Shooting stars on schedule: synchronising re-entry of particles launched from a satellite

The ALE-3 mission plans to create artificial ‘shooting stars’ in the upper atmosphere by launching pellets sequentially from a satellite in low Earth orbit. How does one arrange for all pellets to re-enter over the intended location simultaneously? Starting with conservation of energy and angular momentum, I derive a version of Kepler’s equation that gives time as a function of radial distance in an orbit, which can be used to find the transfer time from the pellet’s ejection to its re-entry as a function of its launch velocity. I show that for a given pellet ejection speed, there is a launch angle from the satellite that results in the fastest transfer time. I use these results to determine the pellets’ launch times and velocities for simultaneous arrival at a desired re-entry point. These results can be applied to de-orbiting any set of objects launched from a satellite, and use concepts that can be covered in an advanced undergraduate course in physics or aerospace engineering.


Introduction
One of the triumphs of classical physics was to explain Kepler's laws of planetary motion as a consequence of Newton's laws of motion and universal gravitation [1,2].Specifically, the orbits of satellites around the Earth can be derived from conserved quantities such as mechanical energy and angular momentum.Changing the velocity of an orbiting object will affect these quantities and change its resulting conic section path.
However, relating position and time for an orbiting object is not straightforward.Many textbooks derive equations for this Kepler problem using a geometrical method similar to that first described by Johannes Kepler himself [2].This requires the definition of new angular measures (the mean, true, and eccentric 'anomalies') to locate the object on its orbit.Instead, in this paper I provide a direct relationship between time and distance from the central body, using the same conserved quantities that define its path.
I apply these results to the ALE-3 mission [3] that will become operational this decade.This mission is a successor to ALE-1 and ALE-2, which had the same goals but experienced malfunctions in orbit.The ALE-3 satellite is a commercial venture of Tokyo-based Astro Live Experiences, Inc.It will carry hundreds of 1 cm diameter pellets that it can launch sequentially from low-Earth orbit, to cause them to re-enter the atmosphere above a specific location to produce 'artificial shooting stars' visible from the ground at night.Each display, consisting of 30-40 pellets launched at intervals of 30 s, is designed to provide entertainment to a population within about 200 km of the target location.While this is the mission's primary goal, given the pellets' known physical properties and launch conditions, tracking their reentry should also allow scientists to probe the properties of the mesosphere between 60 and 100 km altitude.In order to produce simultaneous re-entry, when should the ejection sequence start and end, and with what velocity should each pellet be ejected from the satellite?
In section 2 I use conserved physical quantities to relate a satellite's instantaneous position and velocity to its orbital elements.I derive a version of the Kepler equation that directly relates time and distance from the central body, without the need for geometrical constructs.In section 3 I use these results to predict the orbital elements and transfer times of pellets ejected from a circular orbit.I show that for a given ejection speed, there is an optimal launch angle that results in a minimum transfer time to a specified lower altitude.
In section 4 I apply those results to the ejection of pellets from the ALE-3 spacecraft in a polar circular orbit.After examining purely retrograde ejections, I show how to arrange for sequentially launched pellets to arrive simultaneously over a given location on Earth, working backwards from the last pellet to be ejected.In section 5 I discuss the real world effects on these results, and in section 6 I provide suggestions for future applications.

Relevant equations of orbital motion
Consider a satellite in orbit around a spherical planet of much greater mass M, such that we can neglect any effects of the satellite on it, and use its centre as our inertial reference frame.Since the gravitational acceleration of a satellite is independent of its mass, the form of its orbit depends on the conserved specific mechanical energy and specific angular momentum where 'specific' means 'per unit mass' and v r and v θ are the radial and azimuthal components of the satellite's velocity, respectively.(We should properly assign these conserved quantities to the system of satellite and planet, but for brevity I refer to these as associated with the satellite.)Another conserved quantity for Keplerian orbits is the Runge-Lenz vector [4], which can be made dimensionless to form the eccentricity vector, using a vector product identity to simplify.This vector aligns with the focus-to-periapsis direction, and its conservation shows that orbits in an inverse-square field do not precess.Like the specific energy and angular momentum, it can be calculated at any point in the object's orbit.

