moonbounce.jl

### A Pluto.jl notebook ###
# v0.17.5

using Markdown
using InteractiveUtils

# ╔═╡ e910b348-9e5a-40d9-be9a-e78aacc47743
using Pkg; Pkg.activate("."); Pkg.instantiate()

# ╔═╡ feb48b54-7423-11ec-2426-23da81f3c079
using SatelliteToolbox, Dates

# ╔═╡ c3e74733-4f49-4849-8d64-5def3f5392e9
using LinearAlgebra,Roots

# ╔═╡ 1bd4c734-e245-4027-ac70-2d8043922cbe
using Plots, Plots.PlotMeasures

# ╔═╡ 274cb358-741c-4b43-bd6a-f96f3453c7fa
md"Download some Earth orientation parameters:"

# ╔═╡ 5c813a37-6916-4214-bb0a-380ca33a767f
eop_IAU1980 = get_iers_eop();

# ╔═╡ 5568e66d-8d62-4b78-825a-ec9e382752ea
md"### Some Helpful Definitions"

# ╔═╡ a8e6acad-212e-4cbe-b2b9-03b7292636f0
"""
Speed of light in a vacuum
"""
c = 299792458 # m/s

# ╔═╡ cedb1362-47a2-44ed-abd8-34b37c9d009e
"""
The frequency of the LoRa signal used in the experiment was between 430 and 440 MHz
"""
fLoRa = 435e6 # MHz

# ╔═╡ e271d16d-d600-41c5-9d58-3b56d1683c04
"""
Approximate latitude, longitue, and altitude of the Dwingeloo Radio Telescope used to transmit the signal
"""
dwingelooLLA = (52.8121, 6.3971, 0) # 0 meters elevation

# ╔═╡ 9b2af5c2-42ea-4b4f-a2e7-9b67a98da790
"""
    earthLLA_r(latdeg, lngdeg, altkm, jd)

J2000 position on Julian day `jd` of a point on the Earth's surface specified by a latitude, longitude, and altitude above the geoid in kilometers.
"""
function earthLLA_r(latdeg, lngdeg, altkm, jd)

	rITRFtoJ2K = r_ecef_to_eci(ITRF(), J2000(), jd, eop_IAU1980)
	itrfPos = geodetic_to_ecef(deg2rad(latdeg), deg2rad(lngdeg), altkm*1000)

	posJ2K = rITRFtoJ2K * itrfPos
end

# ╔═╡ b4fc572f-a9e8-4dac-b7b8-023a4229429d
"""
    earthLLA_rv(latdeg, lngdeg, altkm, jd)

Position and velocity of a point on the Earth's surface on the Julian day `jd` in the J2000 frame. Velocity is derived from a forward finite difference of position.
"""
function earthLLA_rv(latdeg, lngdeg, altkm, jd)
	dt = 1 # seconds
	pos1 = earthLLA_r(latdeg, lngdeg, altkm, jd)
	pos2 = earthLLA_r(latdeg, lngdeg, altkm, jd+dt/86400)

	pos1, (pos2-pos1)/dt
end

# ╔═╡ 11e63381-a19c-4202-96bb-3588c9d8d3a5
"""
    moon_rv(jd)

Position and velocity of the Moon at the Julian date `jd` in the J2000 frame (ignoring conversion between UTC time and Barycentric Dynamical Time). Velocity
is derived from a forward finite difference of position.
"""
function moon_rv(jd)
	R = r_eci_to_eci(MOD(), J2000(), jd)

	dt = 1 # seconds
	
	jd2 = jd + dt/86400

	# convert MOD positions to J2000
	# TODO: convert JD_UTC to JD_TBD
	pos =  R*moon_position_i(jd)
	pos2 = R*moon_position_i(jd2)

	pos, (pos2-pos)/dt
end

# ╔═╡ 57fc4f13-2218-43fe-8193-350f6b6db12e
"""

    moonLLA_r(latdeg, lngdeg, altkm, jd)

J2000 position on Julian day `jd` of a point on the Moon's surface specified by a latitude, longitude, and an altitude above the Moon's average radius in kilometers.

