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The companion could be a main sequence star, white dwarf, neutron star or black hole.

We detected pulsar-pulsar binaryhere several decades ago. But if one of the pulsar does not radiate towards us, we may be not able to tell it is a pulsar or a black hole.

If the companion is not a main sequence star, how to know the nature of the companion in a neutron star binary?

Of the binary systems with one single neutron star, is that possible there is another neutron star or a hidden black hole?

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For something to be possible in this universe, doesn't even have to be consistent with the physics we know. Astronomical surprises have often led to revisions in our theories. Thus, all the rare situations you described are possible.

How do we detect such evasive systems?

There is no definite answer. Based on the data we have about the system, you can get creative while doing data analysis and come up with new methods. That is one of the reasons science continues to be fun.

Maybe someone who has had some experience with the concerned matter can expand.

Also, such systems are rare due to a physical reason. Mass transfer between binary systems ensures that one of the stars get evolved earlier than the other. The other star either ends up being a low mass star and evolves accordingly in most cases. The remaining cases account for your rare description of systems.

See Algol Paradox for an interesting read.

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Could you please summarize the stellar types of the other star(non main sequence) in a neutron star binary? How did we find this kind of system generally? – questionhang May 20 '14 at 7:35
This provides no answer to the question. – Rob Jeffries May 1 '15 at 17:08

One way to figure out if one (or both) of the objects is a black hole, neutron star, white dwarf, or other compact object would be to try to measure its mass. For example, a neutron star and a white dwarf are both compact stellar remnants. However, there is one decisive factor that determines which type of stellar remnant a progenitor star will become: the remnant's mass.

The Chandrasekhar limit, named after Subramanyan Chandrasekhar, is the maximum mass a white dwarf can have. It is about 1.4 solar masses - quite a lot, if you think about it. If the stellar remnant weighs more than this, it will become a neutron star. There is a similar limit for the mass of neutron stars, the Tolman-Oppenheimer-Volkoff limit. Stellar remnants with masses greater than this will become black holes.

So if you know the mass of one of the objects and know it to be a stellar remnant, you should be able to figure out which type it is. How would you measure the mass? Well, you could study the orbit of its companion star to try and determine the stellar remnant's effect on its orbit. Another way would be to study gravitational waves emitted by the system. These waves can only be emitted under certain circumstances - for example, in a system of binary neutron stars - see, for example, the Hulse-Taylor binary system, also known as PSR B1913+16. The power radiated as these waves, as well as the orbital decay, depend on the masses of the objects. While detecting gravitational waves is an incredibly difficult task, there are several detectors planned or already in operation, such as LISA and LIGO.

While there are other ways to differentiate between such stellar remnants, using the mass of the object(s) can be useful.

Also, a neutron star can indeed have another neutron star as a companion - again, see the Hulse-Taylor binary system.

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Finding the mass may tell you what it is not, but can be ambiguous. If I find the companion mass to be 1.3 solar masses - what is it? Wie also don't know what the maximum mass of a neutron star is, only that it is more than 2 solar masses, but probably less than 3 and a bit solar masses (a limit imposed by GR). – Rob Jeffries May 1 '15 at 17:05

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