What is the cycle for when the maximum amount of the Moon's apparent area is sunlit?

The Moon is defined as full when its ecliptic longitude is 180° distant
from the Sun's.

But what is the cycle for when the maximum proportion of the Moon's
apparent area is sunlit, and for observers at what latitudes of the Earth's
surface is this maximum realised?

Note that the plane of the Moon's orbit around the Earth is inclined at an
angle of 5° to the plane of the Earth's around the Sun. To get a handle on
why this affects the Moon's apparent fullness, imagine if it were 90°.
Given the inclination I am sceptical as to whether the proportion ever
reaches 50%.

On Sat, 11 Jan 2020 20:43:08 -0000 (UTC), Harold Davis
wrote:

The Moon is defined as full when its ecliptic longitude is 180° distant
from the Sun's.

But what is the cycle for when the maximum proportion of the Moon's
apparent area is sunlit, and for observers at what latitudes of the Earth's
surface is this maximum realised?

Note that the plane of the Moon's orbit around the Earth is inclined at an
angle of 5° to the plane of the Earth's around the Sun. To get a handle on
why this affects the Moon's apparent fullness, imagine if it were 90°.
Given the inclination I am sceptical as to whether the proportion ever
reaches 50%.

Since the Sun's diameter exceeds the Moon's, the Sun always
illuminates slightly more than 50% of the Moon's surface. (To
visualize, draw the external common tangents from a larger circle to a
smaller one.) Using 98M miles as the distance, 432K miles as the
Sun's radius, and 1K miles for the Moon's, the Sun illuminates almost
7 miles into the opposite hemisphere.

Any observer a finite distance from the Moon can only ever see less
than 50% of the Moon's surface. The further away the observer is, the
closer the observable percentage is to 50. (To visualize, draw the
two tangents from an external point to a circle.) However, when the
Moon is in the plane of the ecliptic, an observer at the north pole
can see approximately 12 miles over the Moon's north pole into the
opposite hemisphere. So he can see some 5 miles beyond the
terminator. Similar geometry causes his view of the Moon's south pose
to fall short of the terminator.

However, for a person standing in line with the ecliptic, his view in
any direction falls 4 miles short of the terminator and that is the
maximum you seek.

When the Moon is within 4K miles of the ecliptic, an observer on Earth
at he same offset from the ecliptic will have the view described
above.

When the Moon is more than 4K miles away from the ecliptic, all
Earth-bound observers see some non-illuminated portion and therefore
do not see the maximum you want.

Not only is the Moon's orbit tilted with respect to the ecliptic, but
the direction of the tilt wobbles like a child's top. I have no idea
how to factor that into an equation that would compute the time
between one visible maximum and another..

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Barry Schwarz wrote in
:

On Sat, 11 Jan 2020 20:43:08 -0000 (UTC), Harold Davis
wrote:

The Moon is defined as full when its ecliptic longitude is 180°
distant from the Sun's.

But what is the cycle for when the maximum proportion of the Moon's
apparent area is sunlit, and for observers at what latitudes of the
Earth's surface is this maximum realised?

Note that the plane of the Moon's orbit around the Earth is inclined
at an angle of 5° to the plane of the Earth's around the Sun. To get a
handle on why this affects the Moon's apparent fullness, imagine if it
were 90°. Given the inclination I am sceptical as to whether the
proportion ever reaches 50%.

Since the Sun's diameter exceeds the Moon's, the Sun always
illuminates slightly more than 50% of the Moon's surface. (To
visualize, draw the external common tangents from a larger circle to a
smaller one.) Using 98M miles as the distance, 432K miles as the
Sun's radius, and 1K miles for the Moon's, the Sun illuminates almost
7 miles into the opposite hemisphere.

Any observer a finite distance from the Moon can only ever see less
than 50% of the Moon's surface. The further away the observer is, the
closer the observable percentage is to 50. (To visualize, draw the
two tangents from an external point to a circle.) However, when the
Moon is in the plane of the ecliptic, an observer at the north pole
can see approximately 12 miles over the Moon's north pole into the
opposite hemisphere. So he can see some 5 miles beyond the
terminator. Similar geometry causes his view of the Moon's south pose
to fall short of the terminator.

