Introduction: Feel the Motion of the Earth!
The Earth we live on rotates at a staggering speed while revolving around the Sun. Yet, as we go about our daily lives with our feet firmly on the ground, we do not feel the Earth moving at all! It feels completely natural to see it as "the stars and the Sun moving across the sky" (the geocentric model).
So, how did ancient and historical scientists discover irrefutable proof that "the Earth itself is moving" (the heliocentric model)?
In this chapter, we will clearly organize high-frequency topics on the Common Test: the evidence for rotation (Foucault's pendulum), the evidence for revolution (annual parallax and annual aberration), as well as the mechanisms of time and the solar calendar that directly shape our everyday lives. It might feel tricky at first, but if you master these key concepts one by one, you can turn this topic into a guaranteed scoring area!
1. Earth's Rotation and Its Evidence
What Is Earth's Rotation?
The movement of the Earth rotating once a day on its rotational axis from west to east is called rotation.
Because the Earth rotates from west to east, the Sun and stars appear to rise in the east and set in the west (this is called diurnal motion). However, diurnal motion itself can also be explained by assuming that "the celestial sphere is rotating," so it does not serve as decisive proof of Earth's rotation. The definitive proof that the Earth rotates came from an experiment conducted by the French physicist Léon Foucault in 1851.
Evidence for Rotation: Foucault's Pendulum
When you swing a pendulum made with an extremely long wire and a heavy bob in a wide arc, you can observe that the pendulum's plane of oscillation (the plane along which it swings) gradually rotates over time. This is Foucault's pendulum.
Unless an external force is applied, a pendulum tends to maintain its "original plane of oscillation" due to the law of inertia. Even so, when viewed from the ground, the plane of oscillation appears to rotate slowly. This is nothing less than proof that "it is not the pendulum that is rotating, but rather the ground (the Earth) beneath our very feet that is turning!" This serves as irrefutable proof of Earth's rotation!
Differences in the Rotation of the Plane of Oscillation by Latitude
The apparent rotation speed and direction of the plane of oscillation depend on the latitude \(\phi\) (phi) of the observation site.
• The North Pole and South Pole:
At the North Pole, the Earth rotates counterclockwise directly beneath the pendulum. Therefore, to an observer on the ground, the plane of oscillation rotates clockwise, completing exactly 360 degrees (one full turn) in 24 hours (\(15^\circ\) per hour). At the South Pole, it rotates in the opposite direction—counterclockwise—completing one turn in 24 hours.
• The Equator (latitude \(0^\circ\)):
At the equator, the Earth's surface moves parallel to the rotational axis, so there is no twisting rotation of the plane of oscillation. Consequently, at the equator, the plane of oscillation does not rotate at all.
• Mid-latitudes (latitude \(\phi\)):
At a location with latitude \(\phi\), the angle \(\theta\) through which the plane of oscillation rotates in one hour is expressed by the following formula:
\(\theta = 15^\circ \times \sin \phi\)
Therefore, the time \(T\) it takes for the plane of oscillation to make one complete rotation (\(360^\circ\)) is given by:
\(T = \frac{24}{\sin \phi}\) (hours)
For example, at latitude \(30^\circ\text{N}\), since \(\sin 30^\circ = 0.5\), it takes \(T = \frac{24}{0.5} = 48\) hours to complete one rotation.
[Key Takeaway] Direction of Rotation of Foucault's Pendulum
• Northern Hemisphere: Rotates clockwise (to the right when viewed from above)
• Southern Hemisphere: Rotates counterclockwise (to the left when viewed from above)
• Equator: Does not rotate (the period is infinite)
* Remember the rhythm: "Clockwise in the North, counterclockwise in the South, and stationary at the equator!"
[Common Mistake]
Some students mistakenly think that "it completes one rotation in 12 hours at the North Pole." Since the Earth's rotational period is 24 hours, the period of rotation at the North Pole is also 24 hours. You can also confirm this from the formula: \(T = \frac{24}{\sin 90^\circ} = 24\) hours.
2. Earth's Revolution and Its Evidence
What Is Earth's Revolution?
The motion of the Earth orbiting the Sun counterclockwise (as viewed from above the North Pole) once a year is called revolution.