Standard relations between orbital elements and conserved quantities
Kepler's 1st law tells us that a bound satellite orbit is an ellipse with the planet's centre of mass at the focus closer to periapsis, where the satellite is closest (and moving fastest).Many textbooks derive the ellipse from the inverse square law force of gravity (e.g.[1,2]), while a simpler method using the eccentricity vector of equation (3) can be found in this journal [4].The elliptical orbit can be described in polar coordinates (r, θ) with an origin at the planet's centre where r is the satellite's distance from the centre and θ 0 is the angular coordinate of the orbit's periapsis.The size and shape of the ellipse are defined by its semi-major axis a and eccentricity e, which give a periapsis and apoapsis distance r P = a (1 − e) and r A = a(1 + e), respectively, as shown in figure 1.I refer the reader to references [1] and [2] that derive the remaining relationships between conserved quantities and orbital elements for this section.The semi-major axis of the ellipse is directly related to the orbital specific energy by Setting θ − θ 0 = ±90°, we see that the semi-latus rectum p of the ellipse is given by the numerator in equation (4).This distance can be shown to relate to the specific orbital angular momentum by Therefore, the eccentricity of the ellipse can be expressed in terms of conserved quantities as For the bound elliptical orbits considered here, ε < 0 so a > 0, and e < 1 unless we also have h = 0, a radial trajectory with zero angular momentum.

Calculating orbital elements from position and velocity
To find the orbital elements in terms of the satellite's instantaneous position and velocity, substitute equations (1) and (2) into (5) and (7) For the special case of a circular orbit, c r = 0 and c θ = 1 so that a = r and e = 0. Otherwise, for an elliptical orbit we can use equations (3) or (4) to find the true anomaly (θ − θ 0 ) and thereby the periapsis angular coordinate θ 0 , completing the definition of the orbit.The online supplement provides more details.Geometry of a a satellite's elliptical orbit.r is the distance from the centre of attraction at one focus, θ is the satellite's angular coordinate, and θ 0 the angular coordinate of periapsis.The geometrical centre of the ellipse is marked with an open circle, r P and r A are the periapsis and apoapsis distances, respectively, a is the semimajor axis of the ellipse, and p is its semi-latus rectum.

Time versus distance
Substituting equation (2) into (1) we obtain an expression for the radial speed at any point in the orbit The last term is a 'centrifugal barrier' that reduces the radial speed even as the satellite falls to a smaller radius.At the periapsis of the orbit, v r = 0 so we can express the specific angular momentum h in terms of the specific energy ε and periapsis distance r P there as While there is no closed-form solution for r(t) when a > 0, we can integrate the reciprocal of equation ( 12) to find time since periapsis as a function of position, i.e.
Setting t(r P ) = 0 we obtain the solution (for bound orbits with a > 0) The interested reader can verify that equation (14) satisfies equation (13), which was obtained from conservation laws only.Setting r = r A = 2a − r P gives the transfer time from apoapisis to periapsis as which is one half of the orbital period, in agreement with Newton's version of Kepler's third law.
Since r P = a(1 − e) we can also write equation (14) in terms of the eccentricity e For a given orbital radius r, equations ( 14) or (15) tells us the time elapsed since periapsis for an ascending satellite, and by symmetry the time until periapsis for a descending one.For this paper I shall use the latter interpretation.
For comparison with traditional derivations [2,5], the angles called eccentric anomaly E and mean anomaly M, both measured from periapsis, are given by Unlike its angular counterpart, equation (15) includes purely radial trajectories, which have h = 0, e = 1, and hence r P = 0 and r A = 2a.However, equations (15) and ( 16) become undefined for a circular orbit (e = 0 and r = r P = a).In that case the time between two points on a circular orbit can be found from the constant angular speed w = = v r GM r .

Ejecting pellets for re-entry
In this section I shall use the results derived above to calculate the transfer time of a pellet ejected from a 400 km altitude circular orbit to re-enter Earth's atmosphere, given its ejection velocity relative to the ALE-3 satellite.

