Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

Green- Apis

Red- Orbit line

Blue- Equator ground track of orbit line

CONTENTS

1) Introduction

2) Assumptions for inputs for calculations 1

3) Notes on the flyby calculation

4) Terms in the flyby calculation

5)The flyby scenario (simple version)

6) Assumptions for inputs 2

7) The flyby scenario (longer version)

8) The numbered stages in detail

9) Flyby relative speeds and duration

10) Delta v budget for flyby and escape

11) After escape: hibernation and return via the quasi orbit

12) Flyby vs crash means more data transmission

13) Conclusion

14) Appendix- glossary

1) INTRODUCTION

On September 30th 2016, the Rosetta mission team are going to crash-land the Rosetta orbiter on 67P. Marco Parigi suggested on the Rosetta blog that the Rosetta orbiter should do a close flyby and then boost to escape, hibernate, and return when the comet approaches perihelion again in 2021. I’ve suggested several times on the Rosetta blog about studying Apis at very close quarters as I believe it to be the the most primordial region on the comet, based on the findings in stretch theory.

I’ve now put together a dare-devil flyby scenario for September 30th 2016 that would satisfy both objectives. Instead of crashing, Rosetta would overfly Apis at 200 metres’ altitude, taking 26 minutes and 40 seconds to complete the Apis ground track. She would then boost to escape and return in 2020. She would return largely of her own accord because she’d be injected into a quasi orbit of 67P on escape.

Of course, nothing in this post would be new to anyone looking at the comet and calculating potential landing or flyby scenarios. It’s basically a Hohmann transfer injection from a circular orbit. The injection orbit is an ellipse. A full Hohmann transfer would circularise the orbit again at the lower radius but we would fly on and perform the escape burn from the comet. However, the way in which it’s done requires only 1 m/sec of delta v, including return and reacquisition in 2021. This proves that despite fuel being low, the delta v budget for all these manoeuvres is a fraction of a per cent of the delta v fuel supply that Rosetta arrived with in 2014.

There may be legitimate objections to the height of the initial circular orbit suggested here or the fact that Rosetta’s approach takes more than 16 hours from such a low height. This is done to minimise the orbital speed at flyby without having to do correction burns to slow Rosetta at the last minute (correction at 1.7 km altitude is planned for the controlled crash slated for September 30th 2016). Slow approach from a low altitude increases perturbations from the uneven gravity field but the characterisation of those perturbations is addressed in detail in the twin post to this part. It’s Part 58, linked here:

https://scute1133site.wordpress.com/2016/08/31/part-58-the-case-for-a-dare-devil-flyby-of-apis/

The data from the characterisation would be fed into this flyby set up, adjusting the size of the flyby ellipse by a small amount. Any residual, uncharacterised anomalies would result in a flyby that was a few metres higher or lower than the 200-metre nominal altitude and a slightly earlier or later arrival time over Apis.

2) ASSUMPTIONS FOR INPUTS FOR CALCULATIONS

67P’s mass: 1e13 kg.

67P’s rotational period: 12.7000 hours.

67P’s obliquity: 52° (angle of its rotation axis to its orbital plane).

Apis radius of rotation: 2600 metres.

Apis width: 400 metres (as defined by the equator ground track across Apis, which is essentially across its width).

These inputs vary in their accuracy but this doesn’t detract from the basic setup for the flyby scenario. Any adjustment to inputs simply trickles down through the calculations rather like for a spreadsheet. This changes the orbital elements involved subtly and the timeline slightly but the scenario remains essentially the same.

The twin post also addresses a wide range of objections that may be raised to the flyby scenario. These range from not taking into consideration 67P’s gravitational anomalies (which would be overlaid on the flyby scenario) to the fact that escape to a hibernation orbit at a higher aphelion would be too cold for the panels/batteries to cope. Also, transmission of data and time needed on the Deep Space Network. If you have any objections, it would be as well to read the twin post too, linked above.

3) NOTES ON THE FLYBY CALCULATION.

All calculations were done on paper using a calculator. No computer software of any sort was used. This is stated only to emphasise that I established in advance what the best flyby scenario was, with recourse to stretch theory, and worked backwards from there. There’s just one scenario that stands out above all others, dictating strict parameters for Rosetta’s position and velocity at all times during flyby and escape, so having established that, I worked backwards systematically from there. That’s as opposed to playing around with orbital software until something looks roughly doable after ten thousand Monte Carlo runs. The strict parameters were:

1) approaching the comet along the extended equatorial plane (the rotation plane).

2) overflying Apis at 200 metres’ altitude and at the lowest possible relative speed (along an equator ground track). Apis is chosen for a large number of reasons but “with recourse to stretch theory”, it is established as being the most primordial region on the comet’s surface. This is due to it being the only section of the 29km^2 surface that hasn’t been, torn, scraped, delaminated, stretched, sheared or flung from the comet. It sits in splendid isolation, comprising the flattened tip of the long axis, as if that was its role during the stretching process. And that’s because it really was its role during stretch. It escaped harm by keeping out of trouble, riding on the stretching tip. Everything else snapped away from Apis via flexion forces rounding the stretching tip and left a passive rectangular tablet experiencing no stresses at all, just stuck on the tip (see the Paleo Rotation Plane Adjustment page for more details on this).

3) a 52° plane change burn (a noncoplanar transfer burn).

4) followed soon after by a parabolic escape orbit burn to ensure Rosetta is sent exactly south (or north) of 67P’s orbital plane. Her direction vector velocity component would therefore be orthogonal to 67P’s orbital plane.

5) followed by a very small correction burn to push the parabolic escape speed residual (which is actually zero) a fraction higher to ensure true escape at extremely slow speed and still orthogonal to 67P’s orbital plane. Rosetta would thus enter a heliocentric orbit that doubles up as a quasi orbit of 67P. This ensures automatic return to the comet for reacquisition in January 2021. The automatic return means Rosetta does this with a theoretical zero delta v budget after the small correction burn stated above. 67P’s heliocentric position (its true anomaly) at the time of acquisition will be the same as for August 2014, in other words, the perfect position for Rosetta to resume scientific analysis.

I may post up the actual calculations in due course but it’s rather tedious to type up let alone read equations without proper maths notation symbols. That was the case for the “Spin-up Calcs” page with about one tenth of the calculations as here. In the meantime, the flyby scenario outlined below just gives the resultant outputs: orbital elements, orbital speeds, escape speeds, delta v burns and delta v budgets. In other words, it contains all the actions that Rosetta would need to perform to execute the flyby, escape and 2021 return.

4) TERMS IN THE FLYBY CALCULATION (a full glossary of terms is at the bottom).

Zero point- this is the periapsis (closest approach point) of the flyby ellipse. It’s positioned such that it will be directly above the mid point of Apis rotating under it when the timeline reads T=0. The zero point is 200 metres above the Apis mid point when the mid point rotates instantaneously under it. Rosetta will fly through the zero point at T=0 as well so both the Apis midpoint and Rosetta meet up at/below the zero point at T=0. The zero point is planned as a fixed point in space with respect to the cometcentric reference frame as well as the heliocentric, ecliptic reference frame. It’s determined in advance and is positioned on the line along which 67P’s equatorial plane intersects 67P’s orbital plane. The heliocentric ecliptic reference frame constraint is only relevant to the escape burns and return to 67P in 2021. You can forget about it being of any importance at all to the actual flyby setup. It’s only well after flyby that it comes into its own.

T minus (number of hours) is time before T=0

T plus (number of hours) is time after T=0. The timeline runs from T= minus 79.9810 hours to T plus 300 hours.

T = 0. This is the time at which Rosetta flies through the periapsis of the flyby ellipse. It’s also the time of Rosetta’s ground track crossing of the mid-point of Apis, in a prograde direction (west to east) along the equator. The equator is therefore Rosetta’s ground track. This is because she’ll be orbiting in the equatorial plane. Thus, Rosetta crosses through the zero point (periapsis) at T = 0 and at that same instant the Apis mid point is vertically below her as it also rotates west to east.

Periapsis- the closest point of approach in the flyby orbital ellipse. It’s at one end of the long axis (semimajor axis) of the ellipse. It constitutes the zero point that’s described above. So the periapsis and zero point are one and the same, at the same fixed point in the cometcentric reference frame (more on this further below).

Apoapsis- the highest point of the flyby ellipse. This is by definition at the opposite end of the ellipse from the periapsis and so a line drawn between periapsis and apoapsis will run through the centre of gravity of 67P. From the point of view of the centre of gravity, the two ends of the ellipse are 180° apart, in opposite directions. Since the centre of gravity will be used for the origin of a polar coordinate system, below, it’s useful to remember this, even when Rosetta is in its preparatory circular orbit that doesn’t posses an apoapsis (constant altitude/radius). When on this circular orbit, Rosetta does kiss the future flyby ellipse’s apoapsis twice, once on circular orbit insertion and a second time (after one orbit) at flyby injection.

