycling duration of a bit more than one orbit (thermal cycles timescales). Based on these assumptions, we consider the IR background to be weighted with the changing radiation levels present every chop/nod acquisition, due to the different pointing in the sky exposed to different levels of zodiacal light flux ([13] [14]). We simulate the combined effect by considering that a weighted “deep space” temperature cools down the CubeSat differently, depending on the position over the orbit. Every acquisition, its magnitude is calculated by assigning a different intensity to the diffuse IR background based on the probability of pointing to different patches of the sky (higher if observing near the ecliptic, lower when perpendicular to it).

In particular, during every chop/nod sequence, the background will vary as the cubesat points at different areas of the sky, thus also cooling/heating the optics differently with respect to the nominal observation mode. Overall, this follows the temperature fluctuations of the cubesat and results in a small change between the background used for calibration and the effective one (of the true scene). In the simulation we consider that the temperature cyclically change around its reference point by ±10K with a temperature change rate of maximum 0.02K/s. Then we run through a Monte-Carlo analysis to determine the effect of daily acquiring 100 different chop/nod background for different drifting temperature profiles. Finally, the radiometric stability of the background correction is weighted between the moving average of each group of acquisitions and the oscillation of the nominal background while observing (Figure 8).

Figure 8:

extract of the MC simulation showing a group of consecutive IR background calibrations, measured at different levels of illumination coming from the zodiacal light. The values are expressed in terms of equivalent blackbody radiance in Kelvin. The blue circles represent the effective calibration temperature as the system’s temperature changes depending on the levels of “deep space” exposure. The red lines instead indicate the CubeSat’s temperature at the start and end of an acquisition sequence, which can be identified by the yellow ellipse. In total four consecutive calibrations are shown in the graph.

Expected radiometric stability

Overall, by considering the calibration and mid-term effects listed above, one can summarize the main contributors to radiometric instability to those of Table 6, which for the detector configuration listed in section 2 leads to a stability ranging between 0.2% for low signals and 0.01% for high signals, as shown in Figure 9. The main reason for the large instability at low signals lies in the background fluctuation of the optical elements between daily calibrations – in fact if one accounts only for the calibration errors the instability reaches a minimum of 0.005% for signals close to the average level of the IR background (~10 NEP).

Table 6

list of principal contributors to the radiometric stability of the mid-IR channel, after calibration. All converted in radiance units.

Figure 9:

expected radiometric stability of the signal measured in the infrared channel, after gain and offset calibration. The stability with respect to the input power per pixel is shown, with highlighted in red the signal levels expected during background and hot cover imaging.