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

[PakledHostage]

On 1/2/2023 at 9:47 AM, cubinator said:

Assuming the Moon's orbital period is 2360592 seconds, and that while Mars is rising and setting its own motion through the sky is negligible, 

the Moon's angular velocity is w = 2*pi/T, and Mars' angular size is w*time to rise or set.

 

What happens if you take into account that the moon's orbit is elliptical? It's angular velocity isn't constant throughout the month, how significant is the effect on the results?

[FleshJeb]

On 1/2/2023 at 9:47 AM, cubinator said:

while Mars is rising and setting its own motion through the sky is negligible,

First off, thank you all for doing this project--It's really inspiring.

I love picking apart methodology and finding sources of error:

The refraction problem: As a surveyor, I'm reasonably familiar with the effects of refraction on angular measurement. If this experiment relied on measuring right ascension and declination, yes I think it would be quite significant. (Each pair of observations are about an hour apart, so the sky position will have varied about 15 degrees, leading to different refraction corrections.) However, we're just looking at relative times, so it's not an issue.

The moon topographic problem: Could be an issue, I need to quantify it better. I stared at a picture of the Moon's topography (I'm a professional!) and I think the worst-case scenario might be a difference of 4km. I don't know where you all saw Mars rise and set over the Moon, but the blue crater on the far right side (nearside, obviously) has a pretty big jump on the southern edge:

Spoiler

However, I think the problem is the assumption I quoted:

From https://en.wikipedia.org/wiki/Orbit_of_the_Moon

"The mean angular movement relative to an imaginary observer at the Earth–Moon barycentre is 13.176° per day to the east"

(We'll ignore the fact that it's the barycentre.) 13.176 deg/day * 1 day/86400 sec = 0.00015250 deg/s = 0.54900 arcseconds/sec. Over a 40-ish second observation, the Moon will have moved about 22 arcseconds, and that will definitely cause the times for Mars to rise and set to be different, as it's "pushing" into one, and "pulling" out of the other.

I'm feeling a little brain-dead at the moment, so I'm sorry for not looking at this in more thoroughly, but it appears to be quite significant.

[cubinator]

8 hours ago, PakledHostage said:

What happens if you take into account that the moon's orbit is elliptical? It's angular velocity isn't constant throughout the month, how significant is the effect on the results?

I figured that the Moon's angular velocity wouldn't change by as much as multiple percents in an hour or a few, as it only moved a couple degrees during the whole thing.

I will see if Earth's rotation is enough to create such a difference - the observer is moving in a different direction after an hour, maybe that is not so insignificant.

Edited January 4, 2023 by cubinator

[cubinator]

I did photograph Mars and the Moon before and after the event, so I can verify the topography if needed, but I have been seeing if I can get by without using photos so as to make the determination using even simpler data.

[cubinator]

I calculated the velocity vector at Mars-set of Minnesota based on the vector between my position at t1 and t2, and the velocity vector at Mars-rise using the vector between my position at t3 and t4. 

Then I found the angle between the components of those vectors that are parallel to the YZ plane, because the Moon's motion is assumed to be 0 in the X direction (which is the direction from Earth pointing away from the sun). The angle of Minnesota's motion in that plane changed by 0.15769 radians, and the cosine of that angle is 0.9876.

So, the difference in the observed velocity of the Moon due to Earth's rotation between the observations should be about 1.1%. The observed difference was about 20%.

13 hours ago, FleshJeb said:

(We'll ignore the fact that it's the barycentre.) 13.176 deg/day * 1 day/86400 sec = 0.00015250 deg/s = 0.54900 arcseconds/sec. Over a 40-ish second observation, the Moon will have moved about 22 arcseconds, and that will definitely cause the times for Mars to rise and set to be different, as it's "pushing" into one, and "pulling" out of the other.

I'm not yet sure why you think this would cause the rise and set times to be different - I would think that Mars should come out from the Moon the same way it went in, because the angular velocity of the Moon and Mars are basically constant.

[cubinator]

I think that if it clipped something on the outskirts of Mare Orientale it could have gotten 3 km or so, I'm going to try to figure out if the angle Mars enters the Moon at amplifies that effect.

[cubinator]

If Mars went behind the Moon at a 45 degree angle to the surface, and there was a 3 km dip between the first contact and second contact, then the additional time due to that dip would be 3*sqrt(2) = 4.25 seconds. 