5) THE FLYBY AND ESCAPE SCENARIO TIMELINE (LESS DETAILED VERSION WITH LESS EXPLANATION- FULL VERSION LOWER DOWN).

All times are measured from the T=0 time when Rosetta flies through the flyby ellipse periapsis, directly above the Apis midpoint.

At T minus 79.9810 hours (timed at engine burn shut-off):

Circular orbit insertion. Radius 9594 metres. Orbital period 63.5 hours (exactly five times 67P’s 12.7-hour rotation). Orbital speed 0.2637 m/sec. Apis mid point is at 252.828° from the zero point measured in the prograde direction of rotation. All angles around 67P’s 360° rotation circle and Rosetta’s orbit are measured in the prograde direction of rotation from the zero point acting as 0°. The origin of this polar coordinate system is the centre of the comet’s rotation axis i.e. its centre of gravity. The polar coordinate system is sufficient for characterising Rosetta’s position in relation to 67P’s rotation and specifically her ground track position along the equator in relation to the angle of Apis from the zero point. The polar coordinate system is also sufficient for pinpointing the periapsis and apoapsis of Rosetta’s orbital ellipse because their polar angle is the same as the true anomaly angle used for orbital position: 0° for periapsis and 180° for apoapsis. Thus we can dispense with referring to the true anomaly and just refer to the angle from the zero point. It’s basically just a 360° protractor with the centre of gravity at the centre and “0°” stamped at the zero point.

At circular orbit insertion, Rosetta is at 180° from the zero point at burn shut-off. So a straight line between Rosetta and the zero point will run through 67P’s centre of gravity. Crucially, the orbital plane is in the extended equatorial plane of 67P so its ground track will always run along the equator from this point onwards until well after the flyby. Rosetta’s orbit is prograde i.e. with the comet’s rotation.

From T minus 79.9810 hours to T minus 16.4810 hours:

67P rotates five times (or an apparent four times) under Rosetta during her 63.5-hour orbit. ‘Dry run’ soundings are taken when Rosetta is at 270°, 0° and 90° from the zero point. These are dummy runs for flyby orbit injection. The Apis angle and orbit altitude are measured at those three locations in the orbit. On each occasion, the Apis mid point should be 72.828° ahead of Rosetta and Rosetta’s altitude should be 9594 m. If there’s any drift, actual flyby injection at 180° from zero point is performed a few seconds early or late according to the anomaly. This small advance/retard operation is called a reset here. The reset means the zero point is thus shunted forwards/backwards a few metres and and timeline is reset accordingly to 16.4810 hours with seconds being added or taken away. The Apis angle is determined with fiduciary points along the equator vertically below Rosetta (as was done for sub-second rotation period determination).

At T minus 16.4810 hours (timed at engine burn shut-off):

Flyby ellipse orbit injection. Delta v burn is 0.0864 m/sec retrograde (i.e. directed tangential to prograde orbit direction to achieve a retrograde delta v). This is done to reduce Rosetta’s orbital speed from 0.2637 m/sec to 0.1773 m/sec. Rosetta is at 180° from the zero point at burn shut-off i.e. she’s at the apoapsis of the flyby orbit which commences at that instant. Apoapsis radius is 9594 metres. This is because she was injected from the circular orbit. Periapsis is 2800 metres, 200 metres more than the 2600-m radius of rotation for Apis. Periapsis is at the zero point which was fixed long before the flyby scenario was initiated. Ellipse semimajor axis is 6197 metres, semiminor axis is 5183 metres, ellipse eccentricity is 0.548. Apis is again at 252.828° from the zero point because Rosetta orbited once during which period 67P rotated under her 5 times.

T = 0: Apis midpoint flyover. The Apis flyby commences at T = minus 0.2222 hours (13 minutes and 19.5 seconds) as the Rosetta ground track acquires the western rim of Apis. Rosetta passes over the mid point at T = 0 and her ground track crosses the eastern rim of Apis at T plus 0.2222 hours. Total Apis flyby time is 26 minutes and 39 seconds. Rosetta orbital speed at zero point (T = 0) is 0.6075 m/sec. Apis tangential speed is 0.3574 m/sec so the transit speed of Rosetta across Apis, west to east, is 0.2501 m/sec. This speed ignores a very slight slowing of Rosetta in the 13 minutes either side of periapsis which has the effect of slowing the stated transit speed and lengthening the transit time by a few seconds. However, this balances either side of the zero point so all values right on the zero point at T = 0 remain nominal and Rosetta overflies the Apis mid point on time.

T plus 16.4810 hours (timed at burn shut-off). Note ‘plus’ meaning after flyby zero point crossing:

Noncoplanar transfer burn to change Rosetta’s orbital inclination by 52°. Delta v burn is 0.1554 m/sec orthogonal to Rosetta’s orbital plane. This results in a 52° inclination change only, while prograde orbital speed of 0.1773 m/sec remains the same (apoapsis speed as for flyby ellipse injection). The 52° inclination change takes Rosetta’s orbital inclination from 0° (around the equator line) to, unsurprisingly, 52°. This is the same as saying her new orbital plane is now at 52° to the old one around the equator.

Since 67P’s equatorial plane is already at 38° to the orbital plane on which 67P orbits the sun, by adding 52°, Rosetta is orbiting 67P in such a way that her orbital plane is now at exactly 90° to 67P’s orbital plane. The obliquity of 67P is 52° and is by definition the angle its rotation axis makes with the 67P orbital plane around the sun. This means the 67P equatorial plane is at 38° (90°-52°) to its orbital plane around the sun. Hence the need for an inclination change that’s the same as the obliquity value in order get Rosetta orbiting at 90° to 67P’s orbital plane. The noncoplanar transfer burn is performed when Rosetta is back at apoapsis which is why the orbital speed is 0.1773 m/sec, as it was exactly one orbit ago when flyby injection shut-off occurred. The orbital speed can’t increase or decrease if the delta v burn is a sideways push at 90° (orthogonal) to the orbital plane. A sideways push just gives rise to a change of direction, which is an angle change, which is an inclination change in this case of orbits. In the case of supermarket trolleys travelling at 0.1773 m/sec and a child pushing on it exactly sideways, it will eventually give rise to a 52° aisle change with no change in forward speed. It’s an exact read-across, there’s nothing very exotic about this.

T plus 27.5960 hours (timed at burn shut-off at the semiminor axis vertex at orbit radius, 5183 metres):

Inject to parabolic ‘escape’. Delta v is 0.1204 m/sec prograde i.e. directed backwards along a tangent to the retrograde orbit line. This gives rise to an increase in prograde orbital speed from 0.3871 m/sec to 0.5075 m/sec, escape speed for this altitude. Escape is put in inverted commas above because a parabola puts Rosetta on the cusp of escape only. A parabola is used for escape for a reason, instead of a hyperbola which implies excess speed for true escape. This reason is explained in the more detailed flyby explanation under the relevant subheading below.

T plus 300 hours (hopefully after a long window for flyby data transmission):

Escape injection. A delta v burn of 0.03 m/sec is performed. It needs to be the slowest possible speed that ensures escape with no unintended return, decaying orbit and eventual crash during hibernation. That’s why 3 cm/sec is chosen. It’s much easier to get it right when several hundred km from the comet because it’s a larger vector value in relation to any perturbing gravitational accelerations. It’s a prograde burn in the same direction of travel away from 67P as for the parabola but with any necessary adjustments. Those adjustments are made after assessing that speed vector component away from 67P (which should be near zero) and the direction vector component which should be 90° to 67P’s orbital plane. In effect it would be a stand-alone burn from an almost stationary point in cometcentric space. But it would in effect be directed down a continuation of the parabola arm. There will be a celestial sphere reference point for this direction vector that the star trackers can lock onto for the burn.

6) ASSUMPTIONS FOR INPUTS

1) The rotation period of 67P is assumed to be 12.7000hours. The four decimal places implies sub-second accuracy so as to be in keeping with other inputs like the Apis angle measured to three decimal places. This rotation period value may change slightly. It’s changing all the time but ESOC, with the help of JPL, are characterising these rotation period changes right up to the last day. Any change in rotation period can simply be fed into the calculations above and we watch it trickle through like a spreadsheet input and output. The outputs would change by very little and this flyby scenario would remain essentially unchanged. The output changes would point to where the intitial conditions needed tweaking, mainly the initial orbit height and delta v burn for flyby orbit injection (slowdown rocket burn to start the flyby ellipse).

