Csdrhrt

The title pretty much sums it up. I'm not old enough to have experienced any of the previous manned moon landings. Given that Big Falcon Rocket will go around the moon in 2023, I'm wondering if one will be able to see the whole rocket or at least its second stage from earth when it's approaching the moon - but with my plain eyes (I know camera technology's good).

I'm not expecting to see it like a plane or something, but maybe reflections, thrusters, ...? A tiny dot getting closer and closer to the moon, knowing there are humans in there - that'd be cool...

Thanks in advance!

tl;dr: at an apparent magnitude of about +18.5 you need a several meter telescope and a dark sky. Hubble can do it too. So by reflected sunlight, definitely not by eye. The exhaust from a Methalox (CH4 + LOX) engine barely makes any light in the visible, so no help there.

Starting with the math from this answer:

I'm going to characterize the 55 x 9 meter white 2nd stage as a 22 meter diffuse sphere(ical cow) with albedo 0.3 for order of magnitude estimation.

The expression for absolute magnitude $M_{Abs}$ by rearranging the equation here is:

$$M_{Abs} = 5 left(log_{10}(1329) -frac{1}{2}log_{10}(text{albedo}) -log_{10}(D_{km})right)$$

For the "spherical cow" spacecraft that turns out to be an absolute magnitude of +25.

Knowing the absolute magnitude of an object, you calculate the apparent magnitude $m$ using:

$$ m = M_{Abs} + 2.5 log_{10}left(frac{d_{SR} d_{RE}}{1 text{AU}^2 O(1)}right), $$

where $d_{SR}$ and $d_{RE}$ are the Sun-Roadster and Roadster-Earth distances, each normalized by 1 AU, and the factor $O(1)$ is the phase integral, of order unity, taking into account the angular difference between the direction of illumination and the direction of viewing. In an order of magnitude calculation, this only becomes really significant when the body moves between the Sun and the viewer. See https://en.wikipedia.org/wiki/Absolute_magnitude#Solar_System_bodies_(H).

In this case, replace Roadster with BFR. Since $d_{SR}$ and $d_{RE}$ are 1.0 and 0.0025 AU respectively, and ignoring the phase integral, the apparent magnitude is about -6.5 lower or +18.5 magnitude.

I think that can be seen with the Hubble Space Telescope above the atmosphere in a fairly short exposure. On the ground you'd need a several meter telescope and a dark night, but you can do it. If you want to use your eye, better find one of these telescopes that has an active eyepiece!

BFR's Raptor engines burn "methalox" (methane and liquid oxygen) CH4 + O2. The products are CO2 and H2O both gases. The brightness from the exhaust from many launches are from carbon soot glowing when kerosene (RP-1) is burned, not methane.

Here is a screenshot from the video Blue Origin BE-4 Engine Compilation during the daytime, so you can have a reference to brightness. I haven't found an outdoor firing of the Raptor yet:

If you want an easy way to think about it, imagine how bright it might be in low Earth orbit, 240 miles up (which is just a bit lower than the International Space Station). However bright that is, it will be only a millionth as bright when it’s near the Moon, 240,000 miles away (and so a thousand times the distance). Is it likely that it will be bright enough in low orbit that you can still see it when it’s a millionth as bright as that?

Rocket engines burning in vacuum produce large exhaust plumes that rapidly spread outward to 'fill' the vacuum. This means that for a distance of many dozens of meters - or more - they produce a visible cloud of gas that can catch sunlight and produce very large, visible artifacts in a clear night sky. So - engine burns near Earth (departure, or arrivals burning into orbit) will be quite easily visible to those who are in night and under the burns.

For example, this GLONASS launch trail taken from a thread full of rocket plume images.

Mid course burns may be visible as well, though much less dramatic. Arrival and departure burns happening in the vicinity of the moon are not likely to be visible at all to the naked eye due to great distance.