Lunar reflections throughout SLIM touchdown – Daniel Estévez
In my previous post, I appeared on the Doppler of the SLIM S-band telemetry sign throughout its touchdown on the Moon. I confirmed some waterfall plots of the sign across the residual service. In these, a mirrored image on the lunar floor was seen. The next determine reveals a waterfall of the sign across the residual service, after performing Doppler correction and utilizing a PLL to lock to the residual service. I used to be intrigued by the patterns made by these reflections, specifically by some bands that appear to be a ‘1’ form (essentially the most outstanding occurs at 14:58).
On this submit I research the geometry of the lunar reflection and discover what causes these bands.
Drawback define
The variable that we have now to separate reflections from totally different factors of the Moon’s floor is Doppler. This is just one variable. For the reason that lunar floor is two-dimensional, the issue of figuring out the place the reflection comes from is undetermined. At any given prompt there’s a entire curve of factors on the lunar floor that each one have the identical reflection Doppler. Nevertheless, not all is misplaced, even when at some prompt all of the factors in these curve have the identical reflection Doppler, as time passes the factors can have totally different Doppler versus time curves. So by assuming that there’s a sure correlation in reflection power over time for every level, we would be capable to establish particular person factors on the lunar floor from their Doppler trajectories. Wanting again on the determine above, the query turns into: “do the upper depth patterns within the reflection match the Doppler trajectories of factors on the lunar floor?”.
Doppler formulation
Right here we’re solely within the frequency distinction between a mirrored sign and the direct line-of-sight sign. That is what’s plotted within the vertical axis of the determine above. We are able to make some approximations when computing the Doppler of the mirrored sign.
An correct reflection Doppler calculation includes data of the trajectories of the transmitter (A(t)) and receiver (B(t)) in an inertial body. Assuming we’re in a position to decide the purpose (P(t)) the place the sign that arrives to the receiver at time (t) has bounced off (the bounce occurs at time (t – c^{-1}|B(t) – P(t)|)), then the Doppler is[-frac{f_c}{c}frac{d}{dt}left[|B(t) – P(t)| + |P(t) – A(t – tau(t))|right],]the place (f_c) is the service frequency and (tau(t)) is the answer to the equation[|B(t) – P(t)| + |P(t) – A(t – tau(t))| = ctau(t).]This formulation is difficult to work with, and it additionally makes the questionable assumption that we’re in a position to meaningfully establish a trajectory (P(t)) for the purpose the place the reflection happens. This is sensible when the reflection occurs at a well-defined level, such because the specular reflection level, however not in a extra normal scenario the place the reflection is unfold over an space.
One approximation that we are able to make is to disregard the finite propagation velocity of sunshine. This quantities to saying that (A(t – tau(t))) is roughly equal to (A(t)), which permits us to disregard (tau(t)) fully. Within the case of SLIM, (tau(t)) is on the order of 1 second, and is dominated by the space between the Moon and the Earth. A extra correct approximation would take a hard and fast worth (tau_0) akin to the Moon-Earth distance when the touchdown occurred, and exchange (A(t-tau(t))) by (A(t-tau_0)).
As soon as we ignore the finite propagation velocity of sunshine, the reflection Doppler formulation includes solely factors on the similar time prompt (t). Due to this fact, as an alternative of doing the calculations in an inertial body, we are able to do the calculations in a non-inertial body. In what follows we’ll ignore the propagation velocity of sunshine and assume that (A(t)), (B(t)) and (P(t)) are given in lunar body-fixed coordinates, since this may simplify the calculations.
Since[frac{d}{dt}|v(t)| = frac{langle v'(t), v(t)rangle},]we have now that the reflection Doppler is[-frac{f_c}{c}left[frac{langle B'(t) – P'(t), B(t) – P(t)rangle}B(t)-P(t) +frac{langle P'(t) – A'(t), P(t) – A(t)rangle}P(t)-A(t)right].]We assume that the vector (P'(t)) lies within the airplane tangent to the lunar floor at (P(t)) (considering any floor roughness or inclination), and denote by (N(t)) a unit vector regular to this tangent airplane. This assumption follows naturally from the situation that (P(t)) should lie within the lunar floor for all (t). Notice that right here we’re utilizing the truth that (P(t)) is given in lunar body-fixed coordinates. Definitely (P'(t)) wouldn’t lie within the airplane tangent to the floor if (P(t)) is given in an inertial system centred on the Earth-Moon barycentre. We additionally assume that the incoming and mirrored rays are associated by a mirrored image alongside this tangent airplane:[frac{B(t) – P(t)}B(t) – P(t) = frac{P(t) – A(t)} – 2 N(t) frac{langle P(t) – A(t), N(t)rangle}.]Since (langle P'(t), N(t)rangle = 0), this suggests[frac{langle P'(t), B(t) – P(t)rangle}B(t) – P(t) = frac{langle P'(t), P(t) – A(t)rangle}.]Thus, we are able to simplify the above formulation for the reflection Doppler to[-frac{f_c}{c}left[frac{langle B'(t), B(t) – P(t)rangle}B(t)-P(t) -frac{langle A'(t), P(t) – A(t)rangle}P(t)-A(t)right].]