2) The comet’s mass is assumed to be 1E1013 kg. That’s 1 x 10^13 kg or 10,000,000,000,000 kg. It’s actually been updated recently to 9.982E12 kg, which is 99.8% of the value above. But the calculations were already done with the old value before noticing this change. Again, trickle-down would barely change the output.

3) The surface of Apis is assumed to be at a radius of 2600 metres from the rotation axis. 2600 metres is therefore its radius of rotation. This is an informed guess based on measuring the shape model. It’s the least accurate of the estimated inputs. It could be 2500 metres or 2700 metres. Adjustment of the radius of rotation for Apis affects the size and eccentricity (fatness) of the flyby ellipse. This in turn affects the initial orbit height and delta v burn for flyby orbit injection. Again, the effect is minimal though noticeable when compared to the possible changes in (1) and (2).

4) The west to east width of Apis is estimated at 400 metres. The flyby would have its ground track exactly along 67P’s equator, travelling west to east. Apis straddles the equator with the actual equator line just to one side (south) of a line that would bisect Apis. The stated midpoint of Apis is the midpoint of the section of Apis along which the equator line runs. It’s roughly the same width as the rest of Apis, which is a quasi rectangle. Rosetta crosses the width of that rectangle, not its length. Any adjustment to the Apis width would not affect the flyby ellipse calculations or timeline. It would simply lengthen or shorten the flyby.

7) THE FLYBY SCENARIO DESCRIBED IN GREATER DETAIL

There’s a glossary in the appendix at the bottom (numbered “14”). If you find the terms used here difficult to follow, you’ll need to read that section first and then come back. Since this sub-heading is explaining the flyby in greater detail, it necessarily repeats what is presented in the shorter version.

The point in the elliptical flyover orbit where Rosetta will pass over the midpoint of Apis is the periapsis point of that orbit. Since periapsis is a fixed point in space in a cometcentric reference frame (ignoring orbital speed around the sun) this point can be described as our target point. It’s the target for Rosetta to pass through and for the Apis midpoint to pass under together at exactly the same time. It will be called the zero point because it’s also designated as the point where the timeline is zero as well. All times before the zero point passing are called “T minus [time in hours]”. Times after passing through the zero point are called “T plus [time]”. So to summarise:

So the zero point is the periapsis of the flyby ellipse and it’s the point which Rosetta and Apis pass through (Rosetta) and 200 metres under (Apis midpoint) simultaneously.

T minus zero is the time at which Rosetta and the Apis midpoint fly/rotate together through and under the zero point respectively.

The zero point is also used for a related measurement. It’s defined as the zero angle line for the midpoint of Apis as it rotates in its 360° circular path. As the midpoint passes under the zero point we can start counting 1°, 2°, 3°, 90°, 180°, 270° etc. It will be at 359.99°, 1.27 seconds before passing through the zero point again and starting a new cycle. Since two decimal places don’t get us under one-second accuracy, three decimal places are used so as to be in keeping with other calculations. Hence, as you’ll see below, Apis needs to be at 252.828° when flyby orbit injection is performed. The Apis midpoint angle is sometimes just called the Apis angle for brevity.

The zero point is fixed in the celestial reference frame so a line from 67P’s centre of gravity to the zero point is permanently directed at a particular star or point on the celestial sphere, regardless of comet rotation or orbit speed/direction. The celestial reference frame is the heliocentric ecliptic reference frame as opposed to the Equatorial reference frame which is used by astronomers who routinely characterise 67P’s rotation axis angle in relation to the Earth’s Equatorial plane and not the ecliptic plane. All notions of Equatorial RA and declination should be dropped for this flyby and escape. Even the ecliptic plane can confuse somewhat because the key plane throughout this scenario is 67P’s own sun-centred orbital plane. And coming a very close second is its own equatorial plane (rotation plane) that is at 38° to its orbital plane. We’ll see soon that the flyby ellipse’s long axis gets locked along the intersection line of these two planes prior to the whole flyby scenario. This is of course entirely dependent on 67P’s orbital plane and rotation plane. 67P doesn’t care where the Earth is, what angle it’s tilted at or what plane it orbits in. It only cares about where the sun is and consequently, its own orbital plane, along with the angle to that plane in which it chooses to rotate. To 67P, the Earth is just another rock like asteroids are for us. 67P’s orbital plane just happens to be at 7° to the ecliptic which is the orbital plane of our own rock so we use our orbital plane, the ecliptic plane, to define 67P’s plane.

So even though we must start getting into the mindset of 67P’s orbital plane (and equatorial plane) as reigning supreme for this scenario, we do have to define it in some reference frame. And from that definition we will use the same reference frame for fixing the the zero point (periapsis) and the apoapsis. Once that’s done, we leave the ecliptic reference frame and return to the cometcentric reference frame for all our flyby/escape burns, safe in the knowledge that, behind the scenes, the flyby ellipse is configured correctly with respect to 67P’s orbital plane and equatorial plane and that the escape burns will fling Rosetta out at 90° to the orbital plane. That 90° vector puts Rosetta into her own separate heliocentric hibernation orbit. But 67P’s orbital plane is wholly directing the escape vector. That’s because in order for Rosetta’s orbit to be heliocentric and also act as a quasi orbit of 67P she has to escape at exactly 90° to 67P’s orbital plane.

Both 67P’s heliocentric orbital plane and Rosetta’s post-escape, heliocentric orbital plane have an inclination to the ecliptic plane (they’re virtually identical values around 7°) and the ecliptic plane is just a common plane to which they can both be referred to hence its being called a reference frame. The ecliptic is the 2D reference plane in the 3D ecliptic reference frame. The ecliptic plane is the x,y plane and adding the third axis, z, above and below the x,y plane, we have a 3D reference frame. It’s the preferred choice because all solar system orbiting bodies and their orbital elements are based on the ecliptic plane (specifically their orbital inclination and longitude of ascending node).

The ecliptic reference frame is used simply to define the coordinates of the zero point (periapsis), the apoapsis, and consequently, the flyby ellipse’s major axis that runs between them. These are all locked onto 67P’s orbital plane. But this locking is done along the line where its equatorial (rotation) plane intersects that orbital plane. This intersection line is a fixed line in space. It is entirely dependent on what 67P is doing and is established for the flyby ellipse well in advance of the whole flyby scenario. Since it’s fixed in space, it remains fixed throughout the flyby scenario, which is why we can call the zero point a fixed point as well. It’s because the zero point can be placed on this line, 2800 metres from 67P’s centre of gravity. The coordinates of the line get measured in the ecliptic reference frame and that measurement amounts to a fixed celestial coordinate value on the celestial sphere in that reference frame (or two coordinate values, one for each end of the line, produced to infinity). The zero point can then be fixed in space, 2800 metres along the plane intersection line from 67P’s centre of gravity. Then we place the apoapsis 12,742 metres along the intersection line, on the other side of the centre of gravity. 12,742 metres is the length of the flyby ellipse (its major axis) so this second point is its apoapsis. The centre of gravity is on the major axis by definition anyway, so the major axis of the flyby ellipse therefore gets locked exactly along our intersection line of 67P’s orbit plane and equatorial plane.

At that point we can return to the cometcentric reference frame using the zero point as our periapsis target for Apis flyby and the flyby ellipse plane as our main reference plane. Because we fixed the flyby ellipse major axis on the plane intersection line from the outset, we know that the delta v burns we perform in this reference frame will eventually lead to Rosetta escaping at 90° to 67P’s orbital plane.

Since the flyby ellipse is in the rotation plane of the comet and the rotation plane is the same as the equatorial plane, the reference plane for flyby is the equatorial plane. As mentioned above, angular measurements are made around the flyby ellipse from the zero point as 0° and the centre of gravity as the origin.

Of course, 67P is moving along its orbit all this time with the equatorial plane intersection line sliding along the orbital plane but when projected onto the celestial sphere which is considered as infinitely far away, there’s no change in the intersection line’s celestial coordinates. This is how we obtain a zero point that’s fixed in relation to the comet even though the comet and zero point are in reality, moving round 67P’s orbit. It’s a constructed cometcentric reference frame for the purposes of the flyby which simply strips out the heliocentric orbital speed of 67P’s centre of gravity, leaving just the excess speeds of Rosetta going round and round the centre of gravity. This is just like here on Earth where we have a geocentric reference frame for watching NEO’s fly past or changing the orbits of satellites. We’ve stripped out the Earth’s orbital speed of 30km/sec leaving just the relative excess speed and direction of the approaching NEO or orbiting satellites.

There are actually two choices for the placement of the zero point along the line from the centre of gravity: 2800 metres in one direction or 2800 metres in the other. It doesn’t matter which. One choice will result in sending Rosetta exactly north of 67P’s orbital plane after escape. The other choice will send her exactly south.

