Gravitational Interactions and Ocean Tides: Seventh Grade Science
Ocean tides are rhythmic planetary rises and falls of ocean sea levels caused by the gravitational interactions between Earth, the Moon, and the Sun.
The Gravitational Cause of Ocean Tides
In 1687, Sir Isaac Newton formulated the equilibrium theory of tides, demonstrating that tides are gravitational phenomena governed by Newton's Universal Law of Gravitation. Every particle of matter attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them (F = G * m1 * m2 / r^2).
Because Earth has physical dimensions (a diameter of 12,742 kilometers), the gravitational pull of the Moon is not uniform across the planet:
- Sub-Lunar Point (Near Side): The side of Earth facing directly toward the Moon is closest to the Moon. The Moon's gravitational pull is strongest here, pulling ocean water outward toward the Moon to create a high-tide tidal bulge.
- Anti-Podal Point (Far Side): The side of Earth facing directly away from the Moon is farthest from the Moon. The Moon's gravitational pull is weakest here. Earth's solid rocky center is pulled more strongly toward the Moon than the distant water on the far side. Effectively, Earth is pulled away from the water, creating a second high-tide tidal bulge on the opposite side.
- Between Bulges: At 90 degrees to the Moon-Earth line, water is drawn away to feed the two bulges, creating low tides.
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| THE DUAL TIDAL BULGE ARCHITECTURE |
| |
| LOW TIDE THE MOON |
| v | |
| ( ) v |
| (====[ E ]====) <========================================> (MOON) |
| ^ ( A ) ^ |
| | ( R ) | NEAR-SIDE BULGE: Gravitational pull strongest |
| | ( T ) | (Water pulled toward Moon) |
| | ( H ) |
| | |
| FAR-SIDE BULGE: Gravitational pull weakest |
| (Earth pulled away from ocean water) |
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Why the Moon Dominates Over the Sun
The Sun possesses approximately 27 million times more mass than the Moon. However, the Sun is located roughly 390 times farther away from Earth than the Moon (150 million km vs. 384,400 km).
Because tidal-generating force is a differential force that depends on the difference in gravitational pull across Earth's diameter, the tidal force decreases with the cube of distance (1 / r^3), rather than the square of distance (1 / r^2):
- The Moon's tidal-generating force is proportional to M_moon / (r_moon)^3.
- The Sun's tidal-generating force is proportional to M_sun / (r_sun)^3.
Calculating this ratio reveals that the Moon's tidal force is approximately 2.16 times greater than the Sun's. The Moon is the primary driver of Earth's tides, while the Sun acts as a secondary modulating force.
Spring Tides Versus Neap Tides
As the Moon revolves around Earth, its alignment relative to the Sun changes, producing two distinct tidal cycles every synodic month:
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| SPRING TIDES VERSUS NEAP TIDES |
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| SPRING TIDES (Max Tidal Range): NEAP TIDES (Min Tidal Range): |
| Alignment: SYZYGY (Line: 180 deg) Alignment: QUADRATURE (90 deg)|
| Occurs at: NEW MOON & FULL MOON Occurs at: 1ST & 3RD QUARTERS |
| |
| [Sun] ===> [Moon] ===> [Earth] [Moon] |
| ^ |
| Solar & Lunar bulges REINFORCE: | |
| - Extreme Highest High Tides [Sun] ===> [Earth] |
| - Extreme Lowest Low Tides |
| Solar & Lunar bulges OPPOSE: |
| - Modest Low High Tides |
| - Mild High Low Tides |
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Spring Tides: Maximum Tidal Range
Spring tides have nothing to do with the spring season; the name derives from the German word springen, meaning "to leap up." Spring tides occur twice each month during New Moon and Full Moon phases, when the Sun, Earth, and Moon align in a straight line (syzygy).
The solar tidal bulge and lunar tidal bulge align in the same axis, constructively reinforcing one another. This produces the maximum tidal range: exceptionally high high tides and exceptionally low low tides.
Neap Tides: Minimum Tidal Range
Neap tides occur twice each month during the First Quarter and Third Quarter Moon phases, when the Sun and Moon are at right angles (90 degrees, quadrature) relative to Earth.
The solar tidal bulge opposes and partially cancels the lunar tidal bulge. This produces the minimum tidal range: lower-than-average high tides and higher-than-average low tides, resulting in mild, subdued tidal swings.