If Mars went in at a shallow angle of 30 degrees, the effect of the dip would be 3 / cos(60*) = 6 seconds. Thus, a shallow entry angle amplifies the time difference due to lunar terrain greatly, enough to potentially account for a large part of the observed difference. A shallower angle also would cause Mars to pass over a longer stretch of the landscape, so shallower slopes would be able to make a difference as long as they have the altitude change overall.

Edited January 4, 2023 by cubinator

[cubinator]

I can find roughly what the angle is based on the time spent behind the Moon and the known angular velocity and size of the Moon. I still need to account for Mars' angular velocity later, but it's enough to have a rough estimate of the amplitude of the effect of lunar mountains.

Edited January 4, 2023 by cubinator

[FleshJeb]

42 minutes ago, cubinator said:

I'm not yet sure why you think this would cause the rise and set times to be different - I would think that Mars should come out from the Moon the same way it went in, because the angular velocity of the Moon and Mars are basically constant.

You're right, I was reconstructing the experiment in my head late at night and somehow didn't realize it's the motion of the moon (plus observer velocity) CAUSING the rise and set. Which is why my number is approximately the same as your rise/set angles. So the source of my cognitive error is that my skull is an oblate spheroid.

I understand the how the terrain heights and entry angle to the moon are having an effect, but I'm not sure what you're doing with the quantification--The units aren't working out (km rather than seconds), but I'm assuming you have some unwritten steps here.

(I'm including the below for the benefit of readers who are less familiar with the science of measurement--I'm sure this is something you already know.)
Generally speaking, the terrain issue is a "noisy/non-systematic error" and going to be difficult or impossible to quantify in a conclusive way. It's analogous to me looking down 500 feet of road on a hot day and trying to get an accurate angular measurement through heat shimmer/distortion. I can "eyeball average" the sight picture and take a guess at the error, but I'm better off not relying on it in my primary analysis and treating that measurement as corroborative rather than conclusive. The fun of doing science on a budget!

Hmm, I thought the observer motion would be the source of error, but it doesn't appear so from your analysis. I'm at the point where I need to draw a diagram of the the whole experiment to make sure I'm visualizing it correctly, but I don't have access to AutoCAD at the moment. (I'm actually procrastinating on quitting my job today.)

[K^2]

9 hours ago, cubinator said:

I will see if Earth's rotation is enough to create such a difference - the observer is moving in a different direction after an hour, maybe that is not so insignificant.

The Earth-Moon / Moon-Mars is a significant lever. I wouldn't be surprised. The latitude difference isn't great but I was taking measurements close to horizon, so the difference in apparent motion of Mars relative to Moon due to Earth's rotation would change a lot more for me between the two measurements than for your measurements, taken much higher in the sky, and therefore, with the apparent motion being close to maximum on both ends.

You might have to bite the bullet and do a numerical integration of the occlusion duration by honestly computing both the Mars' and the Moon's apparent positions in the sky over some small time step intervals.

I don't think the topography of the Moon will matter, since it's a 1 part in a thousand in the worst case, and I don't think we have that sort of the precision in measurement. Treat all three bodies, Earth, Moon, and Mars, as perfect spheres of the mean radius and just do the position computations as precisely as possible, and I think that's the best chance to get useful results out of this.

[cubinator]

23 minutes ago, K^2 said:

The Earth-Moon / Moon-Mars is a significant lever. I wouldn't be surprised. The latitude difference isn't great but I was taking measurements close to horizon, so the difference in apparent motion of Mars relative to Moon due to Earth's rotation would change a lot more for me between the two measurements than for your measurements, taken much higher in the sky, and therefore, with the apparent motion being close to maximum on both ends.

You might have to bite the bullet and do a numerical integration of the occlusion duration by honestly computing both the Mars' and the Moon's apparent positions in the sky over some small time step intervals.

I don't think the topography of the Moon will matter, since it's a 1 part in a thousand in the worst case, and I don't think we have that sort of the precision in measurement. Treat all three bodies, Earth, Moon, and Mars, as perfect spheres of the mean radius and just do the position computations as precisely as possible, and I think that's the best chance to get useful results out of this.

That's basically what I'm doing, the problem is I need the Moon's position and velocity rather exactly. Although I'm not to the point of doing numerical integration, since I already know where the few really important moments are. It should take care of the Earthly rotational motion problem, anyway.

