the sun - What constellation does sun occupy by location and date?

You can see for yourself for any date and location using the free Stellarium program (http://stellarium.org/).

The constellation that the Sun is in as seen from Earth does not depend noticeably on the precise location of the observer on Earth. It depends mostly on the time of year. Here are the approximate days of the year at which the Sun enters a new constellation:

• Aries: April 18


• Taurus: May 14


• Gemini: June 21


• Cancer: July 20


• Leo: August 10


• Virgo: September 16


• Libra: October 31


• Scorpius: November 23


• Ophiuchus: November 29


• Sagittarius: December 17


• Capricornus: January 19


• Aquarius: February 16


• Pisces: March 12

The "approximate" part is that the Sun may be up to one day earlier or later than the quoted dates, just like the start of the seasons may be one day earlier or later than the average. (Actually, it is the calendar that is early or late with respect to the phenomenon.) Also, because of the precession of the equinoxes, these dates shift by about 1 day every 70 years.

See http://aa.quae.nl/en/antwoorden/sterrenbeelden.html#6.

Can earth escape sun's gravity with the help of a black hole heading towards our solar system?

I'm thinking, this is the gist of your question

Earth feels a zero net force. Will it help earth to fly away?

First, zero net gravitational force between two large objects is certainly possible, well, not exactly zero, not for more than an instant anyway, but close to zero, absolutely possible, but whether it leads to an object flying away is more complicated. It depends on the relative motion of the 3 objects.

The Moon, for example, orbits the Earth, but from the Moon's point of view, the Sun is about 333,000 times more massive than the Earth and about 388 times further away, (on average) when the Moon is between the Earth and the Sun (390 times when the Moon is on the opposite side of the Earth, again on average. There's some variation in there).

Because gravitation drops with the square of the distance, 388 times more distant means about 151,000 times less G force at that relative distance, but with 333,000 times more mass, the Moon actually experiences over twice the gravitational tug from the Sun than it gets from the Earth, so, even though, from the Moon's surface, the Earth is much larger than the Sun, the sun's mass is sufficient to exert the greater gravitational pull.

So, if, by some magical power, you were to grab a hold of the Earth and stop it from moving and grab a hold of the Moon and stop it too, then let the Moon go, the Moon would fall more towards the Sun than the Earth cause the gravitational pull in that direction is over twice as much. (Ask this great magical being not to let go of the Earth, because if he does, the Earth would fall into the Sun too).

That's not quite the same as your scenario but it points out that zero net gravitation doesn't govern where an object ends up. The Moon orbits both the Earth and the Sun, and it's in a stable orbit around the earth even though it's feeling more gravitation from the Sun. That's because the Moon is inside the stable part of the Earth's Hill Sphere.

In your scenario, however, a passing object the mass of another star could certainly pull the Earth away from the Sun. It wouldn't even need to achieve a net zero gravitation to accomplish that, nor would it need to be nearly so massive.

The picture below covers the Earth orbiting around the Sun. If you bring the net Gravity to zero, in theory the "F" in the diagram shrinks to zero and the Earth continues straight in direction V for that time period, increasing it's distance from the Sun. Source

The Earth's tangential velocity relative to the sun is 30 km/s and it's escape velocity is just the square root of 2 times that, about 42.5 km/s, so an acceleration of the Earth of 12.5 km/s or moving the Earth to an orbit a bit outside Mars' orbit and keeping the velocity the same would both work (or some combination of the two).

The model is a bit more complicated because a gravitational assist, which would also happen in your scenario and a gravity assist can work both ways, increasing or decreasing the orbital velocity. It's possible, depending on direction of the pass, that a passing star could push the Earth closer to the Sun, even passing outside, if it slows the earth's velocity by gravity assist. Drawing it away isn't the only possible outcome.

More on gravity assists here, Short and Longer.

As James Kilfinger points out, stars passing that close is extremely extremely rare so this kind of thing, for all practical purposes, virtually never happens. It's much more rare than a dinosaur killing meteor for example. It's hugely unlikely.

amateur observing - Watching the Mercury transit with improvised devices

Mercury's angular diameter on transit day will be 12 arcseconds.
A camera obscura using a 12 mm aperture could resolve it; one lens from +0.75 diopter reading glasses, if you can get them, will project a bright 12 mm image of the Sun at a distance of 1.33 m.
Note that a larger aperture or a shorter focal length will make the Sun image hotter than direct sunlight unless you add a filter.

Test with sunspots before relying on it for Mercury.
If it counts as improvised, projection with 7x35 binoculars easily showed me the 2012 transit of Venus.

early universe - Is it possible to get a glimpse of the Big Bang through gravitation waves?

Gravitational waves from the big bang may be "heard" but not by LIGO. The waves emitted at or around the inflationary epoch of the big bang are expected to be at much lower frequencies (milli-Hz or lower) than those announced today by LIGO. There are various sources of noise that make LIGO insensitive to GWs at frequencies below about 10 Hz.

It will take space-based interferometers like the proposed LISA, with longer interferometer arms and well away from terrestrial sources of noise to stand a chance of detecting such GWs.

