space time - How far into the future can we go by traveling close to a black hole?

The way that you have specified the question, the answer is as far as you like. You simply put your spaceship into any orbit around the black hole and wait.

A more sensible question is what is the largest time dilation factor that can be accomplished - i.e. that maximises your travel time into the future for a given amount of proper time experienced on the spaceship.

This in turn is governed by how close to the black hole you can come and still tolerate the tidal forces. If you don't put a limit on this (your first case), then the answer is again infinite; you can hover as close to the event horizon as you like, using an enormous amount of rocket fuel, and the time dilation (see below) can be arbitrarily large.

Your second case is more realistic. Roughly we can say that the tidal acceleration across a body of length $l$ is given by $GMl/4r^3$, where $M$ is the black hole mass and $r$ is the distance from the black hole. If we make this acceleration equal to say $1 g$, and your body length $l sim 1$m, then for a $5M_{odot}$ black hole $r simeq 2500$ km (well outside the Schwarzschild radius of 15 km).

If you could "hover" at this radius, then the time dilation factor would be
$$frac{tau}{tau_0} = left( 1 - frac{2GM}{rc^2}right)^{1/2},$$
where $tau$ is the time interval on a clock on the spaceship and $tau_0$ is the time interval well away from the black hole.

For $M=5M_{odot}$ and $r = 2500$ km, this factor is 0.9970.

If the spaceship is in a circular orbit, the factor is $(1 - 3GM/rc^2)^{1/2} = 0.9955$.

These factors are perhaps not as big as you might have imagined!

interstellar travel - Detecting motion

It is part of General Relativity that if you are in a (small) closed box, you cannot tell the difference between acceleration and a gravitational field.

I suppose you could argue that you would explain Earth's gravity to the occupants of the biosphere as a 1g acceleration away from the Earth for 50 years, a slow swivel of the spacecraft, and then deceleration at 1g for 50 years.

It is hard to say whether such a deception is possible without more details, but in principle it could be done if you have some way of obscuring the initial part of the journey - launch etc, where you would think some non-uniform acceleration would be inevitable.

There is also the question of how far you are going and whether there is a feasible propulsion system. An acceleration of 1g for only a year gets you up to relativistic speeds. You could easily travel to other galaxies in 50 years at this acceleration. So, if the destination was supposed to be a nearby star then the deception fails miserably, unless you try to convince the passengers that you are executing some sort of weird trajectory to get there.

Additionally, how big is the biosphere? What about tides due to the Sun and moon? Any sizeable body of water would show up tides (and indeed possibly the coriolis force due to the Earth's rotation).

star formation - How to tell a nebula from a galaxy?

Galaxies are a large organised collection of stars (& nebulae) outside of our own galaxy, very distant. They shine by their own light.

Nebulae are clouds of gas & debris from, usually, a stellar explosion (Nova, supernova, etc), within our own galaxy, lit up by nearby stars (maybe internal ones) - but the gas is not glowing with it's own light, generally.

So a galaxy is going to be a bright clump of glowing stars, regular in shape - be that elliptical, barred or other, generally they will appear symmetrical & regular. A nebula will be much more amorphous & patchy, possibly with shapes from the clouds of gas (c.f. Horsehead, Orion), and you will be able to see individual stars in it - unlike a galaxy where they are too tightly packed & distant to make out individual stars.

optics - Is it practical to hand grind a convex parabolic or hyperbolic mirror?

If you're asking in regards to testing methods, as indicated in the comments, the simplest setup for interferometrically testing convex conic mirrors is with a Hindle Test, shown below in a figure from the University of Arizona College of Optical Sciences. This setup can achieve a perfect null after adjusting the reference sphere to be focused at the focus of the test optic - the catch is that the sphere needs to be larger than your test optic, with a hole through it as shown.