However, for a person standing in line with the ecliptic, his view in
any direction falls 4 miles short of the terminator and that is the
maximum you seek.

When the Moon is within 4K miles of the ecliptic, an observer on Earth
at he same offset from the ecliptic will have the view described
above.

When the Moon is more than 4K miles away from the ecliptic, all
Earth-bound observers see some non-illuminated portion and therefore
do not see the maximum you want.

Not only is the Moon's orbit tilted with respect to the ecliptic, but
the direction of the tilt wobbles like a child's top. I have no idea
how to factor that into an equation that would compute the time
between one visible maximum and another..

Hi Barry,

I am very grateful to you for this detailed reply.

Can you possibly tell me what proportion of the Moon's apparent disc is
sunlit for an observer at Earth latitude theta at the moment of "full"
moon if the Earth-Moon line is inclined at say 2.5 degrees to the plane
of the Earth's orbit around the Sun? (I'm taking 2.5 degrees as an
"average" of 0 degrees and the maximum of 5 degrees.) That would give a
handle on how "full" an "average" full moon is, viewed from Earth
latitude theta. (I'm a bit embarrassed about asking this, because I could
work it out myself, but my geometry is so rusty that it would probably
take me about six hours!)

Thanks again,

Harry

Barry Schwarz wrote in
:

On Sat, 11 Jan 2020 20:43:08 -0000 (UTC), Harold Davis
wrote:

The Moon is defined as full when its ecliptic longitude is 180°
distant from the Sun's.

But what is the cycle for when the maximum proportion of the Moon's
apparent area is sunlit, and for observers at what latitudes of the
Earth's surface is this maximum realised?

Note that the plane of the Moon's orbit around the Earth is inclined
at an angle of 5° to the plane of the Earth's around the Sun. To get a
handle on why this affects the Moon's apparent fullness, imagine if it
were 90°. Given the inclination I am sceptical as to whether the
proportion ever reaches 50%.

Since the Sun's diameter exceeds the Moon's, the Sun always
illuminates slightly more than 50% of the Moon's surface. (To
visualize, draw the external common tangents from a larger circle to a
smaller one.) Using 98M miles as the distance, 432K miles as the
Sun's radius, and 1K miles for the Moon's, the Sun illuminates almost
7 miles into the opposite hemisphere.

Any observer a finite distance from the Moon can only ever see less
than 50% of the Moon's surface. The further away the observer is, the
closer the observable percentage is to 50. (To visualize, draw the
two tangents from an external point to a circle.) However, when the
Moon is in the plane of the ecliptic, an observer at the north pole
can see approximately 12 miles over the Moon's north pole into the
opposite hemisphere. So he can see some 5 miles beyond the
terminator. Similar geometry causes his view of the Moon's south pose
to fall short of the terminator.

However, for a person standing in line with the ecliptic, his view in
any direction falls 4 miles short of the terminator and that is the
maximum you seek.

When the Moon is within 4K miles of the ecliptic, an observer on Earth
at he same offset from the ecliptic will have the view described
above.

When the Moon is more than 4K miles away from the ecliptic, all
Earth-bound observers see some non-illuminated portion and therefore
do not see the maximum you want.

Not only is the Moon's orbit tilted with respect to the ecliptic, but
the direction of the tilt wobbles like a child's top. I have no idea
how to factor that into an equation that would compute the time
between one visible maximum and another..

Barry,

Will it be OK if I copy your post into an answer to the same question
posted at the Stack Exchange astronomy website at

https://tinyurl.com/tze2pwm

?

Harry

On Sun, 12 Jan 2020 14:43:50 -0000 (UTC), Harold Davis
wrote:

Barry Schwarz wrote in
:
Barry,

Will it be OK if I copy your post into an answer to the same question
posted at the Stack Exchange astronomy website at

https://tinyurl.com/tze2pwm

?

Harry

Sure. Maybe somebody there will be able to confirm my calculations.

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On Sun, 12 Jan 2020 14:35:00 -0000 (UTC), Harold Davis
wrote:

Barry Schwarz wrote in
:

On Sat, 11 Jan 2020 20:43:08 -0000 (UTC), Harold Davis
wrote:

The Moon is defined as full when its ecliptic longitude is 180°
distant from the Sun's.