Defining the re-entry altitude
For pellets re-entering the Earth's atmosphere from a 400 km altitude orbit, what altitude should one choose as the lowest for their ballistic motion, where we can neglect aerodynamic drag in comparison to gravity?The 'edge' of space is often assumed to be in the altitude range 80-100 km [6].We expect that atmospheric drag should affect a small particle more than a large space vehicle, due to its lower ballistic number (ratio of mass to frontal area).Therefore we must compare the estimated drag forces at these altitudes to the pellet's weight.The aerodynamic drag force on a pellet can be expressed as I assume a drag coefficient c D ≈ 2.1 for hypersonic re-entry [7] and use a frontal area A = 7.8 × 10 -5 m 2 for a 1 cm diameter pellet.A recent published model atmosphere [8] gives typical densities ρ atm at altitudes of 90, 100, and 110 km of 2.9 × 10 -6 , 0.46 × 10 -6 , and 0.080 × 10 -6 kg m −3 , respectively.Below these altitudes, pellets are expected to ablate and become visible to ground observers [9].
For re-entry speeds v ≈ 7800 m s −1 , the respective drag forces on a pellet at these altitudes are 14, 2.3, and 0.038 mN.For comparison, each pellet of mass m = 2.5 g weighs mg = 24 mN (using g = 9.5 N kg −1 for 90 to 110 km altitude).We can therefore treat each pellet's trajectory as 'ballistic' (following a Keplerian orbit) down to 100 km altitude (a final orbital radius r f = 6478.14km), where the drag force is approximately one tenth of its weight.
As a test of this assumption, after calculating the pellet orbital velocities (section 4), I propagated their orbits numerically from 400 to 80 km using the drag force of equation (16) with the model atmosphere of [8], and found no significant deviation from an elliptical orbit down to ≈90 km altitude.Avoiding the complexity of re-entry dynamics [9], students can use the methods in this section to estimate the 'edge of space' for any orbiting object.

Effect of launch velocity on periapsis
How should a spacecraft in a circular orbit eject an object to transfer it to a lower orbital distance?I assume that the velocity change of the pellet is an impulsive manoeuvre, i.e. occurs with no change in position.In this work, I further limit our analysis of the pellets' motions to coplanar with the satellite's polar orbit-this assumes that the satellite orbit is selected to overfly the target location.In that case a change in the satellite's angular coordinate Δθ equals the change in its geocentric latitude.
Consider a pellet launched at speed u relative to the spacecraft in a direction (elevation angle) f measured relative to the spacecraft's anti-velocity vector, as shown in figure The periapsis distance of the ejected pellet is then given by r P = a(1 − e), i.e.
This is plotted in figure 3 for four possible ejection speeds, using the satellite's orbital speed v C = 7.66855 km s −1 at 400 km altitude (r 0 = 6778.14km).The dotted horizontal line shows the 100 km re-entry altitude (r f = 6478.14km).
Those new to orbital mechanics might suggest that launching the pellet directly downwards towards the Earth's centre (with f = −90°) would provide the fastest re-entry.After all, that would be the case for a projectile launched from an aircraft close to the Earth's surface in a 'flat Earth' approximation.However, in the orbital case a purely radial impulse increases the orbit's specific energy and semi-major axis a of the pellet's path, but has no effect on its specific orbital angular momentum h, so the pellet must still overcome the 'centrifugal barrier' of equation (10) to approach closer.From equation (19) and figure 3, for a pellet fired straight up or down with |f| = 90°, the minimum u to reach 100 km periapsis altitude from a 400 km circular orbit is 0.355 km s −1 , as opposed to only 0.0873 km s −1 for a f = 0°retrograde launch.In the next section I show that the transfer time from circular orbit to re-entry will be reduced if the impulse has a retrograde and a downward component, which reduces both ts orbital energy and angular momentum.