53 minutes ago, FleshJeb said:

You're right, I was reconstructing the experiment in my head late at night and somehow didn't realize it's the motion of the moon (plus observer velocity) CAUSING the rise and set. Which is why my number is approximately the same as your rise/set angles. So the source of my cognitive error is that my skull is an oblate spheroid.

I understand the how the terrain heights and entry angle to the moon are having an effect, but I'm not sure what you're doing with the quantification--The units aren't working out (km rather than seconds), but I'm assuming you have some unwritten steps here.

The unwritten step is that the Moon moves at 1 km/s, so every 1 km of muntain goes by in 1 second, making the units of km and s interchangeable in this setting.

53 minutes ago, FleshJeb said:

(I'm including the below for the benefit of readers who are less familiar with the science of measurement--I'm sure this is something you already know.)
Generally speaking, the terrain issue is a "noisy/non-systematic error" and going to be difficult or impossible to quantify in a conclusive way. It's analogous to me looking down 500 feet of road on a hot day and trying to get an accurate angular measurement through heat shimmer/distortion. I can "eyeball average" the sight picture and take a guess at the error, but I'm better off not relying on it in my primary analysis and treating that measurement as corroborative rather than conclusive. The fun of doing science on a budget!

Hmm, I thought the observer motion would be the source of error, but it doesn't appear so from your analysis. I'm at the point where I need to draw a diagram of the the whole experiment to make sure I'm visualizing it correctly, but I don't have access to AutoCAD at the moment. (I'm actually procrastinating on quitting my job today.)

I have drawn it out in MATLAB and on paper, but if you're anything like me you'll probably spend a lot of time finding the nearest roughly spherical object in your house and rotating it around in your hand while staring at it. Bonus points if you shine lights at it. Let me find my drawing, I think I have a nice-ish one.

Here it is:

I have drawn in the Earth's axis, the Moon's velocity, and the three locations of observation. You can imagine the Moon moving across the face of Earth while Earth slowly rotates. This is what it looked like from Mars, except it was mostly dark because the Sun was shining on Earth from behind.

Edited January 4, 2023 by cubinator

[K^2]

1 hour ago, cubinator said:

That's basically what I'm doing, the problem is I need the Moon's position and velocity rather exactly.

You should have the orbital elements, right? I know it's a chore to track down all the correct datums, and transform the coordinates, but there shouldn't be anything particularly complex there. Just tedious.

To simplify some of the math, I really don't think you need to worry about radial motion of the planets, so if you simply take the Earth's polar coordinates relative to Sol's barycenter (you can look these up on JPL's Horizons) and convert them to cartesian, you can compute Moon's cartesian coordinates to help you work out the angles with respect to the observation points. For Mars, I think it's fair to assume that we know the plane of motion from other observations as well, so again, I'd use the Horizons data for that, and simply assume Mars is moving in a circle of unknown radius r from Sol's barycenter. That puts Mars' cartesian coordinates as a known function of r and t, and if you work out the apparent position from these, you can then fit to the observations. Since you're fitting just one parameter and I don't expect any surprises with local minima, you can probably just do a binary (or golden section if you're feeling fancy) search for the correct r.

[cubinator]

1 hour ago, K^2 said:

You should have the orbital elements, right? I know it's a chore to track down all the correct datums, and transform the coordinates, but there shouldn't be anything particularly complex there. Just tedious.

To simplify some of the math, I really don't think you need to worry about radial motion of the planets, so if you simply take the Earth's polar coordinates relative to Sol's barycenter (you can look these up on JPL's Horizons) and convert them to cartesian, you can compute Moon's cartesian coordinates to help you work out the angles with respect to the observation points. For Mars, I think it's fair to assume that we know the plane of motion from other observations as well, so again, I'd use the Horizons data for that, and simply assume Mars is moving in a circle of unknown radius r from Sol's barycenter. That puts Mars' cartesian coordinates as a known function of r and t, and if you work out the apparent position from these, you can then fit to the observations. Since you're fitting just one parameter and I don't expect any surprises with local minima, you can probably just do a binary (or golden section if you're feeling fancy) search for the correct r.