If they are detected - they might "sound" something like this (if upshifted into the audible range) - from the LIGO website. It sounds like white(ish) noise because of the broad continuum of frequencies expected.

size - How thick can planetary rings be?

There is an explanation for why rings flatten out here. The general mechanism is that particles collide, and gets a very uniform momentum. Thus, any set-up giving unusually thick rings is in essence "cheating".

Here are some ways:

Moons can cause spiral waves in the rings, giving them more of a structure in the z direction. The ones known in Saturn's rings has a modes amplitude of just 10-100 m, but larger Moons can easily increase that.

Another way is simply having massive rings. Then they can not get more flattened, as there are no more empty space to remove.

A tilted ring relative to the Planets orbit around the star is going to experience tidal forces, as long as the radius of the rings is some notable fraction of the planet's orbital radius. From the context that sparked the question, that is not a suitable mechanism though, along with the possibility of having a so low density that particle collisions are rare.

However, more promising:

The halo ring of Jupiter is estimated to be around 12500 km thick (about the same as the diameter of the Earth), and are very fine dust kept from condensing into a disc by both the magnetic fields of Jupiter, and by iterations with the Galilean Moons.

We have four planets with rings in the solar system, so the sample size is quite small. Applying some small-sample-size statistical methodology, in this case an unusual application of the German Tank Problem, we can give a rough but realistic maximum thickness of a ring:

$$N approx m+frac{m}{k}-1$$

Where $m$ is the highest observed value, and $k$ the sample size.

Modified slightly to get a non-integer version that makes some sense, we get:

$$max_{thickness} approx 12500km+frac{12500km}{4} approx 16000km$$

By no means a very certain limit, but at least about what can obtain from what we know.

"Supernova" is the explosion or the resulting celestial body? Is it incorrect to call the explosion "supernova"?

Is it incorrect to call the explosion “supernova”?

Yes and no.

Better said, the explosion is the very first part of a supernova. While the explosion lasts for but a few seconds to a few hundreds of seconds, a supernova can last for hundreds of days. What we see visibly as a supernova are the after effects of that explosion. The explosion can create lots and lots of stuff moving at very high velocities, and lots and lots of highly radioactive nuclei.

If the star had outgassed material prior to the explosion, the highly kinetic material produced by the explosion runs into that previously outgassed material and makes it glow. This takes some time to cool down. The radioactive material produced during the short course of the explosion proper takes time to decay to stable elements. This radioactive decay eventually produces gamma rays, some of which is absorbed by the nearby material, heating it, thereby eventually producing thermal radiation.

For example, a type Ia supernova produces a large amount of nickel-56. This decays to cobalt-56 with a half-life of about 6 days, which in decays to iron-56 with a half-life of about 77 days. The heating that results from these decays is what we see, and because it is so predictable, this is makes type Ia supernovae a fantastic standard candle.

exoplanet - A star a black hole and planets around them

It's possible to have planets orbiting a binary pair of stars, your scenario of a close orbit, sometimes called "short orbit binaries". See here, also posted above in comments. In such a binary-system, nothing can orbit an individual star, but at some distance, plants can orbit and some systems like this have even been observed, listed in the link. The orbital dynamics is the same for a star-black hole short-orbit binary.

Now, there are problems. Black holes form out of very large stars and the formation is one of the biggest explosions in the universe, a Type II supernovas, and that's not very friendly to any planets in orbit. A star might survive it, planets would be harder, though it might be possible for new planets to form from nebula material remaining after the nova (I'm just guessing there).

A black hole could also form from a Neutron star accreting matter, but you still have the problem that the formation of a Neutron Star also only happens out of a Type-II supernova, so such a system has a difficult beginning.

Edit: While some planets have been observed around Neutron stars, these appear to be quite rare. 2 Neutron Stars have been observed with planets, out of over 1,600 Neutron stars observed. A type II nova is very planet unfriendly.

Theoretically a close gravitational capture is possible, but those are very rare, as stars rarely get that close. There's many stars that are known to orbit the super-massive black holes at the center of our galaxy (Andromeda galaxy too), but stars and stellar mass black holes are much more rare. A few have been observed, but they don't appear to be common. Such a system would be easy to observe, so the fact that there are only a few that have been noticed is evidence to them being rare. Here's a few mentions of them. One, Two, Three, Four.

From the 4th article, which is from 2011, so more may be known now, but it says:

Only about 20 binary stellar systems are known to contain a black
hole, out of an estimated population of around 5,000 in the Milky Way
Galaxy.

A 2nd problem is that a star feeding a black hole would create an accretion disk which would be very radioactive and not ideal for life on an orbiting planet. Maybe the planet could have a very thick atmosphere that might protect it, but that would also likely reduce sunlight reaching the surface. The star feeding the black hole would also be losing mass, and over time, grow smaller and provide less light and heat to the planet. Ideally, you'd want it to be a very slow feed. It's pretty far from an optimal life on planet situation.

So it is possible to have planets in this system not to be consumed by
this black hole but just follow their unique orbits

This part is certainly possible. Things can orbit a black hole at a safe distance without any problem. As for life, we don't know how common life is in other solar-systems so nobody can say how likely it might be, but it's theoretically possible, but, in my opinion, pretty far from ideal.