I've defined the Moon frame has +x pointing from Moon center to Earth, +z pointing in the Moon's orbital angular momentum direction, and +y completing the right-handed set.
"""
function moonLLA_r(latdeg, lngdeg, altkm, jd)
	rMoon = 1738.1e3 # meters

	# assumes moon rv is in J2000
	moonpos, moonvel = moon_rv(jd)

	xhat = -normalize(moonpos)
	zhat = normalize(cross(moonpos, moonvel))
	yhat = normalize(cross(zhat, xhat))

	rMoonToJ2000 = [xhat yhat zhat]

	# convert polar angles to a Cartesian position in the Moon frame
	posMoonFrame = (rMoon + altkm*1000) * [cosd(latdeg)*cosd(lngdeg),
										   cosd(latdeg)*sind(lngdeg),
										   sind(latdeg)]
	# conver from my Moon frame to J2000
	posJ2000 = rMoonToJ2000 * posMoonFrame + moonpos
end

# ╔═╡ 73702699-2464-4b22-abc9-422a2285bd4d
"""
    moonLLA_rv(latdeg, lngdeg, altkm, jd)

Position and velocity of a point on the Moons's surface on the Julian day `jd` in the J2000 frame ignoring Barycentric Dynamical Time conversion. Velocity is derived from a forward finite difference of position.
"""
function moonLLA_rv(latdeg, lngdeg, altkm, jd)
	dt = 1 # second
	pos1 = moonLLA_r(latdeg, lngdeg, altkm, jd)
	pos2 = moonLLA_r(latdeg, lngdeg, altkm, jd+dt/86400)

	pos1, (pos2-pos1)/dt
end

# ╔═╡ 461c3f7b-c2a0-4ae2-849b-026978f4dfe3
md"""
## The Main Event: `moonbounce(...)`

With all the position and velocity functions defined, `moonbounce` is able to actually calculate the times of reflection on the Moon and reception back on Earth. The arguments to `moonbounce` are:
- an Earth lat/lng/alt location
- a Moon lat/lng/alt location (relative to my Moon frame defined above)
- a transmission time (as a Julian date)

The functions above make it easy to calculate the positions of (and thus distances and light delays between) points on the Earth and Moon, but we have to be careful because both the Earth and Moon are in motion.

If an Earth point and Moon point are 1.2 light seconds away at a transmission time $t$, it's not correct to say the signal reaches the Moon at $t + (1.2 \text{s})$. Instead, we need to account for the amount that the Moon has moved during the propagation time of the signal! If the Moon is moving slightly away from the transmitter, the arrival time will be something more like $t + (1.20001 \text{s})$. Or if it's moving toward the Earth a bit, the arrival time could be something like $t + (1.19999 \text{s})$. The functions `earthToMoonDelay` and `moonToEarthDelay` below are passed to a quick zero-finding solver to account for this effect.
"""

# ╔═╡ 0e00242a-1023-480f-a187-c5e601712d59
function moonbounce(earthlat, earthlng, earthaltkm,  moonlat, moonlng, moonaltkm,  transmitjd; assumedLightDelaySeconds = 1.0)

	# Cartesian position and velocity of the transmit site at transmission time
	transmitPos, transmitVel = earthLLA_rv(earthlat, earthlng, earthaltkm, transmitjd)

	# calculates the difference between a guess `delaySeconds` and the actual light
	# delay between the transmission site at transmission time and the reflection
	# site `delaySeconds` into the future. Returns zero when `delaySeconds` matches
	# the signal delay between transmission at the transmission site and a 
	# reflection at the reflection site `delaySeconds` later
	function earthToMoonDelay(delaySeconds)
		moonPos, moonVel = moonLLA_rv(moonlat, moonlng, moonaltkm, transmitjd + delaySeconds/86400)

		dist = norm(moonPos - transmitPos)

		dist/c - delaySeconds
	end

	# Earth to Moon reflector light delay in seconds
	e2mSeconds = find_zero(earthToMoonDelay, assumedLightDelaySeconds)

	# time of signal reflection by a reflector on the Moon
	# and position and velocity of the reflector at that time
	reflectionTime = transmitjd + e2mSeconds/86400
	reflectPos, reflectVel = moonLLA_rv(moonlat, moonlng, moonaltkm, reflectionTime)

