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This will live in a removable power box on the trailer.
CaliMobber (as does unseenone) offers good advice, but since you mention that the battery will live in a box presumably with other electronic components, may I suggest an AGM battery type. Deka, Lifeline, Odyssey... They're still a lead-acid battery, but they're sealed. They do require a certain charge profile, however; one that any name-brand "smart" charger has a profile for. An AGM does not outgas (unless blatantly overcharged then it's ruined as a "recombinant" battery), does not require venting, is non-spillable, can be mounted on it's side if necessary and with brands like Odyssey, deliver both engine cranking amperage as well as deep-cycle ability. Not many batteries on the market make that claim. For your proposed setup, yes; you should be looking at a minimum 100Ah capacity.
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An emerging rule of thumb now that AGM batteries are found in more and more solar power battery banks is, "100 watts panel for every 100 amp-hour battery capacity" and that is fast becoming a minimum recommendation. AGM batteries with all their advantages (and increased initial cost) like to be hit with as many amps as possible (within existing battery charger outputs) during the "bulk" charge phase (up to roughly an 80% state-of-charge), then they like a little more than the industry standard for flooded batteries' 14.4 volts for the "absorption" charge phase... more like 14.6 and in the case of Odyssey batteries, 14.7 volts for absorption, and for at least four hours... finalized with a 13.6V "float" charge indefinitely.
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I know 45w is low, but it is what I have for now, so figure it won't hurt to have it hooked up and providing some charge
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Indeed, it is better than nothing. You must be certain, however, to keep that panel in full sun (weather permitting) and care must be taken to see that it not fall under any partial hard shading. Panels when partially shaded are near-useless in output; they will function better than expected, though, during full cloud cover so you can continue to keep it connected in those conditions as well.
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Of further importance, since the 45W panel is all you have at the moment, is to ensure the necessary copper stranded wire sizes to prevent voltage drop. A battery that does not see the necessary voltage due to too-small wiring/cable will not become fully charged.
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Here's a decent Voltage Drop Calculator
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Since you're already on the edge regarding watts input, it is imperative you seek less than 3% losses. Choose "single phase", copper stranded (unless your solar rig uses aluminum wire), "overhead" for Installation, 12 for volts, 3% for minimum volt drop, and 3 amps (the best that panel might produce under absolute optimum conditions)