A sidereal day is the amount of time it takes for the Earth (or, in general, any planet) to make precisely one revolution, relative to the Celestial Sphere. Distant stars make good approximations of fixed points on the celestial sphere, so a sidereal day can be considered to be the time between a star reaching zenith (or it's cloest approach to zenith) and it's reaching zenith again.

A solar day is the time it takes for a planet to make precisely one revolution, relative to the Sun (or whatever star it is orbiting). In other words, this is the time from noon (sundial time) to noon (sundial time).

It is difficult to imagine the difference between the two without a diagram, so here's one. Imagine a planet where the (sidereal) year is only eight solar days (and for simplicity imagine the planet has no axial inclination).

                          \ B
                          A \
                             |   star at zenith for B8
                        /   A8B  - - - - - - - - - - - - -
              sun at  /        A
             zenith             7
            for A9  /           |B
                  /              6
                /                 | A
              /                   B|
            /                      B4A
          /                         B|
        /                            |A
      /                               2
    /                                 |B
  /                                  A|
               sun at zenith for A0   |   star at zenith for B0
* - - - - - - - - - - - - - - - - -  A0B  - - - - - - - - - - - - -

Imagine two observers on opposite sides of the planet at some time 0, so that it's noon for observer A (and consequently midnight for observer B). As time passes, the planet spins and goes around in it's orbit. (In this diagram, the spin and the orbit are both in the same direction - anticlockwise. The Earth's orbit and spin looks the same when viewed from some point to the North of the ecliptic, except of course it spins much faster.) At time 8, the planet has made one complete revolution (relative to the celestial sphere), so the same star that was at zenith for B at time 0 is at zenith once more. The sidereal day is 8 units long. However, as observed from the planet, the position of the sun has changed relative to the celestial sphere, so observer B doesn't see the sun at zenith until time 9. The solar day is 1/8 longer than the sidereal day.

It is no coincidence that 8 is the number of solar days in the year, and 1/8 is the factor by which the solar day differs from the sidereal day. Over the course of a year, the number of sidereal days is precisely one more than the number of solar days. Thus, there are about 365.25 solar days in the year, but there are about 366.25 sidereal days. This difference of 1 can be conceptually grasped by considering a planet that is in tidal lock with the sun - imagine observer A always seeing the sun at zenith in the diagram above, and observer B always on the opposite side. There are no solar days at all, but when the planet makes one revolution in one year, observer B sees the same star at zenith as a year before - so there's one sidereal day where there are no solar days. (If the planet were spinning in the direction opposite the orbit, with more than 1 solar day per year, there would be one fewer sidereal days than solar days. If the planet were spinning in the opposite direction, with less than 1 solar day per year, the sum of the fraction of the solar day per year and the fraction of the sidereal day per year would be 1.)

Earthlings most easily recognise the solar day because the sun is so significant in the sky; it lights up the whole sky by atmospheric scattering and obscures all the other stars, so the solar day is the most significant type of day on Earth. On Neptune or Pluto, however, the sun (at first glance) looks like it's not much more than just another star - brighter than usual, but it does not have such a glare as to make all the other stars invisible. On distant, airless planets, the sidereal day would be more recognisable than the solar day. Neptunians would be more likely to make their calendars, and calculate their dates and times of 'day' by sidereal time rather than solar time.

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