Whilst Rosetta could be sent north or south anyway via the use of inefficient brute force burns, we assume here that fuel is very low so this very careful pinpointing of the zero point/periapsis and apoapsis in advance minimises the delta v and therefore the fuel needed for the 52° plane change after flyby and prior to escape. With the major axis of the flyby ellipse (running between the two fixed apsides) being aligned along the plane intersection line, we ensure that both the 52° plane change burn and the parabolic escape burn are indeed performed with a minimum of delta v. This is because plane changes are best carried out at the lowest speed possible and apoapsis is where Rosetta is slowest. But plane changes are also best carried out at the node crossing of the two planes. So by placing the apoapsis on the node, which is another way of saying the plane intersection line, we kill two birds with one stone.

By the same token, if the ellipse is aligned with its semimajor axis aligned along 67P’s orbital plane and equatorial plane intersection line, it means the minor vertices (midpoints on the ellipse between the two apsides) are uniquely placed. A tangent to the post flyby ellipse curve (after the 52° plane change) at those vertices runs exactly parallel to 67P’s orbital plane. It means the parabolic escape orbit can be performed here with the knowledge that the burn is at 90° to the required escape direction north/south of 67P’s orbital plane. Rosetta is already orbiting in a plane at 90° to 67P’s orbital plane after the plane change so there’s only one direction vector within that plane that needs fixing rather than any direction in 3D space. So by injecting to escape at the semiminor vertex, we ensure the escape isn’t simply in a plane at 90° to 67P’s orbital plane but in a direction that is completely orthogonal to it. The key is that the long arms of the parabola run at 90° to the apex of the parabolic curve. If Rosetta instigates the injection to parabolic escape when travelling parallel to 67P’s orbital plane, she is at the apex of the parabola she’s about to embark on. So she has to end up travelling at 90° to that parallel direction i.e. orthogonal to 67P’s orbital plane. The parabolic trajectory will take her through 67P’s orbital plane a few hours after parabolic injection and she’d end up travelling away on the opposite side of the plane from the side where injection took place.

The zero point would have to be fixed somewhere precise before instigating flyby anyway so placing it and the apoapsis along the plane intersection line doesn’t cost us anything.

Distances for the flyby are in metres and were calculated to two decimal places so as to be consistent with the precision to one second for the time stamps. However, they are rounded to the nearest metre for easy reading. The same goes for orbital speeds that would be to 6 decimal places for sub-second accuracy. This sort of precision won’t be possible in the real life scenario anyway but its easy enough to do so that any errors drift from values that are known to be exact to the second. Despite such precision, an error of 60 seconds wouldn’t compromise the flyby much and the rounded distances and speeds here imply between a 3- and 20-second error. Reinstating the extra decimal places returns us to one-second or sub-second accuracy.

8) THE NUMBERED STAGES IN DETAIL

1) T minus 79.9810 hours:

Rosetta is injected into a circular orbit of radius 9594 metres. This injection would be from whatever orbit Rosetta was on prior to this. At the time of writing she’s performing orbits that go through the 9km altitude point but these are probably ellipses. It would require somewhere around a maximum of 0.2 m/sec for the delta v burn to circularise at 9594 metres as she passes through that point, assuming the new orbit is coplanar with the former orbit. In this flyby scenario, there would be much preparatory work for several days and weeks, all performed in the extended equatorial plane which is the rotation plane. This work, involving characterising the gravity field, would mean insertion to circular orbit at 9594 metres would indeed be coplanar. That’s because, as stated above, the circular orbit and flyby ellipse are both in the extended equatorial plane.

The burn for circular orbit insertion is performed (burn shut-off time to be precise) at T minus 79.9810 hours. Since all calculations are done in seconds, the precision is to the second. Therefore, four decimal places are used for hours to bring the precision to less than a second. So the ‘0’ at the end isn’t superfluous- it could be a ‘1’ or a ‘3’ etc.. Precision doesn’t necessarily mean accuracy. Accuracy depends on other inputs (e.g. Apis radius), calibration and accuracy of measurements (e.g. Rosetta speed, Apis angle) and assumptions such as characterising the gravity field sufficiently.

Crucially, the new circular orbit for Rosetta needs to be in the same plane as the 67P rotation plane (i.e. the extended equatorial plane) and be a prograde orbit, that is, with the comet’s rotation.

The orbital speed is 0.2637 m/sec and is notionally constant because it’s a circular orbit. In reality it would vary a bit due to the gravitational anomalies. However it’s assumed these are largely characterised for the equatorial plane (see below).

The Apis midpoint must be at 252.828° when the circular orbit injection is made and Rosetta has to be at 180°, i.e. watching Apis rotating ahead of her from that position at that moment.

Even though the flyby orbit injection is 63.5 hours away, its zero point has been fixed already as being 180° away from this, the circular orbit injection point. The zero point is along a line from Rosetta at circular orbit injection, through 67P’s centre of gravity and at 2800 metres altitude on the other side. It’s a fixed point (in the cometcentric reference frame and as stated above, the celestial reference frame. Both reference frames strip out the movement/displacement effect of 67P’s orbital speed of circa 15km/sec. So the zero point is fixed, with Rosetta and the comet doing an elaborate dance through and under it for 78.9810 hours before they pass through and under it at the same time.

The zero point is sited where 67P’s extended equatorial plane intersects its heliocentric orbital plane. Hence the zero point which is also the periapsis of the flyby orbit sits in 67P’s orbital plane. The apoapsis point at the other end of the ellipse is also specifically locked on the plane intersection too so it too is sitting in 67P’s orbital plane and that means the line that joins periapsis to apoapsis runs along 67P’s orbital plane too. This is the semimajor axis of the ellipse. This neat geometrical setup proves very useful for the escape injection burn later on.

An orbit of 9594 metres (9.594km) has a period of 63.5 hours which is 5 times the 12.7-hour comet rotation period. This means that the comet rotates four times under Rosetta in one Rosetta orbit at that height. All altitudinal perturbations brought about by the changing comet configuration below Rosetta are thus symmetrical around the orbit. Rosetta should therefore return to the same 9594-metre height even if perturbed by a few metres up and down through the orbit. More importantly, she will arrive at the same point on the circle as where the circular orbit injection was made, 180° from the zero point. This is therefore also a fixed point in the cometcentric and celestial reference frames. When Rosetta passes through it, Apis will again be at 252.828°. Everything will be just as it was 63.5 hours before. That’s because the comet rotated exactly four times under Rosetta while she completed exactly one orbit.

Metres instead of kilometres are used from now on because they are the SI unit in which the calculations are made.

2) T minus 79.9810 hours to T minus 16.4810 hours:

Final checks. These include orbit height stability; final comet rotation rate analysis; Apis angle.

The Apis angle can be ascertained from fiduciary points on the surface rotating beneath Rosetta. Small boulders were used for calculating the 67P rotation rate slowdown and speed-up before and after perihelion in 2015. This was done with sub-second accuracy, which means that the Apis angle can be measured to at least two decimal places if not the three used here for sub-second time precision consistency.

As mentioned above, the Apis angle has to be at 252.828° at flyby orbit injection just as it needed to be when circular orbit injection was performed 63.5 hours earlier. The reason for this will become clear.

Rosetta is injecting to flyby orbit at 180° from the zero point exactly. This is because the injection burn location is the apoapsis of the flyby orbit. Apoapsis is always 180° from periapsis- together they define the two ends of any orbital ellipse. We already know that the periapsis of our flyby ellipse is defined as the zero angle point for the Apis midpoint angle (as well as the zero point ‘target’ for flyby). So we can use the same 360° protractor for our flyby ellipse. Hence its apoapsis is at 180°, exactly opposite the periapsis zero point as you’d expect for any orbital ellipse. It’s the point from which Rosetta is dropped into the flyby ellipse (orbit) and it’s the highest point to which she will return after flyby.

Since Rosetta is dropped from the circular orbit, the apoapsis is the same height as the circular orbit radius, 9594 metres. Since we’ve chosen a periapsis zero point before we even injected to the circular orbit way back at T minus 79.9810 hours, we must inject Rosetta into its flyby ellipse at the correct time and place. That place is the point at which we started with the original circular orbit injection burn. If we’re a few seconds late with flyby injection we’ll still overfly Apis but we won’t overfly the Apis midpoint as we fly through the periapsis zero point. The midpoint will arrive under the zero point ‘on time’ for Apis as only Apis can do. Rosetta will fly through the zero point late. This “few seconds late” scenario is referring to an actual unwanted error of however many seconds as opposed to the carefully reset zero time and zero point described further up if the Apis angle is found to be wrong. The reset is predicted and accounted for but an unwanted error might be due to say, still more uncharacterised remaining residuals in the Apis angle and altitude or the burn duration/shut-off time.