But what is the cycle for when the maximum proportion of the Moon's
apparent area is sunlit, and for observers at what latitudes of the
Earth's surface is this maximum realised?

Note that the plane of the Moon's orbit around the Earth is inclined
at an angle of 5° to the plane of the Earth's around the Sun. To get a
handle on why this affects the Moon's apparent fullness, imagine if it
were 90°. Given the inclination I am sceptical as to whether the
proportion ever reaches 50%.

Since the Sun's diameter exceeds the Moon's, the Sun always
illuminates slightly more than 50% of the Moon's surface. (To
visualize, draw the external common tangents from a larger circle to a
smaller one.) Using 98M miles as the distance, 432K miles as the
Sun's radius, and 1K miles for the Moon's, the Sun illuminates almost
7 miles into the opposite hemisphere.

Any observer a finite distance from the Moon can only ever see less
than 50% of the Moon's surface. The further away the observer is, the
closer the observable percentage is to 50. (To visualize, draw the
two tangents from an external point to a circle.) However, when the
Moon is in the plane of the ecliptic, an observer at the north pole
can see approximately 12 miles over the Moon's north pole into the
opposite hemisphere. So he can see some 5 miles beyond the
terminator. Similar geometry causes his view of the Moon's south pose
to fall short of the terminator.

However, for a person standing in line with the ecliptic, his view in
any direction falls 4 miles short of the terminator and that is the
maximum you seek.

When the Moon is within 4K miles of the ecliptic, an observer on Earth
at he same offset from the ecliptic will have the view described
above.

When the Moon is more than 4K miles away from the ecliptic, all
Earth-bound observers see some non-illuminated portion and therefore
do not see the maximum you want.

Not only is the Moon's orbit tilted with respect to the ecliptic, but
the direction of the tilt wobbles like a child's top. I have no idea
how to factor that into an equation that would compute the time
between one visible maximum and another..

Hi Barry,

I am very grateful to you for this detailed reply.

Can you possibly tell me what proportion of the Moon's apparent disc is
sunlit for an observer at Earth latitude theta at the moment of "full"
moon if the Earth-Moon line is inclined at say 2.5 degrees to the plane
of the Earth's orbit around the Sun? (I'm taking 2.5 degrees as an
"average" of 0 degrees and the maximum of 5 degrees.) That would give a
handle on how "full" an "average" full moon is, viewed from Earth
latitude theta. (I'm a bit embarrassed about asking this, because I could
work it out myself, but my geometry is so rusty that it would probably
take me about six hours!)

Consider a plane that passes through the centers of the Earth, Moon,
and Sun perpendicular to the ecliptic. In that plane, the Sun is
located at (0,0) and is a circle with radius 432e3. The Earth is
located at (98e6,0) and is a circle with radius 4e3. The Moon is
located at (98.25e6,10.9e3) and is a circle with radius 1e3.

The angle from the (center of the) Earth to the (center of the) Moon
is 2.5 degrees. So the angle from the Sun to the Moon is almost .0064
degrees.

Let us define the term boundary to be a line passing through the
center of the Moon that is perpendicular to a particular line of site.
Each of these will divide the Moon into two semicircles.

The first one is the N-S boundary that is perpendicular to the
ecliptic (basically the vertical line x=98.25e6).

The next is the S-M boundary perpendicular to the line from the Sun
to the Moon. Since this line makes an angle of .0064 degrees with the
ecliptic, then the S-M boundary makes the same angle with the N-S
boundary.

The third one is the O-M boundary perpendicular to the line from
the Observer to the Moon. The angle it makes with the N-S boundary
depends on the observer's latitude.

Our observer will also be in this plane at latitude theta.

His distance from the Moon (D) is given by
sqrt(62516e6 - 2e9 * cos(theta))

His elevation (Z) above the ecliptic is
4e3 * sin(theta)

The angle the O-M boundary makes with the N-S boundary is
atan( (10.9e3 - Z) / (250e3 - 4e3 * cos(theta)

From this and the data in my previous message, you should be able to
determine if the observer can view any non-illuminated area beyond the
northern end of the S-M boundary.

But the full answer to your question is much more complicated. For
example, as the observer shifts is gaze to the eastern and western
edges to the Moon's disk, his latitude becomes less and less relevant
and completely irrelevant at due east and west..

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