Thursday, February 19, 2015

HP Prime & TI-84+: Equation of Time (One Way of Calculating It)

Introduction

We normally associate a day that lasts 24 hours long.  However, the true solar is day not constant and varies during the time of the year.  The true solar day is defined as two successive transits through a given point.  For example, think of it as the time it would take for the sun to make a round trip from the highest point in the sky only to return to the same point again. 

What causes the true solar day to not be constant?  The sun moves along the ecliptic and the speed about the ecliptic isn’t constant.  Earth’s orbit around the sun is not a perfect but it is an elliptical orbit.  The speed of the orbit is greatest when the Earth is nearest to the Sun (about January 3), and the slowest when the Earth is furthest away from the sun (about July 3). 

The equation of time describes the difference in time (seconds, minutes, or hours) between the true solar time and time as we normally know it (a day takes 24 hours).  If we used a sundial to measure time and compare it against a mechanical watch, the equation of time would demonstrate the approximate difference. 

For the program and the equation presented in this blog entry, if the result is positive, that means the watch is “slow” compared to the true solar time.  If the result is negative, that means that the watch is “fast” compared to the true solar time. 

I am curious to see if the clock apps on our smart phones follow the mechanical watches or true solar time. 

The equation presented in the program EQT was developed by GS Campbell and JM Norman (An Introduction to Environmental Biophyiscs).  This is one of many ways the equation of time is calculated.  A simple internet research and research through books and articles will show many forms of the equation of time. 

The angle measurement used in this equation is radians.

EQT (in hours) =3600^-1*(−104.7*SIN(F)+596.2*SIN(2*F)+4.3*SIN(3*F)-12.7*SIN(4*F)-
429.3*COS(F)-2*COS(2*F)+19.3*COS(3*F))

where F=π/180*(279.5+360/365*n);

A 365 day year assumed. 


Note:  Please keep in mind that every equation of time is an approximation.

Equation of Time (HP Prime)
HP Prime:  EQT

EXPORT EQT(n)
BEGIN
// n = day number (1 to 365)
// 2015-02-18
// hours, UC Berkeley approx
LOCAL F,E;
// radian mode
HAngle:=0;
F:=π/180*(279.5+360/365*n);
E:=3600^-1*(−104.7*SIN(F)+596.2*SIN(2*F)
+4.3*SIN(3*F)-12.7*SIN(4*F)-
429.3*COS(F)-2*COS(2*F)+19.3*COS(3*F));
RETURN E;
END;


TI-84+:  EQT

Disp “N=DAY NO.”
Prompt N
Radian
π/180*(279.5+360/365*N)→F
3600^-1*(−104.7*SIN(F)+596.2*SIN(2*F)
+4.3*SIN(3*F)-12.7*SIN(4*F)-
429.3*COS(F)-2*COS(2*F)+19.3*COS(3*F))→E
Disp “HOURS:”, E


Resources:

Baldocchi, Dennis.  “Lecture 7, Solar Radiation, Part 3, Earth-Sun Geometry”  September 10, 2012.  Retrieved February 17, 2015.  URL:  http://nature.berkeley.edu/biometlab/espm129/notes/Lecture%207%20Solar%20Radiation%20Part%203%20Earth%20Sun%20Geometry%20notes.pdf

Meeus, Jean.  “Mathematical Astronomy Morsels”  2nd Ed.  Willmann-Bell Inc.:  Richmond, VA  2000 pp. 337-346


This blog is property of Edward Shore – 2015.


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