I'm not looking those orbital elements up, I feel it goes too far against the nature of the project. I am even using my own simplistic measurement of the Moon's distance which I know is slightly inaccurate but that I helped obtain myself. For the purpose of this experiment, I don't know the distance to the Sun or the length of Mars' orbit, or that Earth orbits the Sun. I know the Moon's size and distance, the Earth's axial tilt and the time of day and year, the Moon's period as observed since antiquity, and the coordinates of each observer. My assumptions are very simplistic, so as to show that we can perform science even with simple equipment.

I can find the Moon's position and velocity in the sky using a mere few timestamps and math, and I find beauty in that. All of this could be bypassed by looking things up, but this is more of an art project than an actual experiment. NASA knows the distance to Mars better than I could ever estimate using my backyard telescope, but I am hopefully showing people that they have the power to achieve great things no matter what tools they have in front of them. I'm also showing a fundamental example of the scientific method, in which discoveries are made with an open mind and careful consideration of what can be observed from the world.

Edited January 5, 2023 by cubinator

[kerbiloid]

If plant a stick into the ground as a gnomon, wait for the night, and measure the shadow angle of the stick in reflected Martian light in upper culmination, and do this at once at several places, the shadow lines intersection will show the 3d position of Mars. Thus, the distance also can be computed.

[cubinator]

Argh...I yield! I will use my pictures!

I have been stuck on one simple relationship for almost a month now, and since classes are picking up I figured I'd better at least give you all a number before it's not Thread of the Month anymore. 

Basically, the problem is this. Mars crosses the Moon along a line that doesn't perfectly intersect its center, so the length of the path behind the Moon is some fraction of its diameter which is less than 1 and different for all three observers. 

The relationship between the length of the path for the three locations allows me to calculate the Moon's exact position in space at all relevant times, which I need for the parallax measurement later. Unfortunately, I also need to know the actual lengths of these paths, which I do not yet.

"Hold on," you might say. "Don't you know the length from the duration of occultation and the known velocity of the Moon?"

I actually need to know the velocity of the Moon relative to Mars, which I've never measured. I've been trying to figure it out by using relationships between the positions of observers, the timings from single locations and between multiple ones, and generally flipping the whole thing inside out and shaking it to try to make something useful come out. I think that it's technically possible, but requires some kind of system of equations that I've not mustered the brainpower to fathom.

Meanwhile, the easy answer has been conspicuously sitting in plain sight the whole time. This was a really cool looking event, so guess what I did? I photographed it. I took photos before and after the occultation which clearly show the Moon and Mars, and I've been reluctant to actually use them because I didn't expect to need them for the determination and I didn't take them with the intention of using them as scientific data. 

I can stitch the photos from before and after together using the features on the Moon's surface, and connect the dots to draw Mars' path. From there I can figure out the distance from the center and then use that as a basis to compare the duration of occultation for the other locations and get the Moon's exact position and velocity, along with Mars' angular velocity. This is now my plan. After using this method to find the Moon's path through space and Mars' path through the sky, I will be able to quickly find the expected location of both for an infinitely distant Mars and sift out the difference in timing between observations that results from parallax.

 

Some further explanation and visuals that I didn't know how to fit into this story nicely but are still probably necessary to look at in order to understand it:

Spoiler

This is a graph of the relationship between the observed duration of occultation and the path Mars takes across the Moon. What I know are t_MN, t_CA, and t_UK, and I've drawn them in here in more or less the correct positions. What I don't know is the distance from the center of the crossing line, represented as the horizontal axis. The relationship between the two is circular, and interestingly the Moon/Mars relative velocity V_Moon/Mars is actually the same for all observers, which is really convenient. But I need one of those three black points on the edge of the circle to be defined - that is, I need to know how far from the center Mars crossed the Moon from at least one location in order to define that circle with the other points and find V_Moon/Mars. I can define the MN point using my photos.

Here is a proof that V_Moon/Mars is the same in inertial terms as for any generic moving observer. It shows which values are known and which are unknown. You can also see how finding V_Moon/Mars would allow me to find all the other unknown velocities in this set of vectors. Take a look:

And finally, here is one of my photos, since they are now relevant to this conversation.  

[cubinator]

I got a curve fit for the Moon's edge and Mars' position using two pictures from after the occultation ended. I'll figure out the path length tomorrow.

