A day seems simple: Earth spins once. But "once" relative to what? On the celestial sphere there are two choices, the Sun and the distant stars, and they give two different days.
- A solar day is noon to noon, the Sun back on your meridian: $24$ h on average. Our clocks keep this day.
- A sidereal day ("star day") is one turn relative to the distant stars: $23$ h $56$ min $4$ s.
Why it works
While Earth spins, it also moves along its orbit, about $360^\circ \div 365.25 \approx 1^\circ$ per day, in the same direction as it spins. After one full $360^\circ$ turn the stars are back where they were, but the Sun now lies about $1^\circ$ farther east among them. Earth must turn that extra $1^\circ$ to bring the Sun back to the meridian. At $15^\circ$ per hour, $1^\circ$ takes about 4 minutes, so a solar day is about 4 minutes longer than a sidereal day. In rates: $\dfrac{1}{\text{sidereal day}} = \dfrac{1}{\text{solar day}} + \dfrac{1}{\text{year}}$.
Key idea
A sidereal day (one spin relative to the stars) is 23 h 56 min; a solar day (noon to noon) is 24 h, because Earth also moves about $1^\circ$ along its orbit each day and must turn about $361^\circ$ to face the Sun again.
Worked examples
Worked example: Stars rise earlier
Orion's bright star Rigel rises at 11:00 pm tonight. About when will it rise 30 days from now?
- Each night a star rises about 4 minutes earlier, because the clock keeps solar days and the star keeps sidereal days.
- $30 \times 4 = 120$ min $= 2$ h earlier.
- So about 9:00 pm. That is why the evening sky changes with the seasons.
Answer: About 9:00 pm.
Worked example: Count the spins
How many times does Earth turn relative to the stars in one year?
- In $365.25$ solar days the Sun appears to go once around the sky relative to the stars.
- That means Earth makes one more turn relative to the stars than relative to the Sun.
- $365.25 + 1 = 366.25$ sidereal days per year.
Answer: $366.25$ times.
Common mistake
"A day is one spin of Earth" is only true for the sidereal day. From noon to noon Earth turns about $361^\circ$, not $360^\circ$. Check with the numbers: $366.25$ sidereal days of $23.934$ h make $8766$ h, the same as $365.25$ solar days of $24$ h.
Astronomers set their telescopes by sidereal time, because a star crosses the meridian at the same sidereal time every day: the sidereal time equals the right ascension on your meridian (see right ascension).