Transfer time for a de-orbiting pellet
Equation (18) gives the a pellet's orbital elements as a function of normalized launch speed s and angle f.For downward-angled or purely retrograde impulses (f 0) the transfer time Δt from r 0 to r f can be found from equation (15) as the difference in time to periapsis, i.e.Δt = t(r f ) − t (r 0 ).For impulses angled above the anti-velocity vector (f > 0, away from the Earth, as shown in figure 2), where the pellet passes through apoapsis before falling back down, the transfer time for the segment of the ellipse between these radii is given by subtracting t(r f ) and t(r 0 ) from the orbital period, i.e.  .Periapsis radius r P versus ejection angle f for a pellet ejected from a circular orbit of radius r 0 = 6778.14km (black dashed line), according to equation (19).Impulse magnitudes shown are u = 0.1 km s −1 (green dotted curve), 0.2 km s −1 (magenta dashed curve), 0.3 km s −1 (orange dot-dashed curve), and 0.4 km s −1 (cyan solid curve), The radius r f = 6478.14km for ballistic re-entry at 100 km altitude is shown as a black dotted line.reduced, the optimal ejection angle becomes less negative (i.e 'flatter', more retrograde in direction) and the transfer time increases, up to a maximum 2684 s, which is one half of the pellet's orbital period for a 'inner Hohmann transfer' with f = 0°at the minimum u = 0.0873 km s −1 required to reach a perigee of 100 km altitude [2,10].
In general, the optimal launch angle and minimum Δt must be found numerically from equations (18), (15), and (20), once values of s, r 0 , and r f are specified.

Synchronising pellets' re-entry in time and space
The primary goal of the ALE mission is to provide 'shooting stars on demand' above a given location on Earth [3].To do so, the spacecraft ejects 30-40 pellets one-at-a-time with a controlled launch velocity, at 30 s intervals.How can the impulses be arranged so that all pellets arrive simultaneously at 100 km directly above the target location?

Pure retrograde ejection of pellets
With pure retrograde launches, the ALE-3 spacecraft can create a display either with a simultaneous 'splash' of shooting stars spread across the sky, or a sequential arrival of many pellets at the same position but spread over a time interval, or a combination of both, However, when all the pellets are constrained to an apoapsis radius equal to the satellite's orbital radius r 0 , it becomes impossible to arrange for any two to intercept each other in space and time before re-entry [11].In order for one pellet to intercept another, its launch speed and angle must be adjusted based on its time of launch.(This is analogous to launching a Transfer times to 100 km altitude for a pellet ejected from a circular orbit at 400 km altitude, for the same launch speeds shown in figure 3, calculated from equations ( 17), (14), and (19).The black dot at (0°, 2684 s) represents the transfer time for the minimum possible ejection speed of u = 0.0873 km s −1 to reach 100 km altitude, from equation (19).
projectile on Earth's surface to hit a target, with a specified range and time-of-flight defining the launch velocity uniquely.)

Re-entry of pellets at the same place and time
A challenge for the ALE-3 mission is to eject pellets at 30 s intervals and have them arrive over the target location simultaneously.To achieve this goal, the ALE-3 mission's strategy is to work backwards, by first calculating the release time and angle for the final pellet in the ejection sequence.This pellet is ejected as close as possible to the desired target arrival time, using the launcher's maximum speed and the optimal minimum-time ejection angle for that speed, most easily found by a numerical search, and as shown in figure 4.
There are two advantages to choosing the minimum-time trajectory for the final pellet.First, at the minimum time, small errors in launch angle will cause small errors in arrival time, since as figure 4 shows, the minimum is broad across launch angles.Second, since pellets are launched sequentially, this strategy minimizes the transfer times of all the pellets in the sequence.For a maximum ejection speed of 0.400 km s −1 , the time-minimising launch angle f = −52.37°and the resulting transfer time to 100 km is Δt = 648.9s from equations ( 18) and (20), and as shown by the cyan curve in figure 4.