[cubinator]

This is a nice little set of equations that helps me estimate the Moon's travel path. If you squint very closely, you'll be able to see that it's an equation for a line in terms of theta, the inclination angle of the Moon's velocity at the time. All the "midpt" y and z variables are the locations of the observation sites during the halfway points of the occultation, when Mars was closest to the center of the Moon at a distance which I now know from correlating our timing data with my images.

By the way, I now know how close to the Moon's center Mars got at all three locations.

I made this superposition of two of my images from after the occultation using paint.net:

Spoiler

Then I used LoggerPro to click a bunch of points on the edge of the Moon and generate a circular curve fit to those points.

Then I took two more points for the locations of Mars, and made a linear fit to those points. This gave me essentially a mathematical illustration of the event as I saw it. I took the equation of the Mars line and calculated its tangent distance to the center of the Moon circle, and from there I could find the sector length for the path behind the Moon and relate it to the other locations' sectors using timing data.

I found that the sector of the Moon crossed by Mars was 2908 km in MN, 2670 km in CA, and 2644 km in UK. The minimum distance to the center of the Moon was 950 km in MN, 1111 km in CA, and 1127 km in UK. These distances all correspond to physical distances at the disk of the Moon itself. I can now calculate the angular velocity of Mars.

First I will calculate the linear velocity that Mars would have if it were at the same distance as the Moon, since all my values are in linear terms right now.

C_MN = V_Moon/Mars * t_MN

V_Moon/Mars = 0.733 km/s

V_Mars + V_Moon/Mars = V_Moon

V_Mars = -0.311 km/s = -.298 V_Moon

Now I can convert that "fake" velocity to an actual angular velocity, this is the most important value of the determination so far:

w_Mars = -.7936e-7 rad/s

Now I know how fast Mars moves across the sky, so the difference in timing of the occultations due to the motion of Mars while we waited for the Moon to cover it from each observation location can now be safely extracted. The timing difference that will be left is that due to parallax, which will allow the final distance calculation as well as the calculation of the true velocity of Mars and the size of Mars.

Although, actually, that velocity is not exactly the one I'll use. The calculation I just did leaves out a detail that will change the final result somewhat. The Moon is not exactly collinear with Mars, so I need to use the set of equations at the top of my post to find the Moon's actual velocity vector, and use that to do an actual vector addition rather than a 1D addition. 

[cubinator]

Based on the Moon's velocity, I should have seen the occultation last 2785.4 seconds if Mars were completely stationary in the sky. if I were completely stationary.

Edited January 28, 2023 by cubinator

[cubinator]

I think all that difference is mostly due to Earth's rotation.

[cubinator]

I think that there is still something wrong in how I am dealing with the velocities. Looking back at astronomy programs, it seems that the path across the Moon that I'm predicting isn't what really happened. I need to revisit this with some more  different calculations. Maybe using angular values more in the calculations would help.

[cubinator]

On 1/27/2023 at 11:43 PM, cubinator said:

I found that the sector of the Moon crossed by Mars was 2908 km in MN, 2670 km in CA, and 2644 km in UK. The minimum distance to the center of the Moon was 950 km in MN, 1111 km in CA, and 1127 km in UK. These distances all correspond to physical distances at the disk of the Moon itself. I can now calculate the angular velocity of Mars.

First I will calculate the linear velocity that Mars would have if it were at the same distance as the Moon, since all my values are in linear terms right now.

C_MN = V_Moon/Mars * t_MN

V_Moon/Mars = 0.733 km/s

V_Mars + V_Moon/Mars = V_Moon

V_Mars = -0.311 km/s = -.298 V_Moon

Now I can convert that "fake" velocity to an actual angular velocity, this is the most important value of the determination so far:

w_Mars = -.7936e-7 rad/s

Now I know how fast Mars moves across the sky, so the difference in timing of the occultations due to the motion of Mars while we waited for the Moon to cover it from each observation location can now be safely extracted. The timing difference that will be left is that due to parallax, which will allow the final distance calculation as well as the calculation of the true velocity of Mars and the size of Mars.

I think this is wrong.