Doing an analogous reasoning, we are able to discover that the direct line-of-sight Doppler is roughly equal to[-frac{f_c}{c}left[frac{langle B'(t), B(t) – A(t)rangle} -frac{langle A'(t), B(t) – A(t)rangle}right].]The vectors ((B(t) – P(t))/|B(t)-P(t)|) and ((B(t) – A(t))/|B(t)-A(t)|) are roughly equal, for the reason that receiver on the Earth (B(t)) is far additional away from (P(t)) and (A(t)) than the space between (P(t)) and (A(t)), that are factors near the Moon. We’re solely in regards to the distinction between the reflection Doppler and the direct Doppler, so we are able to ignore the primary phrases within the two formulation above and procure[frac{f_clangle A'(t), P(t) – A(t)rangle}]for the reflection Doppler and[frac{f_clangle A'(t), B(t) – A(t)rangle}]for the direct Doppler. These formulation have the next interpretation: the mirrored Doppler is the same as the projection of the spacecraft velocity vector onto the unit vector that provides the line-of-sight from the spacecraft to the reflection level; the direct Doppler is the same as the projection of the spacecraft velocity vector onto the unit vector that provides the line-of-sight from the spacecraft to the receiver at Earth. For simplicity, we are able to ignore the placement of the receiver on Earth and take as (B(t)) the centre of the Earth.
Specular reflection
In a simplified mannequin the place the lunar floor is assumed to be flat and the receiver very far-off, it’s fairly simple to compute the specular reflection level. The trajectory of the mirrored ray is obtained by reflecting the direct ray alongside a horizontal airplane centred on the transmitter. Particularly, the angle that the mirrored ray makes with this horizontal airplane because it originates from the transmitter is the same as the angle made by the direct ray (however it goes downwards as an alternative of upwards).
On this flat floor mannequin, the distinction between the specular reflection Doppler and the direct Doppler has a pleasant interpretation: it’s proportional to twice the vertical velocity of the spacecraft. It’s because the vectors from the spacecraft to the reflection level and to the receiver are symmetric with respect to the horizontal airplane.
A extra correct mannequin considers the lunar floor as a sphere and doesn’t assume that the space to the receiver is infinite. I’ve already handled on this weblog the issue of calculating the reflection level on a sphere for a ray travelling between two given factors. Within the appendix of this post I computed the reflection level as the purpose that minimizes the size of the reflection path, and solved the minimization downside by brute pressure. It seems that this downside is known as Alhazen’s problem and it doesn’t have a closed kind resolution. It may be solved by discovering the roots of a quartic polynomial or by numerically fixing an equation involving trigonometric capabilities. This short paper describes these two methods to resolve it.
As a comparability of every of the 2 fashions, the next plot provides the space alongside the lunar floor from the subsatellite level to the specular reflection level calculated with the flat floor and receiver at infinity mannequin and with Alhazen’s downside (spherical floor mannequin). We are able to see that there’s a noticeable distinction between the 2 fashions firstly of the recording, when SLIM remains to be at 40 km top, however the distinction turns into a lot smaller as SLIM descends.
The spherical mannequin predicts a specular level that occurs nearer to the spacecraft than the flat mannequin. It’s because because of the curvature of the floor, the floor regular rotates away from the spacecraft as we go additional alongside the floor. Due to this fact, in comparison with a mirrored image on a flat floor, the ray from the transmitter have to be tilted barely extra downwards in order that after the reflection on the spherical floor the outgoing ray has the right angle to succeed in the receiver.
Within the spherical floor mannequin, the distinction between the specular reflection Doppler and the direct Doppler is now not proportional to twice the vertical velocity, however it’s moderately shut so long as the spacecraft just isn’t very excessive. Due to this fact, that is nonetheless a helpful instinct to bear in mind.
Right here is the space alongside the floor from the subsatellite level to the horizon, assuming a spherical floor. This offers the radius of the footprint of the satellite tv for pc. The HORIZONS ephemerides that I’m utilizing for these calculations should not correct sufficient after 15:15 UTC. The truth is, they provide a touchdown location which is a few 5 km above the floor. That is what causes the comparatively massive distance to the horizon on the finish of the plot.
