I would like to know if there is an easy expression to define the path of a point on the Earth (city of Los Angeles is one example) around the Sun?

Question

We are a few hours away from spring equinox. I was thinking if there is a mathematical expression that would define the path of a given point on the Earth, given by longitude and latitude say 34.05, 118.25 Los Angeles, around the Sun. Considering the angle of axis of the Earth, this should look like a stretched slinky or an uneven sinusoidal (because of the superposition of additive and subtractive Earth daily rotation). I guess this path would be on a region of a corrugated cylinder slanted 23.5 to ecliptic plane. I am more interested at an answer at the freshman college level ignoring for example Earth's axial precession. I would imagine there is already expressions or charts available to astronomers, but I am interested more of an elegant, simple solution.

Answer 1

This sketch is an integral part of the answer. Please make sure you have it open.

For the sake of simplicity I have made these assumptions: 1- The orbit of earth around sun is a circle. 2- The projection of Los Angeles (LA) parallel circle onto the ecliptic plane is still circle not ellipse, which is not to far fetched because the minor axis is 92% of major axis and as opposed to planetary orbit its angular velocity is constant with a period of 1/day. 3- The coordinate system has its origin at center of sun and x axis crosses LA on calendar time of spring equinox. 4- rotation of earth is synchronized with annual orbit of earth around sun at a ratio of 365.25 revolutions/year, exactly like the arms of a clock! All of the above assumptions are reasonable within our level of approximation. by inspection of the sketch I have attached it is easily observed the path of city of LA around sun is a meandering course which wobbles around the orbit of earth with adding and subtracting maximum of 4000.cos(34.05) = 3289 miles. This is but small fraction of the ~93000000 miles distance of earth to sun, something in the order of 1/30000. the coordinates of graph are obtained by adding the coordinate of earth in its orbit to that of city of LA on its parallel rotating around the earth axis. $$ X=AU.cos(\theta)-1660.cos(365.25.\theta)$$ and $$ Y=AU.sin(\theta)-1660.sin(365.25.\theta)$$ with 1660 miles being the radius of city of LA's latitude. by checking the sketch you will notice that I have measured the distance from the sun immediately to LA even though its latitude is offset from the center of the earth by about 600 miles, but that is ok because this offset is constant and does not rotate with earth. it's a big gyro. This graph could be refined farther and be given to a computer for plotting. Any help and correction is appreciated.