Firmware Update: 8151
The firmware of the HP Prime has been updated, Firmware
Version 8151 (6/17/2015). The link is
here:
Planetary Orbit and Speed
The program ORBSD calculates approximately:
The distance of a planet (and dwarf planet Pluto) is from
the sun and the speed of the planet, given at any given the number of Earth
days (and partial Earth days) from the perihelion. A planet is at its perihelion is when the
planet is closest to the Sun in its elliptical orbit. Earth is at its perihelion approximately
January 3 to January 5 annually.
Equations
Angle used:
θ = 360° * d/N
d = number of days from perihelion
N = number of days in a planet’s “year”, or number of
days it takes for a planet to make one orbit around the Sun
Distance from Sun (in meters):
r = (a*(1 - ϵ)/(1 + ϵ cos θ ))
a = length of semi-major axis of a planet’s orbit
ϵ = the eccentricity of an orbit (ellipse)
Orbital Speed (in meters/second):
v = √( G * (M_Sun + M_planet) * (2/r – 1/a))
G = Gravitational Constant = 6.63784 * 10^-11 m^3/(s^2
*kg)
M_Sun = Mass of the Sun ≈ 1.9884 * 10^30 kg
M_planet = Mass of the planet
HP Prime ORBSD
EXPORT
ORBSD()
BEGIN
// Orbital distance and
// speed around the Sun
// EWS 2015-06-25
// Initialization
LOCAL lm,pm,la,pa,le,pe;
LOCAL ld,pd;
LOCAL p,sp,θ,d,v,r;
// Planets
sp:={"Mercury","Venus","Earth",
"Mars","Jupiter","Saturn",
"Uranus","Neptune",
"Pluto (Dwarf)"};
// Mass (kg)
lm:={3.29438ᴇ23,4.85749ᴇ24,
5.9722ᴇ24,6.40397ᴇ23,1.89469ᴇ27,
5.67312ᴇ26,8.66437ᴇ25,1.02224ᴇ26,
1.31ᴇ22};
// Semi-Major Axis (m)
la:={57909829824,108209876544,
149594962176,227921734656,
778412012083,1.42673ᴇ12,
2.87097ᴇ12,4.49825ᴇ12,
5.90637ᴇ12};
// Eccentricity (ε)
le:={.206,.007,.017,.093,.048,
.056,.046,.009,.249};
// Days in a year
ld:={87.96899,224.701,365.256,
686.98,4332.58899,10759.22,
30685.4,60189,90465};
// Input
INPUT({{p,sp},d},"Data",
{"Planet:","# Days:"},
{"Planet","# Days after
Perihelion"});
pm:=lm[p];
pa:=la[p];
pe:=le[p];
pd:=ld[p];
θ:=360*d/pd;
HAngle:=1; // degree
// distance
r:=(pa*(1-pe^2))/(1+pe*COS(θ));
// orbit distance
v:=√((1.9884ᴇ30+pm)*6.63784ᴇ−11*
(2/r-1/pa));
// output
PRINT();
PRINT("Distance from the Sun:");
PRINT(STRING(r)+" m");
PRINT("Orbital Speed:");
PRINT(STRING(v)+" m/s");
RETURN {r,v};
END;
Examples
Earth at Perihelion (d = 0):
Distance ≈ 147,051,847,819 m
Orbital Speed ≈ 30,212.81231 m/s
Mars at Aphelion (at about 344 days):
Distance ≈ 249,118,178,096 m
Orbital Speed ≈ 21,921.31299 m/s
Jupiter at Perihelion (0 days):
Distance ≈ 741,048,235,503 m
Orbital Speed ≈ 13,668.77154 m/s
Planetary Data (see sources below)
|
|
Mass (kg)
|
Semi-Major Axis (m)
|
Eccentricity
|
Year (days)
|
|
Mercury
|
3.29438*10^23
|
57,909,829,824
|
.206
|
87.96899
|
|
Venus
|
4.85749*10^24
|
108,209,876,544
|
.007
|
224.701
|
|
Earth
|
5.9722*10^24
|
149,594,962,176
|
.017
|
365.256
|
|
Mars
|
6.40397*10^23
|
227,921,734,656
|
.093
|
686.98
|
|
Jupiter
|
1.89469*10^27
|
778,412,012,083
|
.048
|
4,332.58899
|
|
Saturn
|
5.67312*10^26
|
1.42673*10^12
|
.056
|
10,759.22
|
|
Uranus
|
8.66437*10^25
|
2.87097*10^12
|
.046
|
30,685.4
|
|
Neptune
|
1.02224*10^26
|
4.49825*10^12
|
.009
|
60,189
|
|
Pluto (dwarf planet)
|
1.31*10^22
|
5.90637*10^12
|
.249
|
90,465
|
** approximate values
Sources
Calvert, James B. “Elllipse” http://mysite.du.edu/~jcalvert/math/ellipse.htm 2002. Retrieved
June 20, 2015.
Glover, Thomas J. “Pocket
Ref” Sequoia Publishing, Inc. Littleton,
CO. 2012
“Orbital Speed”
Wikipedia https://en.wikipedia.org/wiki/Orbital_speed Retrieved June 20, 2015
U.S. Navy “Astronomical Constants – The Astronomical Almanac
Online” http://asa.usno.navy.mil/static/files/2015/Astronomical_Constants_2015.pdf
2015.
This blog is property of Edward Shore. 2015.