Two must-know pieces of evidence for revolution that frequently appear on the Common Test are annual parallax and annual aberration. Their names sound similar, but their underlying principles and observational characteristics are completely different. Make sure you can clearly tell them apart!
Evidence for Revolution 1: Annual Parallax
Hold a finger up in front of your eyes and alternate closing your left and right eyes. Notice how the position of your finger seems to shift against the background? That is "parallax."
Because the Earth revolves around the Sun, when we observe relatively nearby stars every six months, their positions appear to shift against the distant background stars. Half of this apparent shift angle is called the annual parallax (symbol \(p\) or \(\pi\)).
• If we consider a right-angled triangle with the radius of Earth's orbit (\(1 \text{ au}\)) as the base looking toward the star, the angle at the star's vertex is the annual parallax \(\pi\).
• The distance to a star with an annual parallax of exactly 1 arcsecond (\(1'' = \frac{1}{3600}^\circ\)) is defined as \(1 \text{ pc}\) (parsec).
• Letting the distance to the star be \(d\) (\(\text{pc}\)) and the annual parallax be \(\pi\) (arcseconds), the following simple relationship holds:
\(d = \frac{1}{\pi}\)
* For example, the distance to a star with an annual parallax of \(0.1''\) is calculated as \(d = \frac{1}{0.1} = 10 \text{ pc}\).
[Apparent Motion on the Celestial Sphere (Annual Parallax)]
When observed over the course of a year, stars trace out small ellipses on the celestial sphere.
• Stars located at the ecliptic poles (perpendicular to the orbital plane): Trace out a circle
• Stars located on the ecliptic plane (in the plane of revolution): Trace out a straight line (back-and-forth motion)
• Stars located anywhere else: Trace out an ellipse
Evidence for Revolution 2: Annual Aberration
Imagine walking with an umbrella on a rainy day. Even if there is no wind, if you start running forward, the rain seems to fall toward you at an angle from the front. Because of this, you need to tilt your umbrella slightly forward to avoid getting wet.
The exact same phenomenon happens with light! Because the Earth is revolving through space at approximately \(30 \text{ km/s}\), starlight coming from the zenith appears tilted slightly in the direction of Earth's orbital motion. This apparent tilt angle in the direction of motion is called annual aberration (or simply aberration of light).
• If \(c\) is the speed of light and \(v\) is Earth's orbital speed, the tilt angle \(\theta\) approximately satisfies \(\tan \theta \approx \frac{v}{c}\).
• The maximum angle of aberration is nearly constant for any observed star at about \(20.5''\) (arcseconds).
• While annual parallax varies depending on the "distance to the star" (more distant stars have smaller parallax), the magnitude of annual aberration does not depend on the distance to the star and is nearly constant for all stars!
[Key Takeaway] Crucial Differences Between Annual Parallax and Annual Aberration
This is a high-yield comparison frequently tested on exams to trip students up!
• Direction of the star's apparent displacement:
Annual parallax: Displaced in the opposite direction to Earth's position relative to the Sun.
Annual aberration: Displaced in the direction of Earth's orbital motion.
* Because of this, there is a \(90^\circ\) phase difference (about 3 months) between the two apparent displacements.
• Effect of distance:
Annual parallax: Inversely proportional to the distance to the star (smaller for more distant stars, making extremely distant stars impossible to measure).
Annual aberration: Constant regardless of distance (about \(20.5''\)). This is because it is determined solely by Earth's orbital speed and the speed of light.
[Fun Fact]
Although annual parallax was theoretically predicted early on, stars are so unimaginably far away that measuring it was extraordinarily difficult. Consequently, annual aberration was actually discovered first in history (by James Bradley in 1728), while annual parallax was not detected until more than a century later in 1838 (by Friedrich Bessel).
3. Earth's Rotation, Revolution, "Time," and "Calendars"
Solar Day vs. Sidereal Day: Why Is a Day 24 Hours Long?
The time it takes from when the Sun culminates (reaches its highest point in the south) until it culminates again the next day is called 1 solar day (averaging 24 hours).
However, if you measure the time it takes for the Earth to rotate exactly \(360^\circ\) (one full turn) relative to distant stars, it actually takes only about 23 hours, 56 minutes, and 4 seconds! This is called 1 sidereal day.