Calculating the previous pellets' velocities
To make a display of 'shooting stars on demand', the ALE-3 satellite ejects 30-40 pellets consecutively with a 30 s delay between launches, so once the final pellet's release time is established, for each of the previous pellets ejected, we have their ejection time and hence location on board the orbiting satellite.We also know the time and location of their re-entry to intercept the final pellet.
Connecting these two events for each pellet with a elliptical trajectory is a standard Lambert problem [12] to determine the elements of each pellet's path (a, e, and θ 0 for these coplanar orbits).From these, the required pellet ejection speed u and angle f can be determined from equation (18).Since the pellets arrive at the same re-entry point simultaneously, the Earth's rotation does not affect the solution.
As Gatland [12] describes, the Lambert method begins by guessing a semi-latus rectum p for the connecting orbit, then calculating the necessary eccentricity e for that orbit to connect the two positions, and calculating the transfer time.The method iterates values of p and e(p) until the desired transfer time is obtained, from which a and the other orbital elements can be found, and hence the pellet launch velocities by solving equation (18).In an online supplement, I provide a prescription for solving the Lambert problem for the pellets considered here, and an Excel spreadsheet for implementing the solution described in [12].
However, I recommend an alternate approach for engaging students, to propagate the pellet's orbits numerically to search for the unique values of u and f that cause each pellet to intercept the final pellet as it arrives 100 km above the target location.This 'shooting method' using direct propagation converts the two-point boundary-value Lambert problem into an initial-value problem.In addition to being easier to visualize than Lambert's method described in [12], the shooting method allows one to (later) incorporate gravitational perturbations and non-gravitational forces (such as drag and radiation pressure) at each time step of the propagation.
A disadvantage of the shooting method, compared to the Lambert method, is that two parameters (s and f) must be varied, as opposed to one (the semi-latus rectum of the connecting ellipse).However, the parameter space for the numerical search is well bounded, once the flight time and launch angle has been found for the final pellet that is ejected at the maximum speed.Pellets launched earlier must have slower launch speeds, but no slower than the Hohmann transfer launch speed that takes half an orbit to reach the final radius at periapsis.The launch angle f must also be 'flatter' than that of the final pellet (i.e. less negative, or possibly slightly positive).The propagation time Δt is specified, so the goal to be sought is to reach the desired change in true anomaly Δθ at the end of this flight time.These constraints can only be satisfied for a unique launch velocity (s and f).
It is straightforward to propagate an orbit over time in many software packages, or even in a spreadsheet program [13].For Keplerian orbits, computation of the conserved quantities in equations (1)-( 3) after each time step provides a check that the trajectory propagation is accurate.Since it was available to me, I chose to use the aerospace industry package Systems Toolkit (STK) [14], which uses an 8th order Runge-Kutta method to propagate orbits, and a gradient search to minimize the distances between the pellets on re-entry.An advantage of STK is that the satellite's and pellets' trajectories can be visualized as they are propagated, I provide an STK animation in the supplementary materials, for trajectories calculated with a final pellet ejected speed of 0.400 km s −1 .A tolerance of ±30 m was used in the search.
Table 1 lists the trajectory properties of four representative pellets launched 300 s apart, which can be found by either method.The orbital elements a, e, and θ 0 are derived from the found launch velocity components using equations (9) and (4).The velocity components at reentry are calculated similarly from the conservation laws equations (1) and (2). Figure 5 shows these trajectories in coordinates of altitude versus change in latitude, which is also the change in true anomaly Δθ of the transfer orbit, since we have assumed a polar orbit for the satellite.
Table 1 and figures 5 and 6 show that the first two pellets in this sequence must be launched at angles above the anti-velocity vector, causing them to reach apogee altitudes >400 km before coming down to 100 km.Apogee alititudes can be computed from table 1 as a(1 + e) − 6378.14 km.

Pellets' motions relative to the satellite
Even for the maximum launch speed of 0.400 km s −1 , the pellets take a significant fraction of the satellite's orbital period 5556 s to fall to 100 km altitude.Figure 6 showing the pellets' Table 1.Trajectories of 4 sample pellets, launched 300 s apart from a satellite in a 400 km altitude circular orbit, for a synchronised re-entry at 100 km altitude.u and f are the pellet's launch speed and angle, respectively.Columns are in reverse time order of pellet ejection.The final pellet speed is set at 0.400 km s −1 ; all other quantities shown depend on this value.Δθ is the change in the pellet's geocentric latitude in time Δt.The last two rows list the radial and azimuthal components of each pellet's re-entry velocity at 100 km altitude, from equations (1) and (2).