LVLH body and most Doppler
In what follows it’s helpful to outline a LVLH (native velocity native horizon) body for SLIM. The +Z vector of this body factors up, within the route of the place vector in Lunar physique mounted coordinates. The +Y vector is outlined because the cross product of +Z and the speed vector in Lunar physique mounted coordinates. This offers a vector that’s regular to the orbital airplane. The +X vector is outlined because the cross product of +Y and +Z. This offers a right-handed system, and furthermore +X factors within the route of the horizontal velocity vector.
The next plot reveals the angles that the speed vector and a vector becoming a member of the spacecraft with some extent within the horizon make with the horizontal XY airplane. We see that for many of the recording the speed vector factors above the horizon. That is in reality what is required if the spacecraft “needs to overlook the bottom”. Within the ultimate a part of the touchdown the spacecraft initiates a steep descent and the speed vector now factors considerably under the horizon.
This geometry has implications for the reflection Doppler. As we have now seen above, we are able to approximate the reflection Doppler because the projection of the speed vector onto the line-of-sight vector to the reflection level. The Doppler is maximal when these two vectors are aligned, however the reflection level must be on the floor. When the speed vector factors under the horizon within the steep descent, it factors to a spot on the floor that provides the utmost Doppler. When the speed vector factors above the horizon, the spot on the floor that provides most Doppler is the purpose within the horizon forward of the spacecraft.
The next plot overlays on the waterfall of the sign reflection the Doppler curves akin to the specular reflection (computed utilizing the spherical floor mannequin), the utmost Doppler (with the restriction that the reflection level have to be on the bottom), and the Doppler on the level within the horizon forward of the spacecraft. The 2 latter curves coincide for many of the recording, as a result of the speed vector factors above the horizon. Even once they don’t coincide in the course of the steep descent, they’re shut.
From this plot it’s clear that the ephemerides are dangerous after 15:15 UTC, for the reason that waterfall reveals reflections that exceed the utmost Doppler in keeping with the ephemerides. Earlier than 15:15 UTC, the contour of the realm within the time-frequency airplane the place there’s a mirrored sign matches the utmost Doppler on floor fairly nicely.
One thing else is noteworthy. The specular reflection Doppler curve doesn’t have any moderately apparent counterpart in what we see within the waterfall of the sign (besides maybe in the course of the steep descent some minutes earlier than 15:15 UTC). Which means that many of the mirrored sign energy that we see doesn’t correspond to the specular reflection. This contrasts with the lunar reflections that we saw from the Lonjiang-2 mission at UHF, which adopted the specular reflection Doppler curve fairly nicely. Some the reason why the specular reflection just isn’t outstanding on this case might be given under.
A selected reflection
The factor that intrigued me most within the waterfall of the mirrored sign is the curved shapes that appear to be a ‘1’. The clearest of those occurs at 14:58 UTC. Perhaps these correspond to reflections on some particular elements of the lunar floor. However which elements? And why do these elements produce a stronger reflection?
What I’ve achieved to reply these questions is to attempt to match the form within the waterfall with the Doppler curve of a specific level within the Moon floor. The best way during which I selected this level is by specifying the second during which the purpose is within the horizon (that is the second during which the purpose first turns into seen from the spacecraft, and so a mirrored image can begin to occur), and the heading thus far on this second (utilizing the LVLH body launched above). By various these two parameters I attempt to match the Doppler curve with the form on the waterfall. Roughly talking, altering the time at which the purpose seems on the horizon strikes the curve left or proper, and altering the heading to the purpose when it’s within the horizon impacts the amplitude and steepness of the curve. I additionally take into account factors that seem within the horizon at a barely earlier and later occasions, and at a barely smaller and bigger heading, with a purpose to receive some error bars for the placement of the purpose.
The next determine reveals a plot of the strongest of those shapes (the one occurring at 14:58 UTC), and the Doppler curve for some extent within the floor that matches this form, in addition to the 2 error bars. The latitude and longitude of the reflection level is given within the determine title.
There’s a lot to unpack within the plot under the waterfall, so let’s go curve by curve. The blue curve reveals the worth that the angle between the conventional of the reflection airplane and the vertical should have in order that the reflection can occur. This isn’t a specular reflection, so the angle just isn’t zero, the floor inflicting the reflection have to be tilted with respect to an ideal spherical floor with a purpose to permit for the mirrored ray to succeed in the Earth. We see that the angle ranges between 10 and 20 levels when the reflection is strongest.