Why does this difference of about 4 minutes occur?
1. While the Earth completes one full rotation (\(360^\circ\)) relative to the stars, it also advances about \(1^\circ\) along its orbit around the Sun (\(360^\circ \div 365\text{ days} \approx 1^\circ\)).
2. Therefore, when the Earth has turned \(360^\circ\), the Sun has not yet returned to due south (culmination)!
3. For the Sun to culminate again, the Earth must rotate an extra amount equal to what it moved along its orbit (about \(1^\circ\)).
4. Since the Earth rotates \(360^\circ\) in 24 hours (1440 minutes), the time it takes to rotate \(1^\circ\) is:
\(1440 \text{ minutes} \div 360 = 4 \text{ minutes}\).
Because of this "extra rotation of about 4 minutes," 1 solar day (24 hours) is about 4 minutes longer than 1 sidereal day (about 23 hours and 56 minutes)!
[Connect the Concepts!] Rising and Setting Times of Constellations
The fact that 1 sidereal day is about 4 minutes shorter than 1 solar day means that "the same constellation rises in the east and culminates about 4 minutes earlier each day."
After one month (30 days), it rises earlier by \(4 \text{ minutes} \times 30 = 120 \text{ minutes} = 2 \text{ hours}\). This is why the constellations visible in the night sky change from season to season!
Standards of Time: Apparent Solar Time, Mean Solar Time, and Standard Time
• Apparent Solar Time: Time based on sundials, where the actual culmination of the Sun marks noon (12:00). However, because Earth's orbit is elliptical and its rotational axis is tilted, the length of an apparent solar day varies slightly throughout the seasons.
• Mean Solar Time: Time based on a fictitious "mean sun" assumed to move along the celestial equator at a constant speed. This provides a uniform 24-hour day throughout the year.
• Japan Standard Time (JST): In Japan, the mean solar time along the meridian of \(135^\circ\text{E}\), which passes through Akashi City in Hyogo Prefecture, is used as the standard time for the entire country.
The Solar Calendar (How Calendars Work)
The time it takes for the Earth to leave the vernal equinox and return to it is called 1 tropical year, which is approximately 365.2422 days long.
If a calendar year were simply set to 365 days, the calendar would drift away from the seasons by about 0.2422 days each year (about 6 hours per year, or roughly 1 full day every 4 years). The rules for leap years were developed to correct for this drift.
In the Gregorian calendar used around the world today, leap years (years with February 29 added, making a 366-day year) are determined according to the following rules:
1. A year that is divisible by 4 is a leap year.
2. However, a year that is divisible by 100 is a common year (365 days).
3. However, a year that is divisible by 400 is still a leap year.
* Under these rules, exactly \(100 - 4 + 1 = 97\) leap years are distributed over a 400-year cycle.
Calculating the average length of a year over 400 years yields:
\(365 + \frac{97}{400} = 365.2425 \text{ days}\)
This is remarkably accurate, producing an error of only about 1 day every 3,000 years compared to the actual tropical year (365.2422 days)!
Summary & Final Pre-Test Check
Finally, let’s recap the essential points so you can easily recall them right before the exam!
• Evidence for Rotation: Foucault's pendulum (the plane of oscillation rotates clockwise in the Northern Hemisphere and counterclockwise in the Southern Hemisphere; does not rotate at the equator; period \(T = \frac{24}{\sin \phi}\)).
• Evidence for Revolution:
1. Annual parallax: Inversely proportional to distance (\(d = \frac{1}{\pi}\)). Apparent displacement is opposite the direction of the Sun.
2. Annual aberration: Constant at a maximum of about \(20.5''\) regardless of distance. Apparent displacement is in the direction of Earth's orbital motion.
• Solar Day vs. Sidereal Day: Because the Earth revolves around the Sun while rotating, 1 solar day is about 4 minutes longer than 1 sidereal day.
• Solar Calendar: To adjust for the fractional remainder of 1 tropical year (about 365.2422 days), there are 97 leap years every 400 years.
If you remember these phenomena together with the fundamental "why" behind them, you'll be able to calmly find the correct answer even when facing data-analysis questions or unfamiliar graphs. Approach your exam with confidence!