Δt (s)
648.9 948.9 1248.9 1548.9 motions relative to the launching satellite for readers to appreciate the effects of the Coriolis and centrifugal accelerations in this rotating reference frame [15].
A surprising result for those new to orbital mechanics is that although the pellets are ejected backwards from the satellite, they re-enter over their target location before the satellite's overflight of it, as shown in figure 6.By the time the synchronised pellets have descended from 400 to 100 km altitude, they are 41.8 km ahead of the satellite.This is due to the conservation of orbital angular momentum and energy, and is a well-known effect in orbital rendezvous [15,16].

Real-world effects
As discussed in section 3.1, even when pellets are launched to arrive at 100 km simultaneously above the target, atmospheric drag will become significant below 90 km altitude.Together with the small differences in re-entry velocities (table 1), this may disperse the final display across a region of sky with trails pointing back towards a 'radiant', as seen with meteor showers.Other effects not considered in this analysis include: (a) Earth's oblateness, nonuniform gravitational field, and solar and lunar perturbations.
These will also affect the launching satellite's orbit.The shooting method described in section 4.3 can incorporate these effects.To test their importance, I used STK's high precision propagator for the pellets' trajectories defined in table 1 at the date and target location shown in the supplement animation.I found that all 4 pellets arrived at 100 km altitude with a ±25 km spread and within 6 s of each other.This would still produce a pleasing display from the ground.Small adjustments to the launch velocities (2 m s −1 and 0.1°) were able to synchronise the pellets' arrival time and location.
Figure 5. Trajectories of the 4 pellets ejected with the properties listed in table 1, showing altitude (vertical scale exaggerated for clarity) versus change in geocentric latitude, with the origin set at the target's location.The black, green, orange, and blue curves show the trajectories of the pellets with Δt = 648.9s, 946.9 s, 1248.9 s, and 1548.9 s, respectively.The launching satellite's circular orbit at 400 km altitude is shown as a gray line.All motions are from left to right; positions are plotted every 60 s.
An STK animation of these trajectories is provided online.
(b) I have assumed a polar circular orbit for the launching satellite.Any eccentricity of the ALE-3 satellite's orbit will causes its altitude and angular velocity to change between ejecting pellets.In this analysis I also neglect out-of-plane velocity components that may need to imparted to the pellets to re-enter above a given location.(c) The effect of the pellets' impulses on the ALE-3 spacecraft.Since each pellet has mass ≈2.5 g and the spacecraft mass is ≈70 kg, from momentum conservation a 400 m s −1 pellet will cause a velocity change of ≈0.014 m s −1 to the satellite, negligible relative to its orbital speed v C = 7668.55m s −1 .(d) Any mechanical errors in launch velocity.The pointing system and launch mechanism of ALE-3 are state-of-the-art for a satellite of this size [17].
Small errors in re-entry time will still make for a pleasing display of 'shooting stars'.However, a small difference in re-entry latitude will appear as a larger offset from the zenith for observers at the target location.For example, consider two pellets that re-enter at 100 km altitude (r f = 6478.14km) displaced by Δθ = 0.1°= 0.0017 radian in geocentric latitude.The arc distance between these positions will be (0.0017 × 6478 km) ≈ 11 km.As observed from sea level 100 km below, their angular separation will then be ≈ arctan(11 km/ 100 km) = 6°, noticeable by spectators but still within their field-of-view.Pellet trajectories in the satellite's reference frame, with the properties listed in table 1.The x and y axes denote the satellite's velocity and anti-Earth (zenithal) directions, respectively.

