Tidal effect of the sun

The Moon makes the largest contribution to the observable tides, but the Sun also has a significant influence on the tides. If Sun, Moon and Earth are in a certain constellation, spring and neap tides occur. Spring tides are tides that are characterized by a particularly high high tide and a particularly low low water level. Neap tides are a form of tide in which high and low water levels are particularly low.

The Sun is significantly heavier than the moon, but it is about 390 times further from the earth. The gravitational acceleration on Earth caused by the Sun is:

\begin{equation} \label{eq:eqn_grav_moon} a_S = \gamma \cdot { m_S \over d^2 } = 6.675\cdot10^{-11} \cdot { {1.988435 \cdot 10^{30} } \over (149.6 \cdot 10^9)^2} = 5.93\cdot10^{-3} {m \over s^{2} } \end{equation}

This value is significantly larger than the gravitational acceleration exerted on the Earth by the Moon:

$$ a_M = 3.31 \cdot 10^{-5} {m \over s^{2} } $$

From this no conclusion can be drawn about the strength of the tidal acceleration, because the Earth is much further away from the Sun than from the Moon. In order to be able to make such a statement, we compare the tidal accelerations of Sun and Moon:

  • Calculate the gravitational acceleration exerted by the sun on a point on the earth's surface.
  • Subtract the gravitational acceleration exerted by the sun on the center of gravity.
\begin{equation} a_{GS} = \gamma \cdot ( { m_S \over (d_{SE} - r_E)^2 } - { m_S \over d_{SE}^2 } ) = 5.052\cdot10^{-7} {m \over s^{2} } \end{equation} \begin{equation} a_{GM} = \gamma \cdot ( { m_M \over (d_{ME} - r_E)^2 } - { m_M \over d_{ME}^2 } ) = 1.123\cdot10^{-6} {m \over s^{2} } \end{equation} where:
  • \(a_{GS}\) Tidal acceleration caused by Sun
  • \(a_{GM}\) Tidal acceleration caused by Moon
  • \(m_S\) Mass of the Sun
  • \(m_M\) Mass of the Moon
  • \(d_{SE}\) Distance Sun-Earth
  • \(d_{ME}\) Distance Moon-Earth
  • \(\gamma\) Constant of Gravity
  • \(r_E\) Radius of Earth

If one compares the tidal accelerations caused by the Sun and the Moon, one can see that that of the Moon is about 2,223 times as strong as that of the Sun.

\begin{equation} a_{GM} = 2.223 \cdot a_{GS} \end{equation}

The animation below shows the tidal forces of Sun and Moon. Since the Sun cannot be represented meaningfully because of its size and distance, it is symbolized here by its light rays. If you look at the Sun and Moon in isolation, you see four tidal bulges. Two are caused by the Moon (white arrows) and two smaller ones by the Sun (orange arrows). On Earth, the effects of Sun and Moon cannot be observed in isolation, because both accelerations are superimposed. The effect of the Sun overlays the effect of the Moon, either constructively (reinforcing) or destructively (weakening).

The tidal acceleration caused by the Moon (white arrows) and the Sun (orange arrows). The blue ellipse is the shape of the resulting tidal bulge obtained by adding both acceleration vectors.

Adding both acceleration vectors gives the resulting tidal acceleration. In the simulation shown here, there is no visualization in vector form; instead, only the convex hull of the tidal acceleration vectors is shown as a blue ellipse. It symbolizes the sea level.

Neap Tides

Neap Tides occur at half Moon when the Sun, Earth and Moon are positioned at a 90 degree angle. Under these circumstances the gravitational bulges of Moon and Sun overlap destructively because the tidal forces of the Sun are acting against the tidal forces of the Moon. At neap tide difference between high tide and low tide are the least.

At half moon when Sun, Earth and Moon are located at 90 degree angle neap tides occur.

Spring Tides

The opposite of neap tides are called spring tides. Spring tides occur when Sun, Earth and Moon line up. Such an arrangement happens at every full Moon and new Moon phase. Then the gravitational forces of Sun and Moon are forming a large tidal bulge. At spring tide the difference between high tide and low tide are at their maximum.

When Earth, Sun and Moon are located on a line the tidal effects of Sun and Moon add up forming so called spring tides.