Tidal Patterns and Daily Rhythms
As Earth rotates on its axis, coastal shorelines move into and out of the ocean tidal bulges. Because the Moon revolves around Earth in the same direction that Earth rotates, Earth must rotate for an extra 50 minutes each day to bring a coastline back under the Moon. This establishes the lunar day of 24 hours and 50 minutes:
- Semi-Diurnal Tides: Two high tides and two low tides of approximately equal height every lunar day, with a high tide roughly every 12 hours and 25 minutes. Common along the Atlantic coasts of North America and Europe.
- Diurnal Tides: One high tide and one low tide per lunar day (e.g., parts of the Gulf of Mexico).
- Mixed Semi-Diurnal Tides: Two high tides and two low tides per lunar day, but with significant inequality in heights (e.g., the Pacific coast of North America).
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| COASTAL TIDAL RECORD CLASSIFICATIONS |
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| SEMI-DIURNAL TIDE: Two equal highs and two equal lows per 24.8 hours |
| ^ /\ /\ |
| Ht | / \ / \ |
| +----+----+------+----+-------------------------------------------> |
| \ / \ / |
| \/ \/ |
| |
| MIXED SEMI-DIURNAL: Two unequal highs and two unequal lows per day |
| ^ /\ _ |
| Ht | / \ /\ / \ |
| +----+----+--+--+-+--+--------------------------------------------> |
| \ / \/ \/ |
| \/ |
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Coastal bathymetry and bay geography can amplify tides through resonance. In the Bay of Fundy in Nova Scotia, the natural oscillation frequency of the funnel-shaped bay matches the incoming tidal cycle, amplifying tides to a world-record tidal range of sixteen meters (fifty-three feet).
Exercises and Solutions
Exercise 1: Calculating the Gravitational Tidal Ratio of the Moon Versus the Sun. The mass of the Sun is approximately 1.989 x 10^30 kilograms, and its distance from Earth is 1.496 x 10^11 meters. The mass of the Moon is approximately 7.348 x 10^22 kilograms, and its distance from Earth is 3.844 x 10^8 meters. Using the differential tidal force proportionality (Tidal Force proportional to Mass / Distance^3), calculate the exact mathematical ratio of the Moon's tidal force compared to the Sun's tidal force, and explain why the less massive Moon exerts greater tidal influence.
Solution 1: The differential tidal force is proportional to M / r^3: For the Moon: Force_moon = (7.348 x 10^22 kg) / (3.844 x 10^8 m)^3 = (7.348 x 10^22) / (5.6795 x 10^25) = 1.2938 x 10^-3. For the Sun: Force_sun = (1.989 x 10^30 kg) / (1.496 x 10^11 m)^3 = (1.989 x 10^30) / (3.348 x 10^33) = 5.9408 x 10^-4. Taking the ratio of lunar to solar tidal force: Ratio = Force_moon / Force_sun = (1.2938 x 10^-3) / (5.9408 x 10^-4) = 2.178. The Moon's tidal force is approximately 2.18 times greater than the Sun's. The Moon exerts greater tidal influence because tidal forces decrease with the cube of distance (r^3). While the Sun is 27 million times more massive than the Moon, it is 389 times farther away. Cubing that distance ratio: (389)^3 = 58,863,869. The distance factor diminishes the Sun's tidal effect by roughly 59 million times, giving the nearby Moon the dominant role in generating Earth's tides.
Exercise 2: Predicting Daily Tidal Sequences from Lunar Day Dynamics. A harbor pilot in Boston Harbor records a morning high tide at exactly 08:00 AM on a Tuesday. The harbor experiences a semi-diurnal tidal regime with two high tides and two low tides each lunar day (24 hours and 50 minutes). (a) Calculate the predicted time of the subsequent low tide that afternoon; (b) Calculate the predicted time of the second high tide that evening; (c) Calculate the predicted time of the morning high tide on Wednesday.
Solution 2: (a) In a semi-diurnal tidal regime, the lunar day is 24 hours and 50 minutes. The interval between a high tide and the subsequent low tide is one-fourth of a lunar day: Interval = (24 hr 50 min) / 4 = 6 hours and 12.5 minutes. Adding 6 hours and 12.5 minutes to the 08:00 AM high tide gives a predicted low tide at 2:12:30 PM (approximately 2:13 PM) Tuesday afternoon. (b) The interval between consecutive high tides is half of a lunar day: Interval = (24 hr 50 min) / 2 = 12 hours and 25 minutes. Adding 12 hours and 25 minutes to the 08:00 AM high tide gives a predicted second high tide at 8:25 PM Tuesday evening. (c) The next morning high tide will occur one full lunar day (24 hours and 50 minutes) after the Tuesday morning high tide: Adding 24 hours and 50 minutes to 08:00 AM Tuesday gives a predicted morning high tide at 08:50 AM Wednesday.