__Question__: I would like to know at what time/Sun angle over a given place on earth

does the shadow of an object becomes the same length of it? How can I

calculate that?

__Answer:__

A
very basic relation (along with some simple observations) allows one to

make the calculation you reference, and also determine one's latitude.

That is:

tan (ALT) = H/ L(m)

where 'ALT' is the maximum altitude of the Sun at the location of the

observer, H is the height of a vertical object (assumed to be planted on

flat ground), and L(m) is the (minimum) length of the shadow

measured.

By example, say H = 0.5 meters, and L(m) is found to be 0.25 meters (which

would occur at or near astronomical NOON or when the Sun is on the

observer's meridian), then:

tan (ALT) = 0.5/ 0.25 = 2

and LAT = tan ^-1(2) (e.g. LAT = arc tan (2) = 63.4 degrees)

[Side note: we can find the observer's LATITUDE from this IF we know the

Sun's declination on the given day. For example, if we are talking about

the day of the winter solstice (Dec. 23), then the Sun's declination angle

d = -23.5 degrees.

Then the latitude of the observer would be:

LAT = 90 + ( d - ALT) = 90 + (-23.5 - 63.4)= 3.1 degrees

That is, the observer is at 3.1 deg North latitude or just above the

equator.

Now, when the shadow of the object

observer's location, one will have:

tan (X) = 0.5/ 0.5 = 1

and X = arc tan (1)= 45 degrees

The difference in angles (ALT - X) =

63.4 - 45 = 18.4 degrees

And this will give a rough idea of the time at the location, say if one

divides by 15 degrees (the number of degrees corresponding to one hour of

time, e.g. the Earth rotates through 15 degrees in one hour).

For this case: 18.4/ 15 = 1.227 or about 1 hour and 14 minutes past

noon. (e.g. 1:14 p.m. local mean time)

make the calculation you reference, and also determine one's latitude.

That is:

tan (ALT) = H/ L(m)

where 'ALT' is the maximum altitude of the Sun at the location of the

observer, H is the height of a vertical object (assumed to be planted on

flat ground), and L(m) is the (minimum) length of the shadow

measured.

By example, say H = 0.5 meters, and L(m) is found to be 0.25 meters (which

would occur at or near astronomical NOON or when the Sun is on the

observer's meridian), then:

tan (ALT) = 0.5/ 0.25 = 2

and LAT = tan ^-1(2) (e.g. LAT = arc tan (2) = 63.4 degrees)

[Side note: we can find the observer's LATITUDE from this IF we know the

Sun's declination on the given day. For example, if we are talking about

the day of the winter solstice (Dec. 23), then the Sun's declination angle

d = -23.5 degrees.

Then the latitude of the observer would be:

LAT = 90 + ( d - ALT) = 90 + (-23.5 - 63.4)= 3.1 degrees

That is, the observer is at 3.1 deg North latitude or just above the

equator.

Now, when the shadow of the object

*equals its height*at the (same)observer's location, one will have:

tan (X) = 0.5/ 0.5 = 1

and X = arc tan (1)= 45 degrees

The difference in angles (ALT - X) =

63.4 - 45 = 18.4 degrees

And this will give a rough idea of the time at the location, say if one

divides by 15 degrees (the number of degrees corresponding to one hour of

time, e.g. the Earth rotates through 15 degrees in one hour).

For this case: 18.4/ 15 = 1.227 or about 1 hour and 14 minutes past

noon. (e.g. 1:14 p.m. local mean time)

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