BDS Software - Stellar System Thoughts 03

Stellar System Thoughts - Page 3

Now, let's calculate orbital circumferance and rate of rotation in the orbital path for each of the planets.

In the following table:

● PPlanetNum = Orbit Number beginning with closest to Star and ending with furthest from Star.

● PID = Planet Name or ID.

● PDistance = Mean Orbital Distance (Semimajor Axis = Radius of Circular Approximation), AU * 1000 = milli-AU.

● PPeriod = Length of Orbital Year, Earth Days.

● PCircum = Circumferance of Orbit, milli-AU = 2 * π * PDistance.

● PRate = Rotation in orbit, micro-radians per Earth Day = 2000000 * π / PPeriod.

● PIPos = Position in orbit at an arbitrary t = 0, micro-radians (as per some old undocumented data).

PPlanetNum PID PDistance PPeriod PCircum PRate PIPos

1 Mercury 387 87.97 2432 71424 636871

2 Venus 723 224.7 4543 27963 4059461

3 Earth 1000 365.25 6283 17202 4220032

4 Mars 1524 687 9576 9146 3476870

5 Flora 2202 1194 13836 5262 3018547

6 Vesta 2362 1325 14841 4742 720472

7 Iris 2385 1345 14985 4672 2455155

8 Metis 2387 1347 14998 4665 676839

9 Hebe 2426 1380 15243 4553 2687109

10 Astraea 2573 1508 16167 4167 2972470

11 Juno 2672 1595 16789 3939 2128429

12 Ceres 2766 1680 17379 3740 4275708

13 Pallas 2772 1686 17417 3727 34732

14 Hygiea 3138 2031 19717 3094 482758

15 Jupiter 5204 4333 32698 1450 942303

16 Saturn 9582 10759 60205 584 3618940

17 Uranus 19229 30799 120819 204 89884

18 Neptune 30104 60190 189149 104 5780007

19 Pluto 39264 90613 246703 69 4852015

PPlanetNum	PID	PDistance	PPeriod	PCircum	PRate	PIPos
1	Mercury	387	87.97	2432	71424	636871
2	Venus	723	224.7	4543	27963	4059461
3	Earth	1000	365.25	6283	17202	4220032
4	Mars	1524	687	9576	9146	3476870
5	Flora	2202	1194	13836	5262	3018547
6	Vesta	2362	1325	14841	4742	720472
7	Iris	2385	1345	14985	4672	2455155
8	Metis	2387	1347	14998	4665	676839
9	Hebe	2426	1380	15243	4553	2687109
10	Astraea	2573	1508	16167	4167	2972470
11	Juno	2672	1595	16789	3939	2128429
12	Ceres	2766	1680	17379	3740	4275708
13	Pallas	2772	1686	17417	3727	34732
14	Hygiea	3138	2031	19717	3094	482758
15	Jupiter	5204	4333	32698	1450	942303
16	Saturn	9582	10759	60205	584	3618940
17	Uranus	19229	30799	120819	204	89884
18	Neptune	30104	60190	189149	104	5780007
19	Pluto	39264	90613	246703	69	4852015

The minimum orbital position is 0° = 0 micro-radians. This is also identical to 360° = 6283185 micro-radians. So, when a planet's PIPos reaches 6283185 micro-radians, that position is immediately changed to 0 micro-radians. For the Solar System, all rotation is set to be counter-clockwise.

Given the PDistance, PIPos, and PRate data, we can calculate the positions of the planets at any given time t > 0, in polar coordinates (r, θ) as:

r = PDistance, milli-AU

θ = ((PIPos + (PRate * t)) / 2000000), radians

where t is expressed in Earth Days.

And, given the PIUseRadius data from the previous page, and the calculated θ values from this page, we can calculate the equivalent image positions, also in polar coordinates (r_i, θ_i) as:

r_i = PIUseRadius, pixels

θ_i = θ, radians

where "pixels" refers to the side-to-side (not the diagonal) dimension of the screen's pixels.

And, we can then convert these polar coordinates to cartesian coordinates (x_i, y_i) through the usual transformation (cf. mathisfun) as follows:

x_i = (r_i * cos(θ_i)), pixels

y_i = -(r_i * sin(θ_i)), pixels

where the pixel dimensions are relative to the center of the Solar System Image at (320, 320) in actual image coordinates. The unary minus sign in the equation for y_i is provided to accomplish the counterclockwise rotation of the planets.

Also, the calculated coordinates are floating point numbers, and we need integers in order to properly describe the position on the 641 x 641 pixel canvas. We will thus express the actual Solar System image coordinates (x, y) -- (cf. Javascript Rounding) as:

x = (mdjRound(x_i) + 316), pixels

y = (mdjRound(y_i) + 316), pixels

We use 316 instead of 320 (the center of the 641 x 641 pixel canvas) to account for the offset of each planet's upper left corner from its own center -- the planets each occupy a 9 x 9 pixel space.

Applying this information, we can see the planets in their initial positions:

which points out several more difficulties:

● The canvas is way too busy. Between the black grid lines and the red orbits, you can hardly see the planets themselves.

● Several of the asteroids overlap and you can't tell where some of them are because they're obscured by others. Specifically, Metis and Hygiea are obscured by Vesta, Astraea is obscured by Flora, and Hebe is obscured by Iris.

● The attempt to move the planets slightly off of their red orbital lines in order to keep them centered in grid squares has proven to be even uglier than anticipated.

● The inner orbits are too close together.

As I may have mentioned before, I'm no artist. But, I think even I can do better than this.

M.D.J. 2018/07/12

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References:

mathisfun. https://www.mathsisfun.com/polar-cartesian-coordinates.html.