When a neutron star is formed it is rotating very rapidly.
a) What are these objects typically called and how do we find them?
b) The first one of these objects found was called a LGM. What does that stand for and why were they called that?
A rotating neutron star is called a pulsar, due to its rotation its emmits electromagnetic radiation at regular intervals. If the beam is pointed towards earth then we can receive the signal and that's how we confirm the existence of a pulsar.
In 1967, when the first time humans observed the pulses from space, they thought it's from some extra terrestrial species. Later on research and receiving more such signals they realized that it is a natural phenomenon. The first pulsar was named as LGM-1 that stands for 'little green man' because it was originally considered to be an alien.
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