Did Vikram have a ‘hard landing’ or did it crash onto the Moon?

Great science results from great contestation. It is in this spirit that I have contested the assertion by the leadership of the Indian Space Research Organisation (ISRO) that its Vikram lunar lander’s failure was a result of “hard landing.”

When a significant aspect of the 9.78 billion rupee (about $140 million) space mission, namely the landing of the Vikram lander and Pragyan rover, failed, it was incumbent on the ISRO leadership to engage in a brutally honest determination of what went wrong rather than offering early assertions such as the mission was over 90 percent successful.

ISRO has time and again proved its talents and skills in conceiving and executing space missions at budgets which no other space agencies are able to. That is a testament to the dedication of its scientists and engineers. At the same time though as a great scientific enterprise it should have the fortitude to accept its flaws. As someone who has spent close to five decades in the United States studying all aspects of reliability of systems, I found it disturbing that the ISRO leadership’s early response to the Vikram crash was woefully defensive.

From what I have seen in the Indian broadcast media, ISRO chief K. Sivan has continued to maintain that Vikram had a hard landing. I strongly disagree based on mathematics and principles such as the coefficient of restitution or COR. It is baffling that this term does not even figure in the sphere of public discussion.

As someone who had calculated within a day of the Vikram failure, the velocity at which it would have crashed and inevitably have been crushed, I take no joy in being vindicated over two months later. My calculations-based conclusions were reported in the Indian media via the IANS wire on September 21 even though they were reached much earlier by me. They bear out what is now being widely acknowledged, albeit after considerable trepidation, even by ISRO.

Based on the dynamics of the fall it was clear to me quite early that Vikram did not have a hard landing as suggested by ISRO but it had, in fact, crash-landed. I had told the IANS wire this, "In my opinion, only the first hit, may have crushed Vikram. There could have been no chance for subsequent bouncing. However, theoretically it is possible to have more bouncing like a tennis ball but Vikram is not a tennis ball."

Mathematics is not beholden to ideology or political expediencies. In this particular case, it clearly produced numbers that suggested a catastrophic end to the lander. Assuming that Vikram was “free falling”, the first crucial number was a velocity of 184 miles an hour at which the lander in all likelihood crashed.  Arriving at the figure was a straightforward calculation. Free fall velocity can be closer to the square root of 2gh. Where g is one-sixth of Earth (32.2 feet per Sec²) h = 2.1 km = approximately 7000 feet. With this free fall velocity is 270 feet per second = 184 miles per hour. Nothing can survive at that velocity.

We are assuming ‘free fall’ but it is not exactly so. It is a minor issue but needs to be mentioned. At 2.1 km height, because of the existing trajectory, there must be initial downward velocity. This is equivalent to forcing the tennis ball downward with a velocity that can rebound greater than ‘free fall’. This downward velocity and coefficient of restitution combined can give us the worst first bounce, worse than mentioned earlier.

It may be reasonable to surmise that only the first hit may have crushed Vikram. There could have been no chance for subsequent bouncing. However, theoretically it is possible to have more bouncing like a tennis ball but Vikram is not a tennis ball. We also have to think in terms of whether the location where it hit was rocky, somewhat rocky or fully powdery like the sand on Earth. That detail would have had discernible impact on whether it bounced or just crashed once and settled down significantly broken.  One has to also determine whether there was consideration in Vikram’s design about “the moon’s ‘coefficient of restitution’ (COR) at the predetermined landing location.       

This is where the COR between 0 and 1 comes into play. For the sake of argument, let’s say it was 0.5.

If the COR is 0.5, the first bounce would be 0.525 km and the next bounce would be 0.13 km and so on. The formula will be the square root of the ratio height after rebound to height for next rebound.

Anything four-legged falling does not necessarily hit the ground with all four legs at the same time depending upon the profile of the Moon’s surface. It is likely that one leg touched first and took all the force due to free falling. If one leg is weakened, the rest cannot help the situation. The deformations that are plastic (permanent deformation) are not recoverable in metal and materials. From anyone making a prediction with common sense, Vikram was in all likelihood destroyed as soon as it hit the lunar ground.

I have offered clear calculations in a slim volume ‘Vikram Lunar Lander: Dead or Alive”, co-authored with the Chicago-based senior Indian journalist Mayank Chhaya.

The whole purpose of detailed and rational feedback is necessary for all future space endeavours for a country like India and an organisation like ISRO that takes particular pride in executing highly cost-effective missions. It does not help to obfuscate failures because in science and technology failures can often be more vital than success because they reveal so much that is unexpected.

[Dr. Bharat Thakkar is a leading expert on reliability of systems and quality control based in Chicago for the past 50 years. He is a mechanical engineer who has as a professor taught at several universities in America]