In industry, it is much more common to use an aperture-stitching interferometer for small quantities of non-research-level optics. Larger, more precise, and higher quantity aspheres may use a set of nulling optics, or a diffractive/holographic element to create a null wavefront, as covered in better detail in the link below, which is a slideset from U of Arizona's optical fabrication and testing course.

http://fp.optics.arizona.edu/jcwyant/Short_Courses/SIRA/7-TestingAsphericSurfaces.pdf

If you're feeling especially ambitious, there is a concept for measuring surface form of a mirror by displaying points on a monitor and using an HD camera to see where the reflections come from, thus telling you the angle of the optical surface at that location. The data is then integrated to form a full surface map. In theory, this system could be developed at low cost, with relatively high performance.

https://www.osapublishing.org/ao/abstract.cfm?uri=ao-13-11-2693

Hope this (or at least some of it) helps!

-J

How do we know which elements are in a galaxy spectra?

Because the wavelength ratio of the lines remains constant despite any Doppler shift.

For example, if the redshift is $z$, all the lines are shifted redward in wavelength by a factor $(1+z)$. This means that a pattern of lines can still be recognisable.

We also have a pretty good idea of what the spectra should look like, which chemical elements will produce visible absorption features with what relative strengths and so on (see below). This usually makes identification of line features straightforward.

Of course if there were just a single line visible in the spectrum (it does happen, usually in high redshift quasars) it can be difficult to pinpoint the redshift.

In terms of analysing what's in a Galaxy, well usually the light is dominated by the mixed spectrum of billions of stars. This spectrum is interpreted and modelled using galaxy evolution models and population synthesis models that predict a spectrum from a given ensemble.

If a Galaxy is near enough, the chemical abundances of its interstellar medium can be estimated from resolved spectra of emission nebulae.

In comparison, interpreting spectra from an individual star is trivial. Hundreds if not thousands of absorption lines can easily be identified and matched with the predictions of very detailed stellar atmosphere models to estimate chemical abundances. These models contain up to millions of possible radiative transitions as well as the various line strengths and broadening processes that affect the spectrum.

surface - Could Pluto and Charon have extra-Solar origin?

(Edit, I think my original conclusion here was wrong, having read up about it).

While it seems probable that Pluto and Charon were formed by collision, I gather it's unlikely that this collision was so recent as to explain their lack of craters and young surface. The Collision, by most articles I've read, happened when the solar system was young, not in the last 100 million years.

Now, it's possible that none of this is certain, but I retract that part of my original post.

But, the creation of Charon by giant collision remains likely See here, and here. That would suggest formation within our solar-system, not from outside.

Pluto/Charon are also among a fairly common orbital region called Trans-Neptunian objects.

According to Wikipedia they suggest the collision that formed Charon happened 4.5 billion years ago.

Sedna (love the story, by the way), has a very elongated and distant orbit which seems more likely with capture of an object. I agree with everything James Kilfinger said. Rogue planet/Rogue object capture may happen from time to time, but the Rogue would need to get quite close to the sun for the sun to have sufficient gravitation to achieve the capture. Highly elongated orbits would seem likely for solar-system orbital capture. Pluto is only slightly elongated and not a good candidate. More massive suns should have significantly greater capture numbers, but that's kinda obvious I suppose.

gravity - Gravitational red hift vs Doppler redshift: Is the universe really expanding?

Ultimately this is an application of Occam's razor

In order for the light to be gravitationally redshifted, it would have to be coming out of a deep gravitational well. For the red shift observed in galaxies to be gravitational, you would have to suppose several things.

First, that the stars in distant galaxies are somehow much denser: more than neutron star dense. (or possibly that entire galaxies are as dense as neutron stars.) A neutron star has a redshift of about z=0.35 distant galaxies have redshift of more than 7. No known object has a gravitational redshift like that.

Secondly that the density is proportional to distance from us. Placing us at a special position in the universe.

It is much simpler to interpret the redshift as a doppler redshift, and therefore an expanding universe