Measuring the distance to Mars using simple equipment (UPDATE: first estimates of distance!!)

[cubinator]

The Moon's angular size is probably different enough from when I last measured it that it'll cause quite a change in the timings, like lasting consistently longer or shorter than expected. Rather than try to find that difference out from the times, I think I'll use my known telescope parameters and the photo to calculate a new angular size for the Moon. That's pretty much how we did it in the Moon parallax experiment, and it should get me more accurate values for the following four values:

-Moon angular size
-Mars angular size
-Moon distance (given Kepler's laws, this might help characterize the Moon's orbit further )
-Mars/Moon angular rate

I just have to double check which eyepiece I was using, so I'm not guessing. I'm probably going to wait around for the Moon to show up so I can do that.

[cubinator]

I've begun to be able to create different scenarios to sweep through. Here's a plot of a range of possible trajectories for the Moon:

This plot represents a 10x10 grid of possible 'initial' positions (at 0 UTC on Dec. 8th), and for each position the path of the Moon is plotted through space for 100 possible 'inclination' angles between -5 and 5 degrees. 

As you can see, this plot is not comprehensive but I can simply add more resolution to the arrays in order to eventually achieve that. I created this simple one in order to not take too much time to compute and to clearly see whether my loops were functioning so far. Looks like they are working as expected!

[cubinator]

I'm now able to sweep through possible trajectories for Moon and Mars together. Mars' distance from the Moon during the MN occultation is known, but its velocity and orientation are not. So the position of Mars is determined partially by the position assigned to the Moon and partially by a swept parameter. Then its direction of motion is swept as well to determine the position during other times.

This plot shows a range of Moon trajectories (the long lines) and a range of their associated possible Mars trajectories (the short lines, within which each set of trajectories converges at the position Mars is in during the midpoint of the MN occultation, and extends left back in time to the start of the CA occultation and right forward to the end of the UK occultation). Mars' exact orientation relative to the Moon is not known (although I could potentially establish it with a really accurate map of the Moon's surface latitudes and longitudes)) so I sweep over possible orientations 0-180 degrees relative to the Moon's center and the ecliptic plane.

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Here is the same figure, zoomed out to show the variety of initial Moon positions calculated. Each Moon position has a set of Mars positions associated with it.

You can think of this plot as what you'd see if you were at the center of Earth, looking out toward the Moon and Mars against the starry background. It's not exactly what was observed by us, since we are in different spots where the Moon appears in slightly different positions against the stars - for instance, the UK is further north, above the ecliptic plane where this plot is, and so the Moon lines would appear to be shifted downward because of parallax with the Moon. You can imagine the 'big lines' as being in front of the 'small lines'.

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The next phase is to make this simulation check for the "target condition", which has to do with the location of the Moon and Mars at the observation times. For a simulation that matches the real positions of the Moon and Mars, the angular distance between them at each recorded time should be exactly one Moon radius for their respective observers, since these are ingress and egress times. Any other simulation will "miss" the times by some substantial amount. Once I do that, I should be able to check combinations of Moon/Mars trajectories for similarity with the real thing.

[cubinator]

For the same parameter sweep as above, I've calculated the total deviation from the observed situation. The graph is a simple 2D line plot, although it does jump around a lot because it is sweeping over so many variables. The horizontal axis is simply an index number for the location within the 5-layer loop, and the vertical axis is the sum across all 8 observations of the difference between one Moon radius and the distance between the Moon and Mars in the simulation at the observation times. A point on this graph where the error is zero represents a combination of Moon-Mars trajectories that exactly recreates the observation times that were measured in real life (in that the observation times correspond with a time where Mars and the Moon are exactly one Moon radius apart). As you can see on the right, all of the values checked during this run had at least some error, and you can see on the left that this set is far from comprehensive. Now that I know the code works, I can run it for longer across more trajectories, to see if I can find the 'sweet spot' where the error is closest to zero. 

Edited August 16, 2023 by cubinator

[cubinator]

I tried measuring the size of the Moon using a photo, but the result was slightly larger than I think should be possible. I had to look up the apparent field of view of my eyepiece in order to calculate this, and since my eyepiece is very old it might not be the same as its modern counterparts advertised online...I may be able to calibrate my measurements by taking a photo of some stars with known distances, like Mizar and Alcor, using the same setup. 