 

A few words on my hiatus:

This is an extremely difficult problem to think about. You have to consider the position of Earth, Moon, and Mars at the same time. You have to consider the position of three different observers on the Earth at different times. Earth is rotating. The Moon is moving. Mars is moving. Everything is at a weird angle. And when I say a weird angle, I actually mean three different weird angles lumped together. A couple of the angles are changing constantly. The rest change non-constantly. Precise positions and velocities turn out to be very important, and they are damn hard to infer based on loosely related data. I am missing some collateral measurements that would have turned out to have been very helpful, because I didn't know I would need them at the time. When I planned this observation, I thought I could measure Mars and the Moon against a background star - a nice common coordinate that would sort out all the insane irregularities between observing locations that I never fathomed. But the star was no match for the Moon's blinding glare, and the three of us - myself, @K^2, and @Starshot - were left measuring against a meaningless void, an infinite ruler with no tick marks. How am I to calculate the distance to Mars in this void? There are so many angles and distances and timings I can get wrong. If my equation for the prediction of the transit times drifts by mere seconds, the final distance estimate could double or quadruple!

Also I was busy graduating college and stuff. And writing an orbital mechanics simulator for a donut planet. Anyway.

I think this problem can still be solved. But it's going to take more work than I first thought. Maybe a numerical simulation. That's not very "do it using only simple technology" but it's still "do it yourself". I've been trying to come up with solutions on and off ever since the occultation. Here are my thoughts on what I think might be a course of action that gets a measurement of the distance to Mars:

 

Now, why I think I don't know Mars' angular rate yet.

I came up with these 'circles' and center distances to figure out the depth of occultation in each location based on the duration. I used a constant Moon angular velocity between the three locations to solve for Mars' angular velocity. In order to be sure, I went to Stellarium to check that the sectors at the different locations were about what I expected. And I found something completely different. The occultation was longest in MN and shortest in the UK. We measured this, we know this for a fact. But in the UK, Stellarium showed Mars traveling right across the middle of the Moon, just about the longest possible sector! How could a sector so much longer than MN's produce the shortest duration occultation of the three?!

The answer, I think, is that the inertial velocity of the observers caused by the rotation of Earth is a significant fraction (up to 40% depending on latitude and time of day) of the Moon's orbital velocity, and causes the Moon's angular rate across the background sky to be different between one location and another. During moonrise (CA) and moonset (UK), the Moon's angular rate will be high because YOU don't have much velocity in the same direction as the Moon. But during the middle of the night (MN), when the Moon is high in the sky, the spinning Earth gives you a huge amount of velocity in the same direction as the Moon's orbit. This slows the Moon's motion across the stars quite a bit, and I think this is what's responsible for the...difference in difference in time.

I can get the angular velocity of Mars if I know the MN-specific absolute angular velocity of the Moon, since I know the occultation depth and duration. I can get the MN-specific absolute angular velocity of the Moon if I know the position of the Moon at ingress and egress. The trouble is that I don't know exactly where the Moon is. There are a couple ways to go about this.

One way would have been to measure the azimuth and altitude of the Moon by hand while I was out there measuring the occultation. I had no idea I would end up needing that, so I didn't do it. Now I'm stuck only sort-of knowing where the Moon is.

I could look up what the Az/Alt was at the time using Stellarium, but that would be tacking on more "taken at face value" numbers to the problem. I'm going to see if I can do without for now.

I can assign an arbitrary location for the Moon that's "close" to where it actually was, and calculate the Mars angular rate based on that theoretical location. It was a full Moon, so the Moon must have been at least fairly close to the Sun-Earth line, and I figure the difference of a few degrees will probably (hopefully) not change the estimate of Mars' angular rate too much. This is the method I'm probably going to try next.

 

It's extremely important for me to get an accurate estimate of Mars' angular rate because it is a significant factor in the timing of occultation in different locations. If I predict Mars to be in the wrong place by one diameter after three hours, the timing prediction will probably be off by around 40 seconds. Since the timing difference due to parallax is probably not more than 40 seconds (corresponding to a distance between observers of 1 Mars diameter), that extra difference from the motion could cause the distance estimate to double.

 

After I estimate Mars' angular rate, I will ditch the simplistic guess of the Moon's location and start sweeping every possible location and velocity of the Moon to find occultation timings that are close to matching what was observed. I hope that I will be able to figure out where Mars is supposed to be in each case - that will be a problem for another day. My hope is that in the end, I will find a Moon trajectory that *almost* matches the observation, and the reason it is not exact is that Mars is assumed in the simulation to be infinitely far away. The only remaining difference between this simulation and reality should be the parallax, and the distance to Mars will be revealed. As you can see, this is extremely complicated and takes a lot of steps that I don't fully understand. I will hopefully make progress on it and eventually solve it. Maybe I will try looking up the Az/Alt of the Moon and using that. I'd still probably have to mess with the Moon's direction of motion anyway, though.

