Good Math, Bad Math

As you’ve hopefully all heard by now, the Mars Phoenix lander made a perfect
landing over the weekend, and is already returning images. NASA managed to not
only achieve a perfect landing, but to use Mars reconnaissance orbiter to catch a
picture of the Phoenix descending with parachutes deployed!

Alas, NASA’s Phoenix press people aren’t nearly as good as its technical people. As an alert reader pointed out, in their press release about capturing
the photo of the probe with parachute deployed, that they said the following:

Phoenix released its parachute at an altitude of about 12.6 kilometers (7.8 miles) and a velocity of 1.7 times the speed of sound.

That looks relative innocuous, right?

Wrong.

The statement about velocity is meaningless.

The speed of sound isn’t a constant. It varies, enormously, depending
on the medium. In air, its speed is dependent on the chemical makeup of
the air, and on its density, temperature, and pressure – among other factors. So what speed of sound are they talking about?

The speed of sound where the probe was entering the Martian atmosphere? That would make sense as a measurement, but be totally uninformative to us back on
earth, since we don’t know the speed of sound in the upper atmosphere of Mars.

The speed of sound on earth? That would be informative to us – since we
have an idea of the speed of sound here, but it wouldn’t make much sense as a measurement there – the point of using the speed of sound would seem to
be related to giving us a sense of the kind of forces acting on the Phoenix
as it decelerates. But the speed of sound on earth doesn’t tell us that – because the kind of shock waves we would expect is dependent on the speed of sound in the atmosphere it’s passing through.

You can talk about speeds compared to the speed of light – because there’s a meaningful upper bound – the speed of light in a vacuum. And that’s what we usually mean when we talk about the speed of light. But with sound, that’s not
true. The speed of sound can vary quite dramatically in different mediums. It’s a big enough difference that it’s part of a common experiment done by
elementary school students! (I can remember doing an experiment in fourth grade science with a wall, where we were measuring when you could hear a rock hit a wall; one person had their ear against the wall; the other was standing a couple of feet away from the wall, and the person with the rock was about 10 feet away. The time difference was noticeable. It was very small – but distinctly noticeable. The speed of sound in air is 1260 feet per second; so a sound takes roughly 1/10th of a second to move 100 feet. The speed of sound in stone is in the range of 21,000 feet per second – which is virtually instantaneous to a human being at a range of 100 feet. So you’re looking at a roughly 1/10th second difference.)

So how fast was the Phoenix moving when it deployed its parachute? I haven’t
a clue. My best guess would be around 580 meters per second – assuming that
they were using the speed of sound in earth atmosphere at standard temperature and pressure. The speed of sound in the Martian atmosphere – which is quite a lot thinner than earth’s – would be slower, so 580 m/s is a decent upper-bound estimate.

Comments

  1. #1 Beetle B.
    May 27, 2008

    I’m sure they meant the “usual” speed of sound in air on Earth – roughly 340 m/s. The press release was probably meant for the layperson, and they often talk of the speed of sound as a constant.

    They could have given an exact number, but that would be somewhat meaningless to most non-scientists. The 340m/s speed of sound is the only generally known speed close to Phoenix’s velocity, so they used it.

    It may be scientifically flawed, but I would argue that it IS better PR than giving the proper speed as a number – because more people would understand it this way.

  2. #2 Mark C. Chu-Carroll
    May 27, 2008

    Beetle:

    Really, you think that for most people, it’s more meaningful to say “1.7 times the speed of sound” than to say “3000 miles per hour”?

  3. #3 droz
    May 27, 2008

    This is a horribly pedantic view of the press release and you are trying to read a little too deep into what the release is saying. It is clear that the writers intended the speed of sound here on Earth under ‘normal’ conditions.

  4. #4 dave
    May 27, 2008

    droz: “It is clear that the writers intended the speed of sound here on Earth under ‘normal’ conditions.”
    Is it? It isn’t to me. When I hear “1.7 times the speed of sound”, the first thing that comes to mind is “So the martians wouldn’t’ve heard it coming at that point”. Why would you reference something happening over there to local properties of the atmosphere over here?

  5. #5 John Miller
    May 27, 2008

    I respectfully disagree that the number is meaningless and that it is relative to STP on earth. I’m certain that the system was design to deploy its parachute when its speed was 1.7 times the shock wave propagation speed of the atmosphere it was in. At the very least it was able to inform controllers what the ratio of its speed to the “speed of sound” was when it deployed the chute. This mach number has huge consequences for aerodynamic craft. The only problem here is that the PR person translated a screen reading “ALT 12600 SPD 1.7 M” into the press report above without enough context to really understand it.

    It