[FleshJeb]

Would it be easier to check it on a table pointed at some graph paper (I assume the focal length is short enough to allow this)? My stuff is only tangentially optical, but I always found it easier to calibrate back in the shop, rather than under field conditions.

[cubinator]

27 minutes ago, FleshJeb said:

Would it be easier to check it on a table pointed at some graph paper (I assume the focal length is short enough to allow this)? My stuff is only tangentially optical, but I always found it easier to calibrate back in the shop, rather than under field conditions.

I was considering shining a laser through the eyepiece to see where the light falls as another method. I would just have to figure out how to mount it at a fixed distance to something. Graph paper is a really good idea.

[FleshJeb]

6 minutes ago, cubinator said:

I was considering shining a laser through the eyepiece to see where the light falls as another method. I would just have to figure out how to mount it at a fixed distance to something. Graph paper is a really good idea.

Can you pull off the objective lens and use the body of the scope (and the range of the focusing mechanism) as your fixed distance?

I'm just spitballing here--Remember that you're talking to a guy who uses a 3-pound sledgehammer to drive tacks to millimeter tolerance.

[cubinator]

Just now, FleshJeb said:

Can you pull off the objective lens and use the body of the scope (and the range of the focusing mechanism) as your fixed distance?

I'm just spitballing here--Remember that you're talking to a guy who uses a 3-pound sledgehammer to drive tacks to millimeter tolerance.

lol nope, mine is a Schmidt-Cassegrain, so no lens. Looks like this covered in frost:

Spoiler

I'm pretty sure I can just set it sideways on my desk or something, with a piece of paper some measured distance away. Then I can draw marks where the laser hits and quickly measure the angle. I can double check it against Mizar/Alcor or such one of these nights too.

[FleshJeb]

13 minutes ago, cubinator said:

lol nope, mine is a Schmidt-Cassegrain, so no lens. Looks like this covered in frost:

Your tube is quite a bit larger than mine.

(Now I wish I hadn't deleted all my work photos)

Spoiler

 

[cubinator]

I measured the eyepiece apparent FOV using the laser and got 65 degrees. This is worsening my problem by causing the estimate for the Moon's size to be even bigger. I'm not sure what's going wrong in my math:

Telescope focal length = 2030 mm

Eyepiece focal length = 12 mm

Telescope magnification = 2030/12 = 169.17

Eyepiece apparent FOV = 65 deg

Telescope FOV = 65/169.17 = .384 deg

 

FOV radius in image = 1271.5 px

Moon radius in image = 2888.5 px

Moon size/FOV size ratio = 2888.5/1271.5 = 2.27

Moon diameter = 2.27*.384 = .873 deg

 

I think that imaging Mizar and Albireo will give me a better unit for angular distance than trying to figure out the FOV itself from unknown design parameters. I should be able to do that tonight or tomorrow.

[cubinator]

I got a photo of Alcor and the next brightest star in view, so I should be able to get my "second opinion" on angular size of my images.

[cubinator]

Based on the scale of the photo of Alcor and HD 116798 (the third star next to Mizar and Alcor) I seem to be getting an image scale of 9.895343242e-5 deg/pixel based on the positions of the stars on the date I imaged them. That gives me a Moon radius of .28583 degrees, which is still too large. Interestingly, the Mars scale predicted by this is very close to Stellarium's answer, and the eyepiece scale that would be predicted by an apparent FOV of 43 degrees. So I'm still not sure why the Moon seems to be too big. I may have made a mistake in my calculations, or picked points that were not appropriately on the edge of the Moon, so I am going to write a script to do the circle radius calculation automatically based on given points, and try it with multiple sets of points and on multiple images. I'm also going to check to make sure that this really is larger than the Moon can ever possibly appear in the sky from Earth. 

[cubinator]

I also found out about another very easy way to measure the actual FOV directly...

1. Put a star on the edge of the eyepiece
2. Wait for it to go all the way across the center to the other side
3. How long did that take? Earth's rotation rate is 15 degrees per hour.

