How to Calculate Tidal Acceleration?

Tidal acceleration is calculated by finding the difference in gravitational acceleration exerted by an external body (like the Moon or Sun) at a specific point on a celestial body (like Earth) and the gravitational acceleration exerted by the same external body at the center of the celestial body. This difference represents the net acceleration experienced due to the tidal force.

Understanding Tidal Acceleration

The tidal force arises because gravity's strength decreases with distance. Points on a celestial body closer to an external gravitational source experience a stronger pull than points farther away. The difference in these gravitational forces creates a tidal bulge on both the near and far sides of the body.

Calculation Steps

Here's a breakdown of how to calculate tidal acceleration:

1. Identify the Gravitational Source: Determine the external body exerting the gravitational force (e.g., the Moon, the Sun).

2. Calculate Gravitational Acceleration at the Center of the Body: Determine the gravitational acceleration vector, g_center, exerted by the external body at the center of the celestial body being affected (e.g., the Earth). This can be calculated using Newton's Law of Universal Gravitation:

   g_center = GM/r²

   Where:

   • G is the gravitational constant (approximately 6.674 × 10^-11 N⋅m²/kg²)
   • M is the mass of the external body (e.g., mass of the Moon)
   • r is the distance between the center of the celestial body (e.g., the Earth) and the center of the external body.

3. Calculate Gravitational Acceleration at the Point of Interest: Determine the gravitational acceleration vector, g_point, exerted by the external body at the specific point on the surface (or within) of the celestial body where you want to calculate the tidal acceleration. This calculation also uses Newton's Law of Universal Gravitation, but the distance 'r' will now be the distance from that point to the external body. Be sure to consider the vector direction.

4. Calculate the Tidal Acceleration: The tidal acceleration, a_tidal, is the vector difference between the gravitational acceleration at the point of interest and the gravitational acceleration at the center of the body:

   a_tidal = g_point - g_center

Important Considerations

• Vector Subtraction: The acceleration values are vectors, meaning they have both magnitude and direction. Correctly performing the vector subtraction is crucial.
• Approximations: Often, simplifying assumptions are made, particularly when dealing with distances much larger than the radius of the celestial body.
• Reference Frame: The tidal acceleration is calculated relative to the center of the celestial body.

Example:

Let's imagine calculating the tidal acceleration on Earth due to the Moon at a point directly facing the Moon.

1. Gravitational Source: The Moon.
2. g_center: Calculate the Moon's gravitational acceleration at Earth's center.
3. g_point: Calculate the Moon's gravitational acceleration at the point on Earth's surface directly facing the Moon. This point is closer to the Moon than Earth's center.
4. a_tidal: Subtract the g_center vector from the g_point vector. The resulting vector points towards the Moon and represents the tidal acceleration at that point. On the opposite side of the Earth, the tidal acceleration vector will point away from the Moon.