Here's the simple formula for working out what proportion of the day an object will remain above the horizon.

Proportion = arccos( tan L * tan d ) / 180

where L is your latitude and d is the declination of the object and all angles are measured in degrees.

Naturally this will generate some imaginary results. Those are the circumpolar objects.

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Thanks 777 geek for a simple formula. You know this, but for the sake of less experienced readers, this gives values for a point, very slow moving (indistinguishable from stationary over the course of the day) source (a star other than the Sun) above the theoretical horizon (not trees and nearby houses and mountains), and makes no correction for atmospheric bending of light near the horizon.

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Forming opinions as we speak

It could be more useful in the 'Astronomical Observing...' section, IMO.

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"Shut up and calculate" R. Feynman

The above given formula requires a correction. A negative sign needs to be placed in front of any one of the symbols between the parentheses.

Below is a version of the formula that also accounts for atmospheric refraction and lunar parallax. Note the positive and negative signs. The result refers to the center of a star or planet, or the upper limb of the Sun or Moon.

H = -0.0099 for stars and planets
H = -0.0145 for the Sun
H = +0.0022 for the Moon

Proportion = arccos ( ( H – sin L * sin D) / ( cos L * cos D) ) / 180

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Curt Renz - "Centaur"
For monthly astronomical calendar visit:
www.CurtRenz.com/astronomical.html

I prefer the expression:

cos(LHA)= (sin(alt) - (sin(lat)*sin(dec)))/(cos(lat)*cos(dec))

where,

LHA is local hour angle
lat is latitude
dec is declination
alt is altitude.

Use -34 minutes of arc for alt for the stars and planets; -50, the sun; +7.5, the moon.

The -34 minute value is for atmospheric refraction, and additional -16 minutes for radius of the sun and moon, and for the moon an average parallax correction of +57.5 added to -50 so that the geocentric coordinates can be used, if topocentric coordinates are used for the moon, retain -50 minutes.

The main advantage to me of making altitude an explicit variable is that I can then note the altitudes of my obstructed horizon; say the minimum altitude is 10 degrees and the maximum altitude is 20 degrees. I use the maximum value, 20 degrees, if I want to be sure that the object has cleared all obstructions. I use the minimum value, 10 degrees, if I want know the earliest possible time that I can see the object rising or the latest time that I can see the object setting. While this method only gives a range of rising or setting times, many times that is enough for me to make a decision as to whether an event is visible or not.

To convert this to a proportion, as stated in the initial formula, divide the LHA by the angular measure of a half circle; ie, if your calculator/computer is working in degrees divide by 180; in radians, pi.

Note that if the right side of the formula is greater than 1 or less than -1, the inverse cosine will be complex; though on many machines an error will be thrown. The physical meaning of this is that the star or planet is circumpolar or anti-circumpolar [anti-circumpolar: that it never rises above the horizon] with regard to the altitude that one has given; but this case should be checked for in the program -- On one programmable calculator, I failed to do so and wondered why the <bleep> that I was getting complex values for the time of sunrise and sunset; an explicit test is always more desirable. (Indeed, it was in tracing the values that that program code gave that I first found out that the trig functions were defined for complex numbers.)

The above given formula is not quite correct. One of the middle parentheses should be placed at the right end. The H values seen in my post are actually the sines of the standard altitudes that you have given. Otherwise, our algorithms are essentially identical.

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Curt Renz - "Centaur"
For monthly astronomical calendar visit:
www.CurtRenz.com/astronomical.html

Check that again; according to my sources the subtraction should be performed before the division -- though, since I do not know how to derive the formula from first principles, I can't say that my sources aren't wrong. Indeed, if one simply substitutes sin(alt) for H in your formula one gets the same formula including order of operations until you divide by 180 to convert LHA to a proportion.

Indeed the subtraction should be performed before the division, as was the case in my formula. My apologies to you; I found a pair of unnecessary parentheses in your formula to be confusing. Upon closer examination I see that your use of parentheses was not really incorrect but overdone. After one pair of parentheses is eliminated, your formula would simply read: cos(LHA)= (sin(alt) - sin(lat)*sin(dec))/(cos(lat)*cos(dec))

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Curt Renz - "Centaur"
For monthly astronomical calendar visit:
www.CurtRenz.com/astronomical.html

Actually, in my programs I have omitted the pair of parentheses that you considered confusing; I simply decided not to assume that all readers who might attempt to use this formula would be as familiar with the default mathematical precedence as I.