Even though the later injection shunts the zero point round slightly beyond its planned position as well, Apis has a rotational angular velocity that’s five times that of Rosetta, orbiting at 9594 metres. So late injection at 9594 m translates to a painfully slow creep-round of the zero point 180° away while Apis races on under and beyond it. So a few seconds late with injection means Rosetta arrives at this shunted-round zero point late and then overflies the Apis midpoint a few seconds later. It will still reach its 200-metre altitude (just before the Apis centre point overfly) but it will be a few metres higher than need be on departure at the Apis eastern edge. None of that matters at all for a few seconds’ delay but it starts to matter when it’s over a minute and is rather unsatisfactory if it’s 5 minutes. If it’s 20 minutes, Rosetta will be surveying eastern Imhotep from 1000-metres-plus instead.

Each of the first three of four comet rotations under Rosetta during the 63.5-hour circular orbit is used as a dry run. The comet rotates 1.25 sidereal (celestial reference frame) rotations for every time it rotates under Rosetta. The apparent rotation period will be 16.6250 hours. Rosetta will be at 270°, 0°, and 90° each time the Apis midpoint is (supposed to be) in the correct angle position ahead. On each dry run, Apis should be 72.828° ahead of Rosetta’s ground track point along the equator (252.828°-180°). This can be checked by using a fiduciary boulder that should pass directly under Rosetta on each dry run. This is made slightly easier since Rosetta is orbiting in the same plane as the rotation plane and its ground track is the equator of the comet. The Apis midpoint is on the equator anyway (this is one of many reasons why Apis is a star candidate for flyby). The fiduciary boulder would also be on the equator. It would be 72.828° of comet longitude behind (west of) the Apis midpoint.

When Rosetta reaches 180° again, where the circular orbit insertion was performed, it’s no longer a dry run but the real thing. If the Apis midpoint was seen to be creeping ahead or slipping behind 72.828° from Rosetta on the dry runs we can ‘go for burn’ a few seconds early or late. That would mean that our zero point that was supposed to be so rigidly fixed is itself shunted backwards or forwards a few metres or tens of metres. That’s the reset if it’s needed because Apis is found to be ahead/behind and dragging the aimed-for zero point with it.

In contrast to the reset, the potential unwanted error at injection, described further above is an unwanted error even though Apis is in the predicted position. So an off-nominal burn when Apis is where we predicted it to be is the is error as opposed to the carefully planned reset. An off-nominal burn means a slightly off-nominal orbital speed which brings Rosetta in early/low or high/late. Late/early burn shut-off with the correct orbital speed means much the same thing but with lesser altitude anomalies.

The reset requires the timeline to have a few seconds added to it or taken away. If Rosetta’s orbit altitude isn’t quite right on the dry run observations, a similar operation is done as for the Apis angle with minor adjustments of a few metres to the size of the flyby ellipse, which affects its orbital period. That would again mean going for burn a few seconds earlier or later.

3) T minus 16.4810 hours (assuming no dry run adjustments or a reset, adjusting the timeline to 16.4810 hours anyway):

Flyby ellipse injection. This is 63.5 hours after circular orbit injection. The comet/Rosetta configuration is identical to the configuration at circular orbit insertion time at T minus 79.9810 hours. This is because 67P has rotated five times under that exact cometcentric point in space (5 x 12.7 =63.5). This is why the 9594-metre orbit radius was chosen. That’s the only radius that would bring Rosetta back to this same point after exactly five comet rotations (and four apparent rotations under Rosetta as she completes her single orbit).

If all is well, Apis is at 252.828° again, and Rosetta is at 9594 metres again.

If the height is correct and with the orbit being circular, the orbital speed is as predicted as well: 0.2637 m/sec.

We know we want to pass over Apis at 200 metres’ altitude. That’s 2800 metres from the centre of gravity (c of g). 2600 metres for Apis radius of rotation plus 200 metres altitude of flyby makes 2800 metres. The c of g is the focus of the new ellipse, the flyby ellipse, just as it was the centre of the circular orbit. Circles are a special case ellipse with both foci merged at the centre so the circular orbit was technically orbiting the focus of an ellipse as well- and the c of g is where the focus is located in both cases.

So at T minus zero, flyby midpoint, we want to be 2800 metres on the opposite side of 67P from where we are now. We’re now at 180°, 9594 metres, T minus 16.4810 hours and ready to inject to flyby. That 2800 metre point is of course the zero point, known and planned-for 63.5 hours ago. But now we’re thinking aloud, trying to work out what size and shape our flyby ellipse will have to be in order for Rosetta to fly through that zero point at all, let alone do it on time.

We already know how far we are from the c of g, or focus, on this side of the comet: 9594 metres. We know the 2800m-radius zero point is diametrically opposite, through the c of g on a straight line (periapsis to apoapsis, 0° to 180°, through the c of g focus). So we just have to add 9594 to 2800 to get that length. That’s the long axis or major axis of the flyby ellipse. So that length is 12394 metres. Dividing in two gives us the semimajor axis, 6197 metres and that acts like a radius as an input for the orbital speed equation.

There’s only one ellipse that can have a 12394-metre major axis and a periapsis of 2800 metres. It has to be fat enough to satisfy those parameters and still be an ellipse (after all, you could fit a circle, which is a very fat ellipse, to those two parameters on graph paper but nothing would orbit along it in space). The fatness of the ellipse is the eccentricy and the eccentricity for this ellipse with this major axis and periapsis is 0.548. The semiminor axis is 5183 metres. The eccentricity and semiminor axis aren’t necessary as inputs but it shows we can define a viable orbital ellipse with just the two main parameters: the major axis 12,394m, and periapsis distance, 2800m.

We know the flyby ellipse apoapsis and the original circular orbit kiss at 9594 metres. They have to because that’s where the flyby orbit injection is performed and Rosetta starts ‘dropping’ away from its circular path into the required ellipse. The fact they kiss means that if you left Rosetta to orbit in that ellipse indefinitely, it would keep kissing the apoapsis on every orbit at 9594 metres’ altitude. It has to do this at some set orbital speed. The apoapsis speed is the slowest speed in the orbit. It’s calculated using just the 6197-metre semimajor axis value and the 9594-metre apoapsis value, along with the comet’s mass and the gravitational constant. That speed is 0.1773 m/sec.

If 0.1773 m/sec is the speed at which Rosetta would always pass through apoapsis then that’s the speed we need her to have when the injection burn shuts off. That speed will send her into the correct ellipse for a 2800-metre periapsis and a 200-metre Apis flyby. So the delta v needed (the amount of slowdown) is the circular orbit speed minus the flyby ellipse speed at apoapsis (when the orbit is momentarily kissing the same 9594 radius as the circle). So delta v for flyby injection is 0.2637 m/sec – 0.1773 m/sec = 0.0864 m/sec. This is a retrograde burn (directed tangentially to the orbit line and prograde (forwards)- see glossary).

So now Rosetta has been injected onto her flyby ellipse and the retrograde burn was shut off when Apis was at 252.828°. The shutdown was at T minus 16.4810 hours. Rosetta is at 180°, Apis is 72.828° ahead of her in the rotational reference frame as measured from the zero point (180° + 72.828° = 253.828°). Apis is also rotating with an angular velocity 7.43 times that of Rosetta when orbiting at 0.1773 m/sec at 9594m and there’s no way Rosetta is going to catch up with it. So that looks like big a problem but it’s not.

The flyby ellipse has an orbital period of 32.9620 hours. We can derive this from the semimajor axis of 6197 metres. Therefore half an orbit is 16.4810 hours. That’s why the flyby injection was performed at T minus 16.4810 hours. It was already known that it would take that long from orbit injection to the flyby zero point, half the ellipse, apoapsis to periapsis. 67P rotates at 12.7 hours per full rotation (that should technically read 12.7000 hours to the four decimal places to show its sub-second precision but it’s a bit pedantic to keep writing it. So if Apis was at the zero point when orbit injection was performed, it would be 3.7810 hours ahead of the zero point when Rosetta flew through after 16.4810 hours of flyby approach. Apis needs to be wound back past the zero point by that same amount, 3.7810 hours so that it is rotating for 12.7 plus 3.7810 hours before passing through the zero point. That means 16.4810 hours of rotation for Apis and 16.4810 hours of orbiting for Rosetta prior to passing through the zero point. So they both pass through together. The 3.7810-hour winding-back for Apis can be expressed as an angle. One full comet rotation takes 12.7 hours. 3.7810 hours as an angle is 3.7810/12.7 x 360° = 107.172°. But this is wind-back in the opposite direction from the angle measurement that was always with the direction of orbit or rotation. When measured in the proper, conventional way it’s an angle of 360°- 107.172° = 252.828°. That angle is measured from the zero point in the direction of rotation and of course was already known 79.9180 hours ago at circular orbit injection.

