# Observation Time

### Observing the Moon

In order to know the feasibility of our experiment in more detail, we must find the length of time for which observations of Earthshine are possible. First, the total length of time in which the entire moon is in shot, regardless of what proportion of this time is useful for our measurements, is dependent on both the amount of time for which the moon is in shot per swathe, and the number of swathes per day which would include the moon, which are dependent on the 'vertical' (roughly North/South) width of field of view, labelled V and the 'horizontal' width of field of view, labelled H. For ease of calculation, we will assume a rectangular view. It is known that the moon has an apparent width of 0.5 degrees from Earth, and the altitude being neglible when compared to the distance between the Earth and moon, we shall use this value.

In each swathe which contains the moon in its horizontal scope, the moon is visible from (V-0.5)/2 degrees above the line from Earth to moon to (V-0.5)/2 degrees below, hence for an arc of V-0.5 degrees. The orbital period being approximately 98 minutes, or 1.63 hours, the satellite orbits at roughly 221 degrees an hour, making the number of hours per useful swathe (V-0.5)/221. Because the moon has a maximum inclination of less than 29 degrees, that is it is never more than 29 degrees above or below the equator, almost the same method can be used for the number of useful swathes per day. First, the moon takes 27.3 days to complete a full orbit, so in a day moves 13.18 from its previous longitude. As the moon's inclination is between 0 and 90 degrees, it progresses relative to the Earth, which means the satellite takes roughly 1.04 days (1498 minutes) to reach the same longitudinal position relative to the moon, rather than the 1 day to reach the same longitudinal position relative to the Earth. The satellite therefore makes 1498/98=~15.3 latitudinal orbits per 'day' relative to the moon, making swathes 24.4 degrees away from each other, centre to centre. As the moon is in shot for H-0.5 degrees, it will be in roughly (H-0.5)/24.4 swathes per 1.04 days, so approximately (H-0.5)(V-0.5)/5390 hours per day. Given a maximum operational lifetime of 8 years, or 2922 days, the moon will be in shot for a maximum of roughly (H-0.5)(V-0.5)/1.84 hours.

Our previous signal to noise calculations have given a maximum possible field of view of 4.93 degrees in each direction, which leaves a maximum possible time in which the moon will be in shot of 4.43^{2}/1.84=10.7 hours. This figure is before accounting for the time in which observing the moon will be useful, which will likely divide the time by at least 20, and yet is still too small to justify the experiment. Clearly, we need to find a solution to increase the time available for observation. Possibilities are listed below:

**Use more than one spectrometer/collection area**-cumulatively they will have a greater time period available, as the moon can be in the shot of one but not the other- This is feasible in terms of space and budget, as the individual spectrometers/collection areas are inexpensive and take up relatively little room
- Objection: at most the observation time can be extended to 2 or 3 times the length. This increase is almost certainly insufficient.

**Allow the collection area to move**-it would 'track' the moon along a single plane of rotation, across each swathe, giving more time- As the main constraint is on the size of the field of view, this allows the time to be increased greatly
- Objection: this adds a tremendous amount of complexity and cost to the experiment, unfeasibly so

**Observe the Earth directly**-as the Earth is both larger and much, much closer the field of view would only contain reflections from Earth, thus reducing the noise greatly- This would also increase the size of the signal, by removing the extra distance to the moon and back and the moon's low albedo
- Objection: Much would have to be altered in the proposal

Clearly, the objection to the last point is far smaller than the fatal flaws in the other two proposals, and so observing Earthshine directly is the path the experiment shall follow.