a) pulsating stars:  an object undergoing changes in size or in temperature (which in turn cause changes in the light output of an object).  Requiring the light variation to be periodic rules out a wide phenomena (supernovas, for example).  However, a pulsating star (which expands or contracts periodically due to non-equilibrium physical conditions inside the star) or recurrent novae (in which a companion star or an accretion disk accretes matter onto another star).

b) rotating stars:  a rotating object with surface irregularities (i.e., either bright or dark spots).  Although rotation is periodic, the rotation period can increase (as in the case of a single pulsar, for example) or decrease (as in the case of a pulsar in a binary system, for example, whose rotation can be sped up by matter accreted from its companion)

c) eclipsing binaries:  an object whose light at the detecting end is reduced or enhanced by interaction with objects that the light encountered along the way to the detecting end (examples of light reduction are eclipses by a companion object or by absorption of light by dust along the way from the object to the detector; an example of brightening due to the presence of another object along the way is gravitational lensing).

let's investigate these possible periodic light variations in more detail.
 

a) pulsating (or oscillating) stars

It is clear that small periods of pulsation (of order seconds) will only occur for stars with small radius and large mass.  The two most well-known objects fitting this description are neutrons (typical values: M = 2 M_sun, R = 10⁴km, city-sized)  and white dwarfs  (typical values:  M = 1 M_sun, R = 10⁷ km, earth-sized).

For white dwarf stability (i.e., for the white dwarf to hold itself together),  the pulsation period will be approximately 20 seconds.

b) rotating stars

                      F_net   =    GmM/R²  -   N  =  ma_c  = mv²/R   =  m (2p R/P)²/R  =   4p² R m/P²

where  N  is the normal force exerted by the star on you and  v  is the rotational speed at the equator.
Since positive is defined in the direction of the centripetal acceleration, the acceleration is positive by definition positive.  For you to remain in contact with the surface of the object clearly requires  N  >  0, or equivalently

                      P_rot  >  2 p (R³/GM)^1/2 

Note that this is almost identical to the condition for pulsation periods obtained above.  The only difference is that the condition for pulsation involves an (approximate) equality, whereas the condition for rotational stability involves an inequality.

As before, small periods of rotation (of order seconds) will only occur for stars with small radius and large mass.  For the typical mass and radius values listed above, 

white dwarf stability (i.e., for the white dwarf to hold itself together),  requires a rotation period exceeding approximately 20 seconds.

neutron star stability requires a rotation period exceeding approximately .0004 seconds.


 

c) eclipsing binary systems

                      G(M₁ + M₂) (P_binary/(2p))²  =  (a₁ +  a₂)³

Clearly, the separation of the stars' center must exceed their combined radii if they are to remain physically distinct.  Therefore

                      P_binary  >  2 p ((R₁ +  R₂)³/G(M₁ + M₂))^1/2

For the 3 combinations of white dwarf of neutron star, the binary periods are limited by

                      white dwarf  +  white dwarf                     P_binary  >   1000 sec   =     15 minutes

                      white dwarf  +  neutron star                    P_binary  >       10 sec

                      neutron star  +  neutron star                   P_binary  >   .001 sec

As an example of the powerfulness of these limiting periods, consider the discovery of the first neutron stars by Jocelyn Bell and her group in 1968.

The objects they found (later named "pulsars") had regular periods of approximately 1 second.    The above analysis immediately rules out rotating white dwarfs and all binaries systems other than those containing 2 neutron stars.  The only possibilities left with such short periods were rotating neutron stars and binaries containing two neuton stars.  The latter alternative can be ruled with general relativity: such a close pair of orbiting neutron stars would emit so much gravitational radiation that the binary period would change noticeably within a short time. 

In sum, simple applications of the laws of physics lead to the correct interpretation for the nature of astrophysical objects that vary in luminosity with a period of a few seconds: rotating neutron stars!