	# like `earthToMoonDelay` above but for the Moon->Earth direction
	function moonToEarthDelay(delaySeconds)
		earthPos, earthVel = earthLLA_rv(earthlat, earthlng, earthaltkm, reflectionTime + delaySeconds/86400)

		dist = norm(earthPos - reflectPos)

		dist/c - delaySeconds
	end

	# Moon to Earth receiver light delay in seconds
	m2eSeconds = find_zero(moonToEarthDelay, assumedLightDelaySeconds)

	# time of reception back on Earth,
	# and the position and velocity of the receiver at that time
	receptionTime = reflectionTime + m2eSeconds/86400
	receptionPos, receptionVel = earthLLA_rv(earthlat, earthlng, earthaltkm, receptionTime)

	(transmission=(time=transmitjd, pos=transmitPos, vel=transmitVel),
	 reflection=(time=reflectionTime, pos=reflectPos,  vel=reflectVel),
	 reception= (time=receptionTime,  pos=receptionPos,vel=receptionVel))
end

# ╔═╡ f3651d01-81bf-4e3e-ab60-cd70541ce24d
"""
    function dopplerFactorBetween(a, b)
Calculates the Doppler effect frequency shift factor of a light wave transmitted from one point and received by another, both with instantaneous velocities.

Based on the equation from the Wikipedia page [Doppler effect](https://en.wikipedia.org/wiki/Doppler_effect#General).
"""
function dopplerFactorBetween(a, b)
	# vector between the two points
	rAtoB = b.pos-a.pos
	rHat = normalize(rAtoB)

	# source velocity relative to the receiver
	vS = dot(a.vel, rHat)

	# receiver velocity, relative to source
	vR = dot(b.vel, -rHat)  # -rHat because we want the direction from b to a

	factor = (c+vR)/(c-vS)
end

# ╔═╡ 5b4fd441-452e-4bf2-b606-b4724909cdd7
md"
## Generating Points on the Moon

I'd like to evenly cover the Moon with points that will act as reflector locations. A great way to evenly distribute a number of points around a sphere is to use, of all things, the Fibonacci spiral. I'm following the procedure described by [extremelearning.com.au](http://extremelearning.com.au/how-to-evenly-distribute-points-on-a-sphere-more-effectively-than-the-canonical-fibonacci-lattice/)

"

# ╔═╡ e019b7ef-3671-4c48-97a0-aed0c0857efc
n = 500

# ╔═╡ c6af0b6f-955f-4ab3-a105-4996aa1a5383
# generate some reflector latitudes and longitudes
refLats, refLngs = let
	ϕ = (1 + 5^0.5)/2
	i = 0:n-1

	lng = rad2deg.(2pi/ϕ * i)
	# colatitude - zero is +z (the north pole), pi is -z (south pole)
	colat = acosd.(1 .- 2/n .* (i.+0.5))

	colat, lng
end

# ╔═╡ 303254ab-7ab0-476e-892c-a2866a48db29
md"""
## Simulate a bounce

With the `moonbounce` function in place, all that needs to be done is turn each of the transmit-reflect-receive sets it produces into a delay-Doppler pair. I'm also returning a "strength" result for each bounce, which hopefully is an approximate indication of how strong a reflection each bounce is. Bouncing off a part of the Moon facing directly at the transmitter should give a strong reflection, while a glancing bounce should be harder to see in the final plot.

Also included here is the handy `angleBetweend` function that returns the angle between two 3D vectors in degrees.
"""

# ╔═╡ 80c79503-c342-4f31-9356-bae9a29aaf4f
"""
    angleBetweend(a,b)
Angle between two (unnormalized) vectors in degrees

Examples:
```jldoctest
julia> using LinearAlgebra

julia> angleBetweend([1,0,0], [0,1,0])
90.0

julia> angleBetweend([1,0,0], [1,1,0])
45.0
```
"""
function angleBetweend(a,b)
	na = normalize(a)
	nb = normalize(b)
	atand(norm(cross(na, nb)), dot(na,nb))
end;

# ╔═╡ e165d46b-af7f-47aa-8fcb-5999606bdb0f
function dopplerDelayPairsAt(jd)

	# generate a Doppler-delay pair a signal transmitted at time t bouncing off each
	# of the reflectors on the Moon (defined above)
	dopplerDelayPairs = map(refLats, refLngs) do moonLat, moonLng
		bounce = moonbounce(dwingelooLLA..., moonLat,moonLng,0, jd)
	
		moonPosAtReflection = moon_rv(bounce.reflection.time)[1]
		reflectionEarthElevation = 90-angleBetweend(bounce.reflection.pos-moonPosAtReflection, bounce.transmission.pos-bounce.reflection.pos)
	
		transmitterMoonElevation = 90-angleBetweend(bounce.reflection.pos-bounce.transmission.pos, bounce.transmission.pos)
	
		if(reflectionEarthElevation > 5 && # skip "reflections" that have a low (or
										   # negative!) elevation on the Moon
			transmitterMoonElevation>10)   # and also skip ones that have too low 
										   # an elevation on the Earth
			
			shift = fLoRa * 
					(dopplerFactorBetween(bounce.transmission, bounce.reflection) *
					 dopplerFactorBetween(bounce.reflection, bounce.reception)
					 - 1)

