Question

The time it takes the Moon to return to a given position as seen against the background of fixed stars, \(27.3\) days, is called a sidereal month. The time interval between identical phases of the Moon is called a lunar month. The lunar month is longer than a sidereal month. Why and by how much?

Short answer

The lunar month is longer than the sidereal month by about 2.23 days because the Moon needs to travel more than 360 degrees to show the same phase due to the Earth's movement around the Sun.

Step-by-step solution

Understanding Sidereal Month

A sidereal month is the time it takes for the Moon to return to the same position against the fixed stars. This is basically the Moon's orbit around Earth, which is about 360 degrees and takes 27.3 days.

Understanding Lunar Month

A lunar month, however, is the time it takes for the Moon to return to the same phase, like from full moon to full moon. During one sidereal month, the Earth has moved along its orbit around the Sun, and the Moon needs to travel more than 360 degrees to catch up and return to the same phase.

Comparing the Two

This extra distance that the Moon needs to travel takes about 2.2 extra days, which makes the lunar month longer than the sidereal month. To calculate this precisely, subtract the duration of the sidereal month (27.3 days) from the average length of the lunar month (29.53 days).

Calculation

The exact duration can be computed as follows: \(29.53 - 27.3 = 2.23\) days. Therefore, the lunar month is longer than a sidereal month by about 2.23 days.

Key concepts

Sidereal Month

The concept of a sidereal month is quite fascinating because it relates to how we observe celestial movement from Earth. When we talk about a sidereal month, we're referring to the time it takes for the Moon to complete one full orbit around the Earth, relative to the fixed stars. These are the stars that don't seem to move across the sky, providing a consistent backdrop.
However, while the Moon is orbiting Earth, Earth is also orbiting the Sun. This dual movement means the Moon needs to make a journey of about 360 degrees around Earth to return to its initial point relative to these fixed stars, taking approximately 27.3 days.
This is what defines a sidereal month: the Moon aligning again with the same stars. While this might seem straightforward, it's crucial to understand this is separate from the Moon's phases, which depend on the relative positions of the Sun, Moon, and Earth.

Moon Phases

Understanding the Moon's phases is key to recognizing why a lunar month is longer than a sidereal month. Moon phases are the varying shapes of the Moon visible from Earth as it reflects sunlight at different angles. These phases begin with the new moon, transition into the waxing crescent, first quarter, waxing gibbous, full moon, and so on.

• The cycle of these phases
• is what marks a lunar month

Interestingly, as the Moon completes one sidereal month, Earth will have moved slightly in its orbit around the Sun. Thus, the Moon must travel a bit further than 360 degrees to realign with the Sun and Earth and recreate the same phase (like a full moon).
This requirement for extra travel explains why the lunar month is longer—about 29.53 days—compared to the sidereal month.

Orbital Mechanics

Orbital mechanics offers the framework to understand why lunar and sidereal months differ. Both Earth and the Moon are in constant motion through space, orbiting the Sun and each other, respectively. Earth travels about 1 degree per day along its orbit. Consequently, while the Moon orbits Earth, aligning with the stars every 27.3 days, Earth’s position has also shifted.
Therefore, the additional journey the Moon must undertake to reach the same point in its orbit relative to the Sun is what extends the lunar month.

• Simply put, the Moon moves about an extra 13 degrees
• or a segment of its orbit beyond the sidereal month
• to catch up with Earth's shifted position

This leads to a total of approximately 29.53 days for a complete lunar month. Grasping these movements within the orbital dance of celestial bodies helps illuminate why these two "months"—sidereal and lunar—are not equal in length.