How to calculate when a star, a planet, the moon, or the sun will rise and set*

1
How to calculate when a star, a planet, the moon,
or the sun will rise and set

• If you know its right ascension and declination
  and your latitude and longitude

2
Notation

• a right ascension
• d declination
• f latitude
• aaltitude

3
We make use of spherical trig

• The standard formula for the altitude of an
  object issin(a) sin(d)sin(f) cos(d) cos(f)
  cos(H)
• If a 0 (the object is on horizon, either
  rising or setting), then this equation
  becomescos(H) - tan(f) tan(d)
• This gives the semi-diurnal arc H the time
  between the object crossing the horizon, and
  crossing the meridian.
• Knowing the Right Ascension of the object, and
  its semi-diurnal arc, we can find the Local
  Sidereal Time of meridian transit, and hence
  calculate its rising and setting times.
• One should put in a correction for atmospheric
  refraction, but we will neglect it for the moment

4
East Lansing f 42.7

• At this time of year, when does Arcturus rise?
• Arcturus a 14h 14 min 19D19
• cos(H) - tan(f) tan(d) -(0.923)(0.351)
  -0.324 H 109 degrees
• In time units H 109/15 hours. Arcturus
  will rise 4.7 hours before it reaches the
  meridian and set 7.3 hours after it passes the
  meridian. That is, it will rise 7.3 hours before
  LST 14H 14M and set 7.3 hours later than that
  time.

5
So when is LST 14H14M on Jan. 20?

• LST at local midnight on Jan. 21 is about
• 4 x 2 8 hours
• So LST 14H14M will come at about 613am local
  time or
• 614 037 EST 650 am. And we predict that
  Arcturus will rise at 7.3 hours before that or at
  about 1130pm.
• Weve neglected refraction and some other small
  effects.

6
Lets calculate when the sun sets

• On Jan. 20 the sun is at a 20H08M d-20D10
• cos(H) - tan(f) tan(d) -(0.923)(-0.367)
  0.339 H 70.2
• 70.2/15 4.7 hours. The sun would set at about
  4.7 hours after local noon. Or at about 4H42min
  37 min 519 pm EST
• The actual setting time for that date is 536pm
• How could we make our calculation more accurate?

7
Corrections

• Refraction varies with your location and even
  the weather but is about 34 arcmin at the horizon
• For the sun to set, the top must disappear. Thus
  even without refraction the center of the sun can
  be 15 arcmin below the horizon before the top
  sets
• So in the formula sin(a) sin(d)sin(f) cos(d)
  cos(f) cos(H)
• For a use -34 15 -49 instead of 0. Then
  we get
• -0.014 (-0.345)(0.678) (0.939)(0.735)cos H
  so H 71.4
• or 4 hours 46 min. When we add the 37 minutes
  that brings our setting time to 523 pm , closer
  but still early
• What have we forgotten?

8
The Equation of Time
9
Taking account of the equation of time

• We find that on Jan. 20 the equation of time is
  about -12 minutes. That is, apparent solar time
  is about 12 minutes earlier than mean solar time.
  When we add 12 minutes to our answer we get
• 523 012 535pm
• Pretty close to the standard value

10
Celebrations all around!