			# convert delay in Julian days to delay in seconds
			delay = (bounce.reception.time - bounce.transmission.time)*86400
	
			shift, delay, sind(reflectionEarthElevation)
		else
			nothing, nothing, nothing
		end
	end

	# filter out all the passes that had an elevation that was too low
	dopplers = filter(x->!isnothing(x), map(x->x[1], dopplerDelayPairs));
	delays = filter(x->!isnothing(x), map(x->x[2], dopplerDelayPairs));
	strengths = filter(x->!isnothing(x), map(x->x[3], dopplerDelayPairs));

	dopplers, delays, strengths
end

# ╔═╡ 21e56be6-dafc-46e6-9831-b258d2d7aedb
# generate a delay-Doppler plot for a transmission at the given Julian date `jd`
function dopplerDelayPlotAt(jd; plotargs...)
	dopplers, delays, strengths = dopplerDelayPairsAt(jd)
	
	title = Dates.format(DateTime(jd_to_date(jd)[1:end-1]...), "u. dd HH:MM")
	
	scatter(dopplers, delays, 
		alpha=strengths,     # first cut at Lambertian-type reflection strength TODO
		label=false,
		xlabel="Frequency shift (Hz)",
		ylabel="Delay since start of transmission (s)",
		markerstrokewidth=0,
		guidefont = 8, tickfont = 8,
		title = title;
		plotargs...
	)
end

# ╔═╡ ece89996-7e66-4923-81d2-549bd332752e
let
	transmitTimes = (DateTime(2021, 10, 5, 3, 24, 55):Dates.Hour(1):DateTime(2021, 10, 5, 18, 24, 55))
	
	bouncePlots = dopplerDelayPlotAt.(date_to_jd.(transmitTimes), aspect_ratio=400, ylabel="Delay (s)", xrot=40)
	
	p = plot(bouncePlots..., layout=(4,4),size=(1400,1000), bottom_margin=20px, left_margin=20px)

	titleplot = plot(title="Moon Bounce Doppler Delay October 5, 2021", grid=false, showaxis=false, xaxis=nothing, yaxis=nothing, bottom_margin=-20Plots.px)
	
	plot(titleplot, p, layout=@layout([A{0.01h}; B]))
end

# ╔═╡ 40f45a40-01d2-4569-8825-63a81779cd99
md"
## A Plot that Matches the Original

By guessing and checking, I found that a transimission time of 12:24:55 UTC on October 5, 2021 produces a plot that is a pretty good match to the one shared by Lacuna [here](https://lacuna.space/lora-moon-bounce/)
"


# ╔═╡ 59550d8b-13de-4d99-9164-b5b0a42fea7e
t = date_to_jd(DateTime(2021, 10, 5, 12, 24, 55))

# ╔═╡ 833c3151-9e37-4f72-9ffa-bf1958d9adc7
bounce = moonbounce(dwingelooLLA..., 0,0,0, t)

# ╔═╡ 7059f0f2-c620-4353-a2c7-2070d90320fe
dopplerFactorBetween(bounce.transmission, bounce.reflection)

# ╔═╡ 494659ca-1258-4f04-a8b3-c7889610ab43
dopplerFactorBetween(bounce.reflection, bounce.reception)

# ╔═╡ 356a3352-4a77-41e2-9206-749ca8d40983
# take two Doppler factors and convert to a frequncy shift in Hertz
shift = fLoRa * (dopplerFactorBetween(bounce.transmission, bounce.reflection) *dopplerFactorBetween(bounce.reflection, bounce.reception) - 1)

# ╔═╡ ce1238ea-0897-4d2b-902a-14da86387833
delay = bounce.reception.time - bounce.transmission.time

# ╔═╡ fd28654f-18aa-4e27-a83d-91f2e6cd81c4
dopplerDelayPlotAt(t)

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