The orange curve provides the elevation of SLIM as seen from the reflection level. This begins at zero when the purpose first seems on the horizon of SLIM, will increase to a most as SLIM passes abeam of this level, after which decreases once more as SLIM leaves the purpose behind and it disappears under the horizon. The inexperienced curve is the elevation of Earth as seen on the reflection level. This may change into related under.
Lastly, the crimson and purple curves give the heading from SLIM to Earth and to the reflection level. The heading is outlined with respect to the LVLH body outlined above. As a result of how the body is outlined, heading will increase counterclockwise, as an alternative of clockwise as is extra ordinary. As a normal rule, to ensure that a powerful reflection to occur, the transmitter, reflector and receiver have to be kind of aligned. These signifies that the 2 headings have to be shut. A mirrored image airplane that could be very tilted can present a mirrored image path during which the heading adjustments considerably. Additionally, scatter by a tough floor (which might be modelled because the floor regular various shortly at a small scale) can change the heading of a mirrored image. However often the reflection can not change the heading a lot. We see that the reflection is simply robust when the distinction between the 2 headings is lower than 10 levels or so. This matches our instinct.
An vital comment is that factors which are symmetric with respect to SLIM’s groundtrack give the identical reflection Doppler curve. This occurs as a result of the angles between SLIM’s velocity vector and the line-of-sight vector from SLIM to every of those two symmetric factors at all times coincide. Right here the purpose I’ve chosen is west of SLIM’s groundtrack. The Earth can also be west of SLIM, since SLIM is travelling in direction of the north within the jap hemisphere of the Moon. Due to this fact, at some prompt this level on floor, and Earth change into aligned. There’s a symmetric level east of SLIM’s groundtrack that has the identical Doppler curve. This level has reverse heading to that of the purpose we’re contemplating. The heading distinction for this level is due to this fact at all times massive, and so a powerful reflection at this level can not occur.
The determine under reveals the reflection factors plotted in LROC QuickMap. Six factors are proven: the reflection level west of SLIM and the 2 factors used as error bars, and the symmetric level east of SLIM with its two error bar factors. When the reflection is most robust, SLIM was south of those factors, seeing the factors at a heading between 10 and 20 levels, and travelling north. The Earth can be to the northwest, at a heading barely above 20 levels.
It’s exceptional that the factors west of SLIM fall fairly near the nortwestern wall of Spallanzani crater. If we think about the geometry, the Earth is at an elevation of round 45 levels at this level of the lunar floor. When the reflection is strongest, SLIM is decrease on the sky of the reflection level, at an elevation of round 20 levels. The northwestern wall of the crater, with a slope of round 20 levels is strictly what is required to mirror the SLIM sign upwards in direction of Earth, as an alternative of to a decrease elevation, as a flat floor would do. The SLDEM2015 slope knowledge in QuickMap reveals that the slope of this crater wall ranges between 15 and 30 levels, so it supplies many surfaces which have the required geometry for the reflection to occur.
Different reflections
The floor level whose reflection Doppler curve was a very good match for the reflection proven above could be very near the northwestern wall of a crater. I’ve defined why this case ought to give a very good reflection geometry, however to attempt to discover if this was only a coincidence, I’ve studied a number of the different reflections which are distinctly seen within the waterfall. Their plots are proven right here.
To automate the evaluation with QuickMap, I’m exporting the reflection factors in a GeoJSON, which might then be loaded in QuickMap. The primary of those reflection factors is on the northwestern wall of the Mutus M crater.
The following reflection is harder to elucidate. The reflection level lies considerably north of the wall of a crater subsequent to Nicolai E. It is likely to be that this wall is what causes the reflections. There are a number of smaller craters across the reflection level, and these additionally present a floor regular which is as required by the reflection geometry. Curiously, the symmetric reflection level east of SLIM lies inside a crater, though I don’t suppose that this crater can flip the heading of the reflection path by round 40 levels, which is what can be wanted to get a mirrored image there.
Lastly, the reflection level for the final reflection matches fairly nicely the northwestern wall of the Lindenau crater.
The next map reveals all of the reflection factors that I’ve recognized. This offers good proof for the truth that a definite reflection seems within the waterfall every time SLIM passes southeast of a crater, because the northwestern crater wall provides a floor that may mirror the sign upwards to Earth.