[darthgently]

33 minutes ago, cubinator said:

I also found out about another very easy way to measure the actual FOV directly...

1. Put a star on the edge of the eyepiece
2. Wait for it to go all the way across the center to the other side
3. How long did that take? Earth's rotation rate is 15 degrees per hour.

Simple, direct, elegant, and as accurate as required.  But you do need to take care the start and end points are directly opposite and you aren't traversing a chord instead of the diameter.  Or just mark the start and end points (using snapshots at start and end) do some math if it isn't close enough to a diameter

...or, start with star in center and stop when it reaches the edge and multiply by 2

[cubinator]

Circle-measuring script confirms the large Moon size I see, at least on the pixel scale. I've got an indexing bug in the error-measuring variables to take my mind off this for now. Once that's taken care of maybe I can play around with the analysis some more.

[cubinator]

This plot shows the deviation from observed conditions for scenarios sweeping over the Moon's 2D position in space, the Moon's direction of motion, Mars' position relative to the Moon, and Mars' direction of motion: That's 5 dimensions.

The error displayed is obtained by measuring the angular distance between Mars and the Moon's center for each observer at their recorded times. This distance then has the Moon's angular radius (as observed) subtracted from it. The new value is added up between all of the observations. A value close to zero means Mars was very close to the edge of the Moon at all recorded times for that scenario.

As you can see, there are not any very close scenarios in the above sweep. I think this is because of some differences in other Moon-related parameters currently kept constant. I have the Earth-Moon distance set at the value it was measured at in September 2018. As we know, the Moon moves around a bit, so it's quite likely it was at a different distance in December 2023. I think the bigger difference, though, is the angular speed of the Moon, which I have set as its average based on the sidereal period. When the Moon is close to Earth it speeds up, and when it is far it slows down, and according to Kepler's second law and orbital values from Wikipedia I calculated that the angular speed at apoapsis is 0.7999 what it is at periapsis. So I really should know how close the Moon was to apoapsis or periapsis in order to know its angular speed.

By playing around with the Moon's distance and angular speed values (specifically, making the Moon more distant and slowing it down), I can reduce the amount of error. The two horizontal lines are where I think I should be in order to start seeing parallax with Mars.

There are a couple ways I could go about this. One is to increase the number of dimensions to sweep from 5 to 7, including the Moon's distance and angular speed. This would increase the computation time and possibly give me multiple combinations of observation-matching results, which may give the distance to Mars but muddle the actual distance to the Moon. The other thing I could do is measure the Moon's apoapsis and periapsis myself by photographing the Moon over the course of a month and measuring the angular size and position in the sky. This would give me a more complete measurement of the Moon's orbit and let me get a better idea of its state during the occultation. Given that it is currently cloudy, I'm going to see about getting a start on Option 1.

[cubinator]

Also, as a heads-up: There will be another occultation of Mars coming up on Sept. 16th! This one will happen during the daytime, and the Moon will only be a thin crescent, so if you're planning to observe this one, be VERY CAREFUL as the Sun will only be about 20 degrees away! If you accidentally look at the Sun through any unprotected magnifying equipment like binoculars or a telescope your eye tissue will come out of it looking like a fried egg! I'd recommend setting up such that you and all your equipment are in the shadow of a large building or structure to your west, so that the Sun only moves deeper into the shadow and you have no chance of pointing at it.

This one's probably not so useful for making parallax measurements, as Mars is so much farther away, but it's an interesting chance to get to see a planet during the daytime. I'll probably set up if it's clear and maybe grab an angular size measurement if I can, but besides that I'll just sit back and enjoy the show!

[cubinator]

This is getting very exciting! I introduced Moon's distance and angular speed to the sweep, and have been repeatedly running it with tweaked ranges of values. I can minimize the upper limit of the error I see, but the lower limit of the error seems to keep hovering at about 0.065 degrees, no matter what I do!

What I expect from this script is that the error between the two most distant locations will be around the angular diameter of Mars (which is a little smaller than the distance between the horizontal lines and zero). Since I'm plotting the sum of all the errors, it makes sense that the plateau might be around 8 times that size.