So it simply means that at flyby orbit injection, Apis is ahead of Rosetta because it still has to rotate 107.172° to the zero point and then another full 360°. Rosetta only has to do half an orbit on its ellipse. That way, they pass through the zero point together, 16.4810 hours after flyby injection.

4) T plus 27.5960 hours (timed at burn shut-off at the semiminor axis vertex at orbit radius, 5183 metres):

Inject to parabolic ‘escape’. This was described above and though it’s repeated here, it’s described in a subtly different way which may aid the understanding of the parabolic injection burn. Delta v for the burn is is 0.1204 m/sec prograde i.e. directed backwards along a tangent to the retrograde orbit line. This gives rise to an increase in orbital speed from 0.3871 m/sec to 0.5075 m/sec, escape speed for this altitude. Escape was put in inverted commas further above because a parabola puts Rosetta on the cusp of escape only. A parabola is used for escape for a reason, instead of a hyperbola which implies excess speed for true escape. That reason is the unique properties of its two open ends which are parallel at infinity (and are so, in effect, at 200-300 km from 67P). Hyperbolas’ arms aren’t parallel and fan out by definition. Those arms, when at some distance from the perturbing body, are essentially straight lines called the asymptotes. If Rosetta were sent along one of them, she’d end up travelling away from 67P at less than 90° to 67P’s orbital plane around the sun. This is not permitted for easy return to 67P in 2021.

Since Rosetta is now orbiting in a plane that’s at 90° to 67P’s heliocentric orbital plane we can exploit this property of the parabola having parallel arms. At the very beginning, we chose both the zero point and the apoapsis to be sited on the intersection line of 67P’s orbital plane and its equatorial plane. That meant both the apsides (periapsis and apoapsis) sat on that plane intersection line. So it also means they both sit on 67P’s orbital plane. They are the semimajor vertices, and their positioning has implications for this semiminor vertex we’re currently at for the parabolic injection burn. Since they both sit on 67P’s orbital plane, it means the long axis of Rosetta’s flyby ellipse and of this new (52° plane changed) ellipse also sits on that plane. The long axis is the semimajor axis. So when Rosetta went through apoapsis on her inclination change burn, the 52° adjustment meant she had to end up travelling at 90° to 67P’s orbital plane. That in turn means this semiminor vertex where the parabolic escape burn is performed is running exactly parallel to 67P’s orbital plane around the sun. Or to be exact a tangent to the orbital ellipse at the vertex point is parallel to 67P’s orbital plane. Now, the new parabola that Rosetta is sent on by the burn, wraps round 67P with the burn point at the apex of the turn. The burn point or semiminor vertex is the point of symmetry for the parabolic curve. A tangent to the parabola curve at this point is at 90° to the two infinitely long, parallel arms. Such a tangent is also parallel to the orbital plane of 67P. Thus the parabola arms are both at 90° to 67P’s orbital plane. Rosetta is immediately set on a path to travel down one of the arms after the burn. Thus, we can be absolutely sure that Rosetta ends up travelling away from 67P on a trajectory that’s at exactly 90° (orthogonal) to 67P’s orbital plane. Only when she’s 200-300 km away down that arm would we take stock, see that she is indeed travelling at 90° and give her a tiny push of 3cm/sec to be sure of escape. If travelling at 88° or 89°, the escape burn of 3cm/sec would get a minuscule sideways vector to correct this. This puts Rosetta on a truly heliocentric orbit which is nevertheless a quasi orbit of 67P. She will, in theory, return to this point above 67P four years and three months later in January 2021- and arrive at 3cm/sec.

5) T plus 300 hours. This time stamp is somewhat arbitrary but it’s at a point where flyby data transmission is hopefully complete and the parabola arm location of Rosetta is now far enough from 67P to be pointing away at effectively 90° to 67P’s orbital plane around the sun. The following narrative for the T plus 300 hours time stamp is just a copy and paste from the simple version of the timeline above. It has no elaborations like 1 to 4 have:

Escape injection. A delta v burn of 0.03 m/sec is performed. It needs to be the slowest possible speed that ensures escape with no unintended return, decaying orbit and eventual crash during hibernation. That’s why 3 cm/sec is chosen. It’s much easier to get it right when several hundred km from the comet because it’s a larger vector value in relation to any perturbing gravitational accelerations. It’s a prograde burn in the same direction of travel away from 67P as for the parabola but with any necessary adjustments. Those adjustments are made after assessing that speed vector component away from 67P (which should be near zero) and the direction vector component which should be 90° to 67P’s orbital plane. In effect it would be a stand-alone burn from an almost stationary point in cometcentric space. But it would in effect be directed down a continuation of the parabola arm. There will be a celestial sphere reference point for this direction vector that the star trackers can lock onto for the burn.

9) RELATIVE SPEEDS DURING FLYBY AND FLYBY DURATION.

Apis has a tangential speed of 0.3574 m/sec. That’s because it rotates in a circle of 2piR in 12.7 hours. 2 pi times 2600m is 16338.4 metres. 12.7 hours is 45,720 seconds. Speed is distance over time which is 16338.4/45,720 = 0.3574 metres per second.

The orbital speed of Rosetta along its flyby ellipse at periapsis is 0.6075 m/sec (quite a lot faster than the 0.1773 m/sec at apoapsis due to falling deeper into 67P’s gravity well). The difference between the Apis tangential speed and the orbital speed of Rosetta is 0.6075 – 0.3574 = 0.2501 m/sec. Apis is about 400 metres wide from east to west. Rosetta is orbiting exactly along and over the equator because if you recall, the original circular orbit was in the same plane as 67P’s plane of rotation, which is the extended equatorial plane. So is the flyby orbit because the flyby injection was directed forwards (prograde) within the circular plane- so there was no sideways push to change planes. That means Rosetta has to come in over and along the equator from west to east, flying over Imhotep. The equator runs through and across Apis from west to east as well, of course. So Rosetta traverses the 400-metre width of Apis at 0.2501 m/sec. This makes the flyby time 1599 seconds, or 26 minutes and 39 seconds. And it should take place at just above 200 metres’ altitude all the way, kissing 200 metres’ altitude at the mid point. It’s just above 200 metres either side because Rosetta’s flyby ellipse is fatter than the rotation circle of Apis and therefore fatter than the rotation circle of the 200 metre altitude point as well. The circle kisses the ellipse periapsis point, i.e. nested inside one end of the flyby ellipse. It kisses that one point only and the ellipse diverges either side of it, but not by much when considering such a small distance as the 400-metre Apis ground track on a 12km long and 0.548 eccentricity ellipse. In reality it’s over 900 metres of circle/ellipse nesting due to the rotation movement of Apis over 1599 seconds. But it’s still essentially just over a 200-metre-altitude flyby all the way.

200 metres was chosen because if there were a small error in the delta v burn to flyby injection or there were unresolved gravitational anomalies, it would hopefully not cause an error of more than 200 metres in altitude so there would just be an even closer flyby and all data would still be sent back after escape. Besides, if there were such a negative altitude anomaly, it would also speed up the approach time (because of being a smaller ellipse). Rosetta would arrive early and overfly Atum. Atum’s radius of rotation is somewhat less than that of Apis, stuck out on the long axis tip as Apis is, so we’d be safe anyway. The opposite scenario of a positive altitude anomaly means Rosetta would be safe anyway. She’d arrive late and overfly eastern Imhotep at an even higher altitude because Imhotep drops away too. This is yet another advantage of targeting Apis: negative altitude anomalies from the nominal 200 metres translate into early arrival and comfortable flyby altitudes as the terrain height drops away from Apis anyway. And positive altitude anomalies can’t translate into a crash risk because Apis has the highest radius of rotation of the whole comet.

In reality, flyby would be a few tens of seconds longer because Rosetta would be just a tad slower than 0.6075 m/sec at the beginning and end of the flyby. The 0.6075 m/sec applies only for an instant at the zero point. 13 minutes before zero point and 13 minutes afterwards Rosetta’s speed would still probably be just above 0.6000 m/sec though.

Apis would rotate through 12.590° during the 26 minute 39 second flyby. As mentioned above, the circle it described would nest quite well with the tighter curve of the flyby orbit ellipse at periapsis, keeping the two curves somewhat close to parallel and minimising Rosetta’s altitude above the nominal 200m lowest point when either side of it.

10) THE DELTA V BUDGET FOR THE APIS FLYBY AND ESCAPE

Flyby orbit injection: 0.0864 m/sec.

Plane change burn at apoapsis and T plus 16.4180 hours: 0.1554 m/sec.