Polarization
A pure query is why the specular reflection is lacking within the waterfall plot. The comparatively flat lunar floor between the craters supplies ample alternatives for a specular reflection to occur. One thing to bear in mind is that the polarization used for this recording is the one akin to the direct sign (nominally RHCP). How a mirrored image interacts with round polarization is determined by the incidence angle. A traditional reflection returns round polarization of the alternative sense. That is nicely know when coping with antenna reflectors. In distinction, a grazing reflection returns round polarization of the identical sense. Typically a mix of RHCP and LHCP might be returned relying on the incidence angle. When the incidence matches Brewster’s angle, the return might be horizontally polarized. This may be understood as 50% RHCP and 50% LHCP.
The report Radar Reflectivity of the Earth’s surface by Katz provides extra particulars about this. The precise behaviour is determined by the floor composition and frequency, however qualitatively it’s at all times as described above. The next determine, taken from this report, provides the reflection coefficients for a clean sea floor at a frequency of 6 GHz. Within the case of SLIM we as an alternative have the lunar floor at 2.1 GHz. I don’t know if the S-band reflectivity of the Moon has been studied on this element.
The reflection coefficient for round polarization is given by the skinny traces. Identical-sense polarization is the dashed line, and opposite-sense polarization is the continual line. We see that there’s solely a big same-sense reflection for small grazing angles. On this case the Brewster angle is round 7 levels. There may be nonetheless some small quantity of vertical polarization returned at this angle, most likely as a result of the floor doesn’t behave as a super mannequin.
Within the case of SLIM, the grazing angle for a specular reflection is the same as the elevation of the Earth on the reflection level. This elevation will increase as SLIM travels north via the southern lunar hemisphere, however generally it ranges between 35 and 60 levels. These angles are most likely nicely above the Brewster angle, so solely a small fraction of same-sense polarization energy is returned. I believe this explains why there isn’t any distinct signal of a specular reflection within the waterfall plot. It might have been very fascinating to file this occasion in twin polarization and evaluate the 2 polarizations.
Even when we take into account non-specular reflections, the grazing angle is bounded under by the geometry. Since SLIM have to be at a non-negative elevation on the reflection level, the minimal grazing angle is (varepsilon_E/2), the place (varepsilon_E) is the elevation of Earth on the reflection level. Extra generally, the grazing angle is (varepsilon_E/2 + varepsilon_S/2), the place (varepsilon_S) is the elevation of SLIM on the reflection level. This reveals why the same-sense polarization waterfall favours reflections removed from SLIM, for which (varepsilon_S) is small. Such reflections want a floor slope of (varepsilon_E/2 – varepsilon_S/2) with a purpose to mirror the sign upwards in direction of Earth. For the reason that heading to the reflection level and the heading to Earth should even be shut, which means that the same-sense polarization waterfall favours reflections on a route which is near the speed vector. These have a Doppler that’s bigger than the specular reflection or the direct sign.
We are able to additionally see that the reflection generally will get weaker as time advances, till the second of the steep descent. This is sensible taking polarization into consideration. As SLIM travels additional north, the Earth is larger up on the sky, so it’s harder to get reflections with small grazing angle which have a major quantity of sense-sense return. There may be one other impact that’s at play right here, since SLIM can also be descending. Which means that the realm that may mirror indicators will get smaller, because the footprint shrinks. Then again, there may be much less free-space path loss within the radar equation. It seems that the free-space path loss discount wins out, so generally a decrease top ought to give extra mirrored energy.
This isn’t the total story of why of all of the attainable non-specular reflection paths solely a few of them are robust within the waterfall. The transmitter antenna sample would additionally play a task right here, however I don’t have any details about this. At the least we all know that till the steep descent the perspective of the spacecraft was kind of the identical, because it was pointing the engine for a retrograde burn. When the steep descent occurs, the perspective adjustments considerably. Maybe this explains why a powerful reflection seems once more, though the Earth is now very excessive within the sky and reflections with small grazing angle should not attainable.
Conclusions
On this submit I’ve proven how some distinct patterns within the waterfall of the sign mirrored off the Moon’s floor throughout SLIM’s touchdown might be matched to the Doppler curves of reflections occurring on the northwestern partitions of craters. As SLIM passes southeast of those craters whereas travelling north on the southern lunar hemisphere, these partitions give the required reflection geometry. Furthermore, in comparison with the specular reflection, these reflections on crater partitions have a considerably smaller grazing angle, which causes a better same-sense polarization return.
The truth that the info was solely recorded in same-sense polarization constrains the realm the place stronger reflections can occur, favouring this type of crater wall reflections. How the transmitter antenna sample illuminates the lunar floor additionally influences which reflections might be strongest, however there may be not sufficient knowledge in regards to the transmitter antenna to make use of this within the interpretation of the remark.
Code and knowledge
The Jupyter notebook and GNU Radio flowgraph used on this submit might be present in this repository.