In the end, it is not enough for me to know how far away the Moon is. I need to know where it is. I think I have enough constraints to figure that out: The timings of the occultation in the three locations, and the depth of occultation in MN. This is a very interesting problem to me, and I'd be very proud of myself if I figured it out. I'm still coming back to it for a few reasons. The distance and size estimates would be really cool and impressive on their own, but I think they also can enable some deeper characterization of the solar system that I've dreamed of doing for a long time. If I know Mars' distance and angular rate, I can get the relative velocity between Earth and Mars. From that and the length of the year, I might be able to get the size of Earth's and Mars' orbits around the Sun, and the Sun's size. The size of Earth's orbit precludes a lot of interesting information that can be gained from observations of the other planets. I will take my time, and I will let you know if I come up with anything!

Booting up MATLAB...again....again!

Edited August 8, 2023 by cubinator

[cubinator]

After placing the Moon at an arbitrary location near the "Full" position and calculating its motion in spherical coordinates as viewed from MN, my code suggests that the angular velocity against the background stars is indeed slowed by MN's motion. The angular velocity of the Moon around Earth's center is 2.6617e-06 rad/s, and the angular velocity around MN during the occultation is around 2.0309e-06 rad/s. From this I should be able to reach an approximate value for Mars' angular rate across the stars, which should hopefully be about the same for everyone since Mars is so far away.

[JoeSchmuckatelli]

This has been fun to watch. 

I couldn't possibly do what you've attempted... But I really appreciate the writeup! 

[K^2]

11 hours ago, cubinator said:

Cubinator, K^2, and Starshot were left measuring against a meaningless void.

That reads like part of a crawl text before the movie starts.

[cubinator]

On 8/8/2023 at 9:30 AM, cubinator said:

and I figure the difference of a few degrees will probably (hopefully) not change the estimate of Mars' angular rate too much.

It does. Placing the Moon in the wrong position by a few hours causes Mars' estimated angular velocity to be different. What's worse, though is that changing the Moon's direction by one degree causes Mars' velocity to change by 5%, where I need it to be accurate within around 0.1% to get anywhere close to detecting parallax. So I need to know not only exactly where the Moon is, but where it's going.

On 8/8/2023 at 9:30 AM, cubinator said:

After I estimate Mars' angular rate, I will ditch the simplistic guess of the Moon's location and start sweeping every possible location and velocity of the Moon to find occultation timings that are close to matching what was observed. I hope that I will be able to figure out where Mars is supposed to be in each case - that will be a problem for another day. My hope is that in the end, I will find a Moon trajectory that *almost* matches the observation, and the reason it is not exact is that Mars is assumed in the simulation to be infinitely far away. The only remaining difference between this simulation and reality should be the parallax, and the distance to Mars will be revealed. As you can see, this is extremely complicated and takes a lot of steps that I don't fully understand. I will hopefully make progress on it and eventually solve it. Maybe I will try looking up the Az/Alt of the Moon and using that. I'd still probably have to mess with the Moon's direction of motion anyway, though.

I *think* I can do this by sweeping over only three variables: the Moon's initial position, the Moon's initial direction of motion, and Mars' exact position during MN occultation (which is actually more of an orientation constraint). That's good, because finding a minimum in four or five dimensional data sounds hard.

Everything else seems to be dependent on those, the most egregious being Mars' velocity which has both a magnitude and direction dependent on its own position, the Moon's velocity, and MN's velocity. I could calculate the velocity first and then the position but I'm guessing it wouldn't be easier.

Once I get all that from those three variables + the observer location information, I should be able to calculate the Moon-Mars angular distance at each of the times we recorded. Since these are all ingress or egress times, that distance should be the same as one Moon radius. If it's not, then that means that the combination of Moon position, Moon velocity, and Mars position is wrong. Repeat with another combination until I find one that matches. My expectation is that it will never match exactly, but will be off by about one Mars diameter in the UK. This is the parallax, and a couple of SOH-CAH-TOAs later the solar system will be my oyster. Hopefully.

And THIS is a pretty straightforward, best case scenario! It probably has logic holes and flaws in it that I'm not seeing yet, but whatever. It's been my project for the day, and I will see where it takes me tomorrow!