Parabolic injection burn from semiminor vertex: 0.1204 m/sec.

Full escape burn along parabola arm: 0.0300m/sec.

Total: 0.3922 m/sec.

To this, we could add about 0.1500 m/sec for any preliminary burn for the initial circular orbit insertion, depending on where Rosetta had come from. This would increase the total to 0.5422 m/sec. With about 0.45 m/sec for the best reacquisition scenario in 2021, the entire budget remains at below 1 m/sec.

11) AFTER ESCAPE: HIBERNATION AND RETURN VIA THE QUASI ORBIT.

0.03 m/sec is 3 centimetres per second. This would still translate to a few thousand kilometres of drift over the 4-year hibernation period. However, with the escape burn performed orthogonal, either north or south with reference to 67P’s orbital plane, Rosetta could be teased into being a quasi-orbiting satellite of 67P. This would greatly minimise the delta v necessary for return to 67P in four years’ time.

The key thing to remember here is that when any perturbation is made on a small body by another larger one, the smaller one always, in theory, crosses back through the point where it was perturbed. In the case of NEO’s it’s the node where their orbit crosses the Earth’s orbit and if the Earth happens to be near that point in its orbit at the same time, the NEO will be perturbed into a new ellipse. But the new ellipse still goes through the node and the NEO will keep passing through the node, once per orbit. It has a memory for that point because that’s where its newly perturbed orbit started. There are glaring exceptions to this, the main one being later perturbations by Venus, Mars and Jupiter. These remote perturbations (remote from the node) cause the otherwise faithful node crossing to drift.

In the case of Rosetta, 67P mimics the perturbing body and Rosetta is mimicking a perturbed body. In reality, the escape burn is doing most of the apparent perturbing as if Rosetta was on nearly the same heliocentric orbit as 67P and 67P perturbed her by 3 cm per second. Wherever it is along the shared heliocentric orbit of 67P and Rosetta that the escape burn is performed, that is the node where their two heliocentric orbits start diverging and will cross next time round. In the case of Rosetta, the burn to escape means 67P no longer affects her, only the sun. She’s in a very slightly inclined heliocentric orbit from 67P and those two orbits cross at the node. If Rosetta sails away at 3 cm per second and does so directly north or south of 67P’s orbital plane on September 30th 2016, then she would be drifting towards this same perturbation point in 6.4 years’ time after one full orbit. She’d be approaching directly from the south of 67P’s orbital plane if she left directly northwards, and vice versa if she had left directly southwards. Of course, 67P would be approaching the perturbation point as well, to cross at the same time, because it’s performed an identical ellipse around the sun to Rosetta’s slightly inclined version of the same orbital ellipse.

Now, hibernation would be for around four years, not 6.4 years. That takes us to September 2020 and 67P/Rosetta will be on the other side of their virtually identical orbital ellipses. Three months after that, their two heliocentric orbits cross a second time. They cross at the diametrically opposite crossing to their September 2016 node. In those three months, Rosetta would be drifting towards 67P from directly above or below its orbital plane and doing so at 3cm per second. She’d be only 500km away at the start point, in theory, but probably a few thousand km away in practice due to escape velocity residuals. That crossover point, reached in January 2021 just happens to be the exact position in the orbit where Rosetta acquired 67P in 2014. This is the sweet spot for comet acquisition as we know from last time around and 67P/Rosetta would be travelling back towards the sun.

So it’s simply a happy coincidence that the 2021 comet acquisition sweet spot is where the two orbits cross back over and Rosetta is passing straight back past 67P to embark on the half of her orbit that’s on the other side of 67P’s orbital plane. As stated above, they would both be on the opposite side of their heliocentric orbital ellipses to the man-made node due to the escape burn. However, this is the opposite node and is defined by the point where the two orbital ellipses cross back over on the opposite side. There are two nodes: an ascending node and a descending node. They’re diametrically opposite in the orbit.

If Rosetta is sent exactly north or south of 67P’s orbital plane she executes an identical ellipse (same semimajor axis, same eccentricity) around the sun and in an identical time. The only difference is that Rosetta’s orbit has a tiny inclination to 67P’s. It crosses 67P’s two nodes where the orbital planes of both bodies cross. From 67P’s point of view, Rosetta appears to be orbiting in a highly elliptical orbit with 67P in the middle, not at one focus as would be the case if Rosetta were truly orbiting. The ellipse would extend about 2000km above and below 67P. In theory, it would be an almost a straight line, straight up above the 67P orbital plane at 90° for about two years and then tracing the same path all the way back again for another two years or so. If that happened, Rosetta would fly straight past the comet at the opposite node, in January 2021, right on the acquisition sweet spot. But the tiniest errors in escape delta v (speed and direction residuals) would turn this into hundreds of kilometres at least. Jupiter’s differential gravitational acceleration adds up to 2000km more but probably somewhat less. These are all small distance anomalies though. It’s why Rosetta would be woken up in at least September 2020 after four years so as to take her bearings, asses the distance anomaly and do an acquisition delta v burn. If woken early enough, the burn would need to be very small, 0.2 m/sec for a 3000km distance anomaly. This saves on delta v budget which means fuel.

The escape vector has to be exactly north or south of 67P’s orbital plane. If it’s angled at all, it throws Rosetta outwards from or inwards to the sun. This changes the orbital eccentricity. This wouldn’t matter if a) they were meeting back up at the perturbation node 6.4 years later and b) the semimajor axis was the same (allows same orbital period) But for meeting up at the opposite node, even if the orbital period is the same, the eccentricity difference fattens the ellipse from the original node point and Rosetta finds herself thousands of kilometres inside or outside 67P’s orbit.

The only way to get the quasi orbit we want is with Rosetta approaching 67P in September 2020 from exactly above or below 67P’s orbital plane. That approach vector is itself a signature that Rosetta is in an identical ellipse around the sun, with an identical period but with an orbital plane that’s at a slight inclination to 67P’s orbital plane. She would approach from above if she’d been sent above 67P’s orbital plane in September 2016 and she’d approach from below if sent below in September 2016. In both cases, she would have spent half an orbit above or below the 67P orbital plane.

One easy part in this scenario is that it doesn’t matter all that much how fast the escape is so long as it’s directly north or south of 67P’s orbital plane. The faster it rises from the plane, the faster it will return on the other side of the orbit in mirror fashion. It won’t change the ellipse size or shape or its orbital period, just its inclination to 67P’s orbit. However, a faster escape and consequent return, say 0.1 or 0.2 m/sec does mean more delta v needed to slow it down on return. It also means that Jupiter can tease them even further apart if their initial separation is bigger. That’s why 3 cm per second was chosen. It reduces the possibility that Rosetta ends up 30,000km away on wake up and having to give her a 2 m/sec delta v just to get her back to the comet in six months and then another 2 m/sec to slow her down. This worst case eventuality is actually an argument for wake up a little earlier in May or June 2020. This would place Rosetta at the same heliocentric orbital position as for the 2014 wake up.

With the quasi orbit, Rosetta might be close enough to 67P to do some useful science without even being given delta v. That’s if it was low on fuel and needed reserves for attitude control to point the antenna at the Earth and then turn back to 67P repeatedly.

12) FLYBY VERSUS CRASH MEANS MORE TRANSMISSION OF DATA.

If the dare-devil flyby gets overly perturbed by the uneven gravity field and crashes, we lose nothing. It won’t be any different from the controlled crash scenario. It will gather and transmit data right up to the point of crashing and transmit no data thereafter. However, I suspect the Rosetta transmitter can’t transmit, in real time, all the data that could be collected before crashing. It has to do it in real time because the mission scientists have ruled out transmission after the crash. I’m not sure, but the low transmission bit rate in September must be a lot less than the data that could be recorded and stored over that last hour or so and then transmitted later over a period of days or weeks. The Deep Space Network (receiving dishes) will be busy with other missions in October, not least of all because Rosetta hasn’t asked for any time on it after the crash. But if it’s just the last last hour or so of very useful close-up data with as many instruments as possible taking data, surely it would be possible to send it back in October and November. It would just be sent in dribs and drabs after the flyby scenario outlined below. That’s what New Horizons is still doing: it’s taking it 16 months to send back a few hours’ worth of data.

13) CONCLUSION

This post concerns itself with the nuts and bolts of the flyby, orbital elements, orbital speeds and delta v burns etc. The twin post expounds on the numerous advantages of opting for a flyby rather than a crash and explains why Apis is a star candidate for that flyby. Its credentials run to a long list that doesn’t just focus on its status as being probably the least processed region on the comet and therefore the closest to being primordial. However, the primordial surface argument, that can only be arrived at by invoking stretch theory, is the most pressing reason to go for the Apis flyby.

