What I have been trying to do is make sense of the landing site selections for the Soviet Mars 2, 3, 6 and 7 landers from a mathematical point of view. In other topics it has been shown that the Soviets used Mariner images to plan the landing sites, and a lot has been written on the technical aspects of these missions, but not much has been written about why a certain landing site was selected and which were the constraints on landing.
Given the completely automated and pre-set landing sequence it looks like the M71 and M73 landers were very constrained in their landing sites, they could only land at a few pre-defined spots and there was only very limited room to alter a landing point once the craft was launched.
If I calculate the situation at the time of Mars 3 and Mars 6 touch down I get the following situation:
Click to view attachment
for Mars 3
Click to view attachment
for Mars 6.
Both images are remarkably similar.
In both images the direction of the sun is shown as the yellow vector, the grey vector marks the motion of Mars, and the red mark shows the landing site. It can be shown that both craft:
1) Landed in the local afternoon (Mars 3: 15:52 LMT and Mars 6: 15:12 LMT)
2) Landed at a latitude below the suns declination (Mars 2 and 3: 25 deg below suns declination, Mars 6: 32 deg)
Data as I calculate it:
CODE
Probe Lat Sun Decl Diff Lon Sun GHA Diff LMT
Mars 2 44 S 20 S -24 58 E
Mars 3 45 S 20 S -25 158 W 144 E 58 W 15:52:00
Mars 6 24 S 8 N -32 19 W 67 W 48 W 15:12:00
Mars 7 50 S ? 8 N -57 28 W ?
Mars 2 44 S 20 S -24 58 E
Mars 3 45 S 20 S -25 158 W 144 E 58 W 15:52:00
Mars 6 24 S 8 N -32 19 W 67 W 48 W 15:12:00
Mars 7 50 S ? 8 N -57 28 W ?
For Mars 2 and Mars 7 I did not (yet?) find exact data on UTC time of closest approach to Mars, so I could not calculate the Sun GHA and local mars time. Mars 7 is the odd one, the latitude of 50 S does not seem to make sense as it is much farther south then all the other landers, however as Mars 7 missed Mars altogether the location of its intended landing site seems disputable.
For Mars 2 and 3 the latitude of the landing point also relates to the inclination of the final orbit of the mothership (Mars 6 and 7 were fly by missions, however the Mars 6 location relates to the inclination of the Mars 5 orbiter).
I am not yet completely sure why it was apparently necessary to land at a latitude 25-30 degrees below the suns declination, but a landing in the local afternoon seems to make sense.
If I calculate the trajectory backwards, then separation of the lander from the mothership was almost at local noon, in other words the spacecraft was almost exactly between Mars and the Sun at the moment of release. I have the feeling that this was required for three reasons:
1) It is mentioned that the automatic navigation system used angle measurements of Mars to calculate the trajectory and to time the moment of release, for angle measurements it is easiest if you have a 'full Mars'.
2) On release of the lander, the spacecraft had to be oriented perpendicular to the flight path, while it is mentioned that it was stabilized using the Sun and the star Canopus. This works only if the craft is between Mars and the Sun, then if you point the sun sensor (and solar panels) at the sun you are already perpendicular to the flightpath.
3) After release and the firing of its solid propellant engine, the lander was oriented with its heatshield pointing in the direction of flight and then spin stabilized for the cruise to Mars. Once again, this works easiest if the craft is almost in between Mars and the sun, you only need to point the spin axis at the sun and the craft will be stabilized with the heatshield in the correct direction!
Remember this was all a completely automated procedure, the lander used gunpowder engines both for the separation burn and for the orientation so everything was completely pre-set, it only worked if conditions were exactly as calculated prior launch of the craft. Once the lander was released there was no more 'steering' or anything, the whole further sequence was completely pre-set and ballistic.
I have not yet completely worked out these constraints, but it looks like the M71 and M73 landers were extremely constrained in their landing sites, the latitude of the landing site was probably completely fixed and longitude was defined by arrival time at Mars, which also could be varied only within small limits. Thus, it was not that they 'wanted' too land at a specific location, but that they didn't have much other choice given the completely automated and pre-set landing sequence. They were very restricted in where they could land those landers.
In order for the sequence to work three items were of utmost importance:
1) On release the mothership had to be in a trajectory with a fly by distance of 1500 km +- 200 km (Perminov)
2) On release the craft had to be orientated perpendicular to the flightpath
3) The lander had to be released at exactly the correct time and distance to Mars.
Click to view attachment
Given an atmospheric entry angle of 12 degrees +/- 5 deg (Perminov) we can calculate the distance X in above image that the lander had to travel perpendicular to the flightpath of the mothership as 1574 km (1526 - 1648 km).
Perminov states that the solid fuel engine of the lander gave an impulse of 120 m/sec (however elsewhere 100 m/sec is mentioned), if this is true, the lander had to be released 3 hours and 54 minutes prior atmospheric entry (or 3:47 till 4:04 given the 5 deg window). Usually a release time of 4 hours 35 minutes is mentioned, which might indicate the pulse was slightly less then 120 m/sec, or the mentioned time is until closest approach to Mars instead as atmospheric entry, but at least it shows our calculation is quite close to the actual situation.
If the lander is released too early it will travel too far perpendicular to the flightpath and enter the atmosphere at too large an angle, on the other hand if the lander is released too late it will enter at a too shallow angle or miss Mars all together.
Similarly, if the mothership is in an approach orbit with significantly less then 1500 km flyby distance, the lander will also enter at a too steep angle, this is what happened to Mars 2 (flyby distance abt 1100 km so abt 400 km offtrack).
Finally, it can be calculated that there was precious little time for communication with the lander after touch down. From what I calculate it looks like in the Mars 2/3 situation the mothership disappeared below the horizon less then 10 minutes after touch down of the lander. This means that most of the images and surface data had to be relayed during the next sol when the mothership again passed the landingsite. Unfortunately, in case of Mars 3, the mothership was inserted in a completely wrong orbit and passed the landingsite only about two weeks later, when (even if it worked correctly) the battery's of the lander were long run out...