Conclusion
The ALE-3 mission intends to deliver 'shooting stars on demand' by sequentially ejecting pellets from low-Earth orbit, causing them to re-enter simultaneously over a specified location.For their ballistic paths from 400 to 100 km altitude, I used conserved quantities (specific energy and angular momentum), and a time versus radius version of the Kepler equation, to find the ejection velocities for each pellet to intercept each other over the target simultaneously.
The reader can use the provided equations and methods to calculate pellet trajectories for different starting and ending orbit altitudes and different maximum ejection speeds.The ALE-3 mission is not the first to eject an object from orbit to a targeted re-entry point.From 1959 to 1972 the classified US Corona spy satellite program delivered film cartridges back to Earth from 120 to 160 km altitude orbits, by firing a solid rocket to eject a small re-entry capsule with u ≈ 0.4 km s −1 [18].(The 1968 film Ice Station Zebra-a favourite of aviation magnate Howard Hughes-was based on a fictionalised story of a wayward cartridge's recovery).
Our results can also be applied to the ejection of objects from a circular orbit around any other (spherical) planetary body.Ejection speeds will scale with the satellite's circular speed = v GM r C 0 while transfer times will scale with its orbital period p r GM 2 .One possible application could be to 'bomb' a specific location on the surface of an asteroid or comet with a sequence of projectiles launched from orbit, to reveal details of the body's composition as a precursor to resource extraction [19].For planets with an atmosphere, one can deliver multiple ballistic re-entry vehicles from orbit to a specific re-entry point.An advantage of synchronising their re-entries is that the planet's rotation will not disperse them over the surface.These methods could be applied to Star Trek escape pods [20], or more ominously, to kinetic weapons launched sequentially from orbit, often referred to as 'rods from God' [21].
In an online supplement, I provide an animation of the example pellets described in table 1 and shown in figures 5 and 6.I also provide the equations for applying the the Clohessy-Wiltshire approximation [15] and Lambert's method [12] (with a spreadsheet), to calculate the de-orbit launch velocity for a pellet to arrive at a given position and time relative to its launch from the satellite.

Figure 1 .
Figure1.Geometry of a a satellite's elliptical orbit.r is the distance from the centre of attraction at one focus, θ is the satellite's angular coordinate, and θ 0 the angular coordinate of periapsis.The geometrical centre of the ellipse is marked with an open circle, r P and r A are the periapsis and apoapsis distances, respectively, a is the semimajor axis of the ellipse, and p is its semi-latus rectum.
2. Let the normalized launch speed be s = u/v C .Since the pellet starts inside the satellite with c r = 0 and c θ = 1, immediately after ejection its normalized velocity components are then c r = s sin f and c θ = 1 − s cos f, respectively.From equation (9) the corresponding orbital elements are

2 .
Geometry of pellet launches from the ALE-3 satellite (ALE-2 satellite shown, since ALE-3 is still under development).

Figure 4
Figure 4 plots these transfer times for typical launch speeds values in the range of the ALE-3 satellite's capabilities.The figure shows that for a given pellet launch speed u = sv C , there is a minimum transfer time that corresponds to the transfer orbit with the fastest average radial speed between altitudes.The time-minimizing launch angle f approaches −90°( radially downwards) as the launch speed increases.Conversely, if the launch speed is

Figure 3
Figure3.Periapsis radius r P versus ejection angle f for a pellet ejected from a circular orbit of radius r 0 = 6778.14km (black dashed line), according to equation(19).Impulse magnitudes shown are u = 0.1 km s −1 (green dotted curve), 0.2 km s −1 (magenta dashed curve), 0.3 km s −1 (orange dot-dashed curve), and 0.4 km s −1 (cyan solid curve), The radius r f = 6478.14km for ballistic re-entry at 100 km altitude is shown as a black dotted line.

Figure 4 .
Figure 4.Transfer times to 100 km altitude for a pellet ejected from a circular orbit at 400 km altitude, for the same launch speeds shown in figure3, calculated from equations (17),(14), and(19).The black dot at (0°, 2684 s) represents the transfer time for the minimum possible ejection speed of u = 0.0873 km s −1 to reach 100 km altitude, from equation(19).

Figure 6 .
Figure 6.Pellet trajectories in the satellite's reference frame, with the properties listed in table 1.The x and y axes denote the satellite's velocity and anti-Earth (zenithal) directions, respectively.