14) APPENDIX- GLOSSARY

This glossary, like the keys to the photos in other parts, is a narrative glossary. It complements the overall explanation outlined above.

Vector- for a velocity vector it’s the speed and direction as opposed to just the speed. For a force vector, e.g. a rocket thrust, it’s just the direction. Velocity is used when a change of orbital speed is brought about by a rocket burn because the change can be faster or slower implying an adding in the forward direction or a subtracting in the negative direction. However, often just ‘speed’ is stated to denote the orbital speed. This is because its direction isn’t needed in many instances. It’s just the prograde speed of Rosetta on its orbit at that point of the orbit and instant in time. It still has a direction of course and we would soon be talking of velocity if we were on a collision course with the comet. The idea it to just miss it though.

Prograde- forwards direction along the orbit line (strictly speaking, the tangent to the curved orbit line at a particular point on the orbit and instant in time).

Retrograde- backwards direction along the orbit line (strictly, a tangent too).

Prograde burn- a rocket engine boost to attain an increase in velocity along the direction of the orbit line. The thrust vector (direction) for a prograde burn is backwards along the retrograde line.

Retrograde burn- a rocket engine boost to attain a decrease in velocity along the direction of the orbit line. The thrust vector for a retrograde burn is forwards along the prograde line so as to slow down the orbiter. This fancy talk of vectors and thrust is no different from what you have to do when pushing your loaded trolley round the supermarket. And the speeds along the orbit (relative to the comet) are identical to trolley speeds: 20 to 70 centimetres per second. The velocity changes to speed up and slow down are no different either. In both cases, it involves a force acting for a finite period over a short distance to bring about a change in velocity.

Flyby Injection burn- this is the retrograde burn needed to slow Rosetta enough for 67P’s gravity to pull it into an elliptical orbit from a circular starting orbit. The burn ‘injects’ it from the circle into the ellipse. It in effect slows Rosetta enough to make her start dropping towards 67P but since there’s a residual element of speed that’s tangential to the radius at that point (the residual prograde speed along the circular orbit), Rosetta can’t drop vertically. She’s drawn into an ellipse. Since the vertical component of gravitational acceleration comes to dominate the shape of the orbit, the residual tangential speed has less influence than it did for a circular orbit. That’s how the circle gets distorted into an ellipse. Notice the word “trajectory” is hardly used at all here or anywhere else in the blog. It’s woolly and tells us nothing about the shape of the path of Rosetta unless preceded with “elliptical”, “circular”, “parabolic” “hyperbolic” etc. These are all curved paths of one sort or another and all are types of orbit. As soon as we use the word ‘trajectory’, we think in terms of a start point, an end point and some path, whatever path, usually straight, between the two. By visualising the elliptical path of Rosetta you can start to see that as it approaches 67P the increasingly curved shape of the ellipse almost meshes with the circular path of the rotating Apis section of crust. If we talk vaguely of the ‘flyby trajectory’, this subtlety will go right over our heads.

Escape injection burn- this is the prograde burn needed to increase Rosetta’s prograde orbital speed to escape speed. In the scenario outlined above, it’s actually the parabolic escape burn, sending Rosetta down one of the parabola’s arms and only to the cusp of escape. It would be executed 27.5960 hours after the Apis flyby and occur at the semiminor vertex of the newly inclined version of the flyby orbit ellipse. Rosetta needs to escape 67P in order for Marco Parigi’s hibernation scenario and subsequent return to take place. Rosetta is given a 0.03 m/sec surplus speed in the escape injection described here. That’s just three centimetres per second. That means it doesn’t drift too far from 67P during hibernation and can still return via the quasi orbit. If she orbits 67P in a true, conventional orbit during hibernation, the orbit may decay and cause Rosetta to crash on the surface.

Delta v- the increase or decrease in velocity attained due to prograde or retrograde burns. Velocity is used because it has direction, either forwards or backwards. The backwards scenario would be because a decrease in velocity is a negative (backwards) change. Often, sideways delta v’s occur to take the orbit out of plane with the current orbital plane (a non-coplanar transfer burn) as was the case for the 52° inclination change delta v at apoapsis, described above.

Delta v budget- the sum of all delta v’s both positive and negative (prograde and retrograde burns) for a particular orbit change scenario or set of changes over time. Delta v budget is important because it’s constrained by available fuel supplies. For this scenario there are two delta v budgets, one for the Apis close flyby, including the inclination change burn and two escape burns after that. Then there’s another budget for reacquisition in 2021. This is nominally a single burn to capture Rosetta in a true orbit around 67P assuming she flies straight back into its arms. However this is so unlikely as to be dismissed. It’s assumed Rosetta will have undergone at least 3,000 km of drift and require 0.2 m/sec delta v to get back in 3 months along with a 0.24 m/sec retro delta v to go into orbit. The two budgets, September 2016 budget and September 2020 to January 2021 budget are added to make one budget. That’s because the flyby is a stand-alone proposition which needs a certain amount of fuel. The reacquisition is another bonus scenario, a rather long shot attempt at waking up Rosetta. Once woken up, it can’t do much without the enough fuel for the reacquisition delta v budget. Both budgets are around 0.5 metres per second, total 1 metre per second.

“c of g” – this is used for brevity for the the centre of gravity of 67P. It’s also the centre of rotation i.e. central along the rotation axis. So all radii of surface features such as Apis are measured from it.

Radius of rotation- this is the same as the radius described above. It’s just a reminder that mentioning radius in this blog usually immediately leads on to rotation and the tangential speed of that surface feature. For Apis, the radius of rotation is estimated at 2600 metres from the shape model dimensions. Its tangential speed is therefore 0.357 metres per second, just about the fastest value for anywhere on the comet, which makes the relative flyby speed more sedate. That’s obviously a big advantage, one of several for targeting Apis. If its radius of rotation is actually 2500 or 2700 metres, we just change the 2600 metre input here and watch the output change by a tiny amount. The scenario remains unchanged in its basic principle. The outputs that would change are periapsis/apoapsis distances (see next two glossary entries below), the size and eccentricity of the ellipse, the orbital speed at different points round the orbit, the orbital period, and the delta v’s required. But they’d all change by very small amounts for a 100-metre change in the rotation radius of Apis, or for that matter, for a 10 to 20-minute adjustment in 67P’s rotation rate. The rotation rate has been seen to vary by that much already over a few months.

Periapsis- the closest point of an elliptical orbit to the comet’s c of g. The c of g of the comet is also the focus of the ellipse that defines the orbit. Satellites always orbit around one of the foci of the ellipse because that’s the point where the gravitational force is directed towards throughout the orbit (being the c of g, of course). We’ll be constructing an elliptical orbit for Rosetta with a 2800-metre periapsis. What we want to do here is make Apis pass right under that 2800-metre periapsis point exactly when Rosetta is going through periapsis as well. That gives us a 200-metre-high flyby. That assumes my informed guess that Apis is 2600 metres from the c of g is correct. It’s important here to imagine the periapsis as one end of the orbital ellipse we construct and also as a target point for Rosetta to arrive at and the centre of Apis to arrive under at the same time. If we strip out the motion of 67P around the sun we have a comet-centric view of the orbit. That way the periapsis can be seen as a point in space that stays locked, just as the ellipse it’s on is locked. The periapsis point can then also be used as a zero point for measuring the angle of Apis from the target (so the angle zero point, the flyby target and the periapsis point are all one and the same thing). Apis rotates in a perfect circle of course, 360°, and passes under Rosetta’s proposed periapsis point every 12.7 hours. We’ll see that Rosetta acquires the western edge of Apis when Apis is about 6° or 13 minutes from the periapsis zero point. Rosetta passes over the centre of Apis when they both go through the periapsis zero point (so they both hit the target together as planned). Rosetta then crosses the eastern edge of Apis when Apis is 6° past the zero point. Rosetta burns to escape a few minutes later.

Apoapsis- the farthest point of an elliptical orbit from the c of g. This is an important point too because it’s the point from which the Apis flyby ellipse is started. It’s the the same radius as the circle Rosetta had been orbiting before the retrograde burn slowed her orbital velocity. This negative delta v means Rosetta slows down along its circular orbital path. That in turn means the downward pull from 67P’s gravity overcomes the ‘centrifugal’ force of the former circular path speed. Rosetta immediately drops away from the circular path and assumes an elliptical path that stays inside the circle. But the ellipse kisses the circle at its highest point from 67P because that’s where the burn happened and so that’s where it started dropping away. That point is the apoapsis. It’s fixed like the periapsis and is exactly 180° opposite the periapsis, at the other end of the ellipse.

PHOTO CREDITS

FOR NAVCAM:

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

To view a copy of this licence please visit:

http://creativecommons.org/licenses/by-sa/3.0/igo/

All dotted annotations by A. Cooper.