I still need to do some work to try to find the best minimization of the individual observations rather than the sum, because some of these combinations of errors don't make much sense. However, it's good enough that I have numbers that I can start to crudely mash together:

I take the physical distance between CA at its t2 and UK at its t4 as a baseline...(8931.7 km)

I subtract the angular error between t4UK and t2CA...(0.0103 deg for the point on the left above and 0.00518 deg for the point on the right)

and now I have Mars appearing in a different place in the sky between CA and UK. Divide baseline by sine of the error and...

 

The point on the left says Mars is 49,786,510 km away, and the point on the right says Mars is 98,801,991 km away! Neither of these are quite correct, but the true value (82 million km, "what NASA wants you to think", etc.) is between these two, which is extremely promising!

Edited September 9, 2023 by cubinator

[cubinator]

I'm going to leave the code running at a high resolution overnight and see if it doesn't grind to a crawl. In the likely case that this doesn't magically produce all the answers, I will need to find a way to sort out the good tries from the bad.

[cubinator]

Update: It crawled to a halt as I expected it would. This is how far it got:

It got through about 2% of one value of Moon distance. You can see it sort of shallow out a bit, but it's not enough to know if there's a better estimate elsewhere. I'll see what this data says. As you can see, there is a lot of calculation effort that is not ultimately needed.

[cubinator]

The issue I'm having with the errors is that they are out of order. For instance, at the rightmost low point on the latest graph, the errors are, in rad/1000:

CA t2 -.2234    CA t4 .0608    MN t1 .1004    MN t2 .0212     MN t3 -.0044     MN t4 .0553    UK t1 .739    UK t4 -.6674

Some are positive, meaning the event is predicted later than observed, and some are negative, meaning the event is predicted earlier. Notice how CA t2 is negative and CA t4 is positive. t2 looks like this:

And t4 looks like this:

So if we are measuring the Moon-Mars center distance at those times, t2 will be smaller than 1 Moon radius (negative error) and t4 will be greater (positive error). Mars' angular size was estimated earlier to be around 20 arcseconds, which is around 0.1 rad/1000, close to the size of this difference in error. I need to implement a better estimate of Mars' angular size that takes into account the offset path behind the Moon and the variable Moon angular rate, and use that to estimate the error. I also need to make a code that, instead of saving every worthless value, only saves the 'good' sets of parameters as it's sweeping through. These changes should make it easier for me to find good estimates of everything, AND easier to make visualizations of the event - like the MS Paint drawings above, but with actual data!

If I calculate the occultation path for each location, I might also be able to constrain the estimate to cases where Mars has the same angular size between all locations. Hmm, maybe I should look back at trying to find an analytical solution...

Edited September 30, 2023 by cubinator

[cubinator]

I've reorganized my code to be a lot neater in general, and generate more useful information.

Instead of storing every possible scenario, it runs the general occultation calculation in a function, and checks each time whether the error is the smallest it's seen. Only then does it append the data onto an array of scenarios. So I can run through 1 million combinations of parameters while only storing (and having to look at) 1400 progressively-smaller ones. 

This should allow me to run the calculation at a high resolution without worry of slowing MATLAB to a halt or generating gigabytes of garbage data. I've also added a couple new calculated parameters that should allow me to write an intuitive visualization of the occultation from all three locations that clearly shows the error in the calculation, and eventually could let you see the parallax with your own eyes! - if my math and our observational accuracy is good enough.

[cubinator]

Here's my summed error result, in radians, for the first run of this code. I ran through 6.8 million scenarios out of 10 million I'd originally set it to go through. 1414 'interesting' scenarios were stored.

I'm not sure why the error is able to jump up occasionally, but the trend is how I'd like it to be. Using the lowest point on this graph, and t2CA and t1UK as baseline yields a distance of 32 million km - this indicates a larger error than expected overall. Smaller error means smaller difference between Mars' position in the sky for different observers, which is caused by greater distance.

So I'm not running into my fear of getting an error that's somehow too small yet. And I haven't taken into account Mars' angled entry and exit behind the Moon yet. So I think I'm in a good position so far.

Edited October 1, 2023 by cubinator