I studied the equation of time recently and understand the Earths keplerian orbit is responsible for it. I understand the keplerian orbit described the Earths elliptical path and it was this alone that created the equation of time.
I have just read that the Earth does not travel at a constant velocity during its orbit of the Sun. I assume this will also contribute to the equation of time.
Please can someone clarify this siuation.
Thanks
Mike
In an orbital ellipse, the velocity varies in orbit. Remember that the force of gravity varies as the radius changes, and the centripety force (mv^2/r) has to change with it. The total Potential energy of position plus the total Kinetic energy is constant while in orbit, but in each position one decreases, one increases.
Bob, Thank you for your reply which I have to admit I do not fully understand. Bit over my head but will study it further. Would it be fair for me to conclude for the time being that both the elliptical orbit as well as the varying velocity of the earth combine to create the equation of time?
Thanks
Mike
yes.
Thank you.
The earth travels in an elliptical shaped orbit around the Sun.
An ellipse is the locus of a point, P, moving in such a way that the sum of its distances from two fixed points, F and F', called foci, is a constant. Draw yourself two perpendicular axes, X and Y with origin O. Locate two points, the foci, on the X axis, equidistant on either side of the origin O, say 1 inch, if you want to draw your own picture. Label the points F on the left and F' on the right. Each represents a focus of the ellipse. Now locate two points, on the X axis, 2 inches on either side of the origin. Label these A on the left and B on the right. AB is the major axis of our ellipse. With a compass, using AO as a radius and F as the center, swing an arc that intersects the +Y axis at C and the -Y axis at D. CD represents the minor axis of our ellipse. Let AO = a, CO = b, and FO = c, the typical terms used to represent the major and minor axes of an ellipse and the distance to the focus from the center O. Note that CF also equals "a" since we used radius AO to locate point B. (As you can see, from the pythagorean theorem, a^2 = b^2 + c^2.) Therefore if you are given the major and minor axes of an ellipse, you can determine the location of the focus points from c = sqrt(a^2 - b^2) where a = semi-major axis and b = semi-minor axis. The eccentricity of the ellipse is e = c/a = <1. The terms a, b, and c are also related by b^2 = a^2(1 -e^2) or e^2 = 1 - b^2/a^2.
That said, an elliptical orbit is the elliptical shaped path of a body, or satellite, around another body having a gravitational field. As Kepler's First Law states, the orbits of all the planets in our solar system, as well as the orbits of all Earth satellites, are ellipses. In the case of the planets, the Sun is the focus of the planetary orbits. In the case of Earth satellites, the Earth is the focus. Geometrically, an ellipse is the locus of a point moving in such a way that the sum of its distances from two fixed points, f and f', called foci, is a constant. Pictorially, draw yourself two perpendicular axes, X and Y, with origin O. Locate the two foci points, on the X axis, equidistant on either side of the origin O, say 1 inch for example, labeling the points f on the right and f' on the left. Each represents a focus of the ellipse. Now locate two points, on the X axis, 2 inches on either side of the origin O. Label these A on the left and B on the right. AB is the major axis of our ellipse. With a compass, using AO as a radius and f as the center, swing an arc that intersects the +Y axis at C and the -Y axis at D. CD represents the minor axis of our ellipse. Note that fB equals AO since we used radius AO to locate point B. (As you can see, from the pythagorean theorem, (fB)^2 = (fO)^2 + (OB)^2.) Therefore if you are given the major and minor axes of an ellipse, you can determine the location of the focus points from fO = sqrt[(fB)a^2 - (OB)^2] where AO = semi-major axis and OB = semi-minor axis. The eccentricity of the ellipse is e = fO/AO = <1. The terms a, b, and c are also related by b^2 = a^2(1 -e^2) or e^2 = 1 - b^2/a^2.
Now, unfortunately, we must change our terminology and parameter designations for use in the area of our orbital description. For our elliptical orbit, and referring to the picture created above, the basic terminology becomes AO = a = the semi-major axis, OB = b = the semi-minor axis, f and f' are the two foci, fO = c = the distance from the foci to the center of the ellipse, e = the eccentricity = c/a, p = a geometric constant of the ellipse called the parameter, sometimes referred to as the semi-latus rectum, and determined from p = a(1 - e^2), point B is now designated as p, the periapsis point, the closest the elliptical path comes to the focus f (not to be confused with p the parameter), point A is now designated as a, the apoapsis point, the farthest the elliptical path extends from the focus f (not to be confused with a the semi-major axis).
The polar equation of our elliptical orbit is defined as r = p/[1 + e(cos(v))] where r is the radius from the focus f to a point on the ellipse, p is our geometric constant, as defined above, e is the eccentricity, and v is the angle (true anomaly) between the radius r and the major axis of the ellipse, with the periapsis end being zero, and measured positively in the direction of flight, counterclockwise. If you know the semi-major and semi-minor axes and the eccentricity, the periapsis distance is calculated from r(p) = p/[1 + e(cos(0))] = p/(1 + e) = a(1 - e). Similarly, the apoapsis distance is calculated from r(a) = p/[1 + e(cos(180))] = p/(1 - e) = a(1 + e). It becomes clear that r(p) + r(a) = 2a and r(a) - r(p) = 2c. These expressions may be combined to yield e = [r(a) - r(p)]/[r(a) + r(p)].
The velocity of the orbiting body at the periapsis is defined by
______________
/ (2µ) r(a)
Vp = \/ --------------------- in feet per second
r(p) [r(a) + r(p)]
The velocity of the orbiting body at the apoapsis is defined by
______________
/ (2µ) r(p)
Va = \/ -------------------- in feet per second
r(a) [r(a) + r(p)]
The velocity at any point on the orbit is given by
_____________
V = \ / 2µ/r - µ/a in feet per second
______
The orbital period is given by T = 2(Pi) / a^3
\/ ------ in seconds
µ
Hello Mike! You're correct that the equation of time is influenced by the Earth's elliptical orbit around the Sun. And yes, the Earth's varying velocity during its orbit also contributes to the equation of time.
To clarify the situation, let's go through how the Earth's orbit affects the equation of time:
1. Elliptical orbit: The Earth's orbit around the Sun is not a perfect circle but an elliptical shape. This means that the distance between the Earth and the Sun varies throughout the year. During the perihelion (when the Earth is closest to the Sun) in early January, the Earth's orbital velocity is slightly faster. Conversely, during the aphelion (when the Earth is farthest from the Sun) in early July, the Earth's orbital velocity is slightly slower.
2. Equal area in equal time: Kepler's second law of planetary motion states that a line joining a planet and the Sun sweeps out an equal area in equal periods of time. When the Earth is closer to the Sun (perihelion), it covers more area in a given time, resulting in an increased angular velocity. Conversely, when the Earth is farther from the Sun (aphelion), it covers less area in the same time, leading to a decreased angular velocity.
3. Combined effects: The combination of the elliptical orbit and the equal area law causes variations in the Earth's apparent solar time compared to mean solar time. Apparent solar time is based on the position of the Sun in the sky, while mean solar time is an average time value. The equation of time is the difference between apparent solar time and mean solar time, resulting from the varying velocities as the Earth moves through its elliptical orbit.
So, both the elliptical shape of the Earth's orbit and its varying velocity contribute to the equation of time, causing variations in how the Sun appears to move across the sky throughout the year. The equation of time is an essential factor to consider when calibrating sundials, determining accurate solar positions, or for astronomical calculations involving precise timing.
I hope this clarifies the situation for you! Let me know if there's anything else I can help you with.