## Spacecraft orbit

horizontal plane

Diagram explaining the local vertical/local horizontal (LVLH) frame of reference. Change in velocity: delta-v and frames of reference

The main purpose of the LOI burn was to slow the spacecraft down by a desired amount. This change in velocity is really a vector quantity because its direction is as important as its magnitude. Known within the industry as delta-v, it is normally resolved into three components (x, y and z) which were given by the next three numbers in Henize's list - minus 28975, minus 07764 and minus 00441. Such was the primitive nature of the Apollo computer that there was no provision in it for entering or displaying a decimal point. Instead, the position of the decimal point was fixed in the programming and on the form. Everyone associated with working with the machine knew where it was in any particular context. In this case, 28975 simply meant that the burn was for a change in velocity along that axis (but, being negative, in the opposite sense to the direction of the axis) of 2,897.5 feet per second.

So what about these three axes? By now it should be clear that coordinate systems were, and are, ubiquitous in spaceflight, and become especially important when dealing with engine burns. The firing of an engine in space results in a change of velocity and it is necessary to define the direction of that change in relation to a frame of reference; a known set of Cartesian coordinates against which it can be plotted. We can use any frame of reference we like but it is customary to use one that makes the calculations easier, and the one that is generally favoured is local vertical/local horizontal (LVLH).

This frame of reference is constructed relative to a line drawn from the spacecraft to the centre of the body it is orbiting, or whose gravitational sphere of influence it has entered. Imagine a point where this line intersects the planet's surface. We can further imagine a flat plane at this point parallel to the horizontal. Obviously, as the spacecraft moves around the planet, the absolute orientation of this plane keeps changing but it provides a useful reference for orbital velocity computation. In this arrangement, the + z axis points towards the planetary centre, the + x axis is in the direction of orbital motion parallel to the local horizontal and the + y axis is perpendicular to the orbital plane.

With this, we can make more sense of the velocity components given in the PAD. The large negative figure for the x component, -2,897.5 feet per second, meant that the burn was largely retrograde, against the spacecraft's motion, which is exactly what would be expected, given that they were trying to lose speed. The figure for y, — 776.4 feet per second, meant that the spacecraft was being pushed sideways as part of the process of ensuring that it ended up in the correct orbital plane for the landing site. The figure for z, -44.1 feet per second, was small in comparison and was away from the Moon's centre. Converted to metric units, these velocities were expressed as -883.2, -236.6 and -13.4 metres per second.

"All zips for roll, all zips for pitch, all zips for yaw" - Again we have to deal with frames of reference for these three numbers, all of which are zero. However, whereas delta-v used local vertical/local horizontal as described above, these numbers were given with respect to the guidance platform, itself aligned to our old friend, the current REFSMMAT. For every burn, mission control gave the crew a set of three angles that represented the attitude of the spacecraft in terms of roll, pitch and yaw directions and these were always stated with respect to the current REFSMMAT. But since the platform's orientation had been aligned to match the spacecraft's calculated orientation for the burn - the so-called LOI REFSMMAT - then the attitude angles for this burn were necessarily all zeros, or 'zips' as Henize put it. This arrangement meant that the FDAI (flight director/attitude indicator) or '8-ball' in front of the crew showed zero in all three axes, providing an easy means of monitoring the direction in which the spacecraft was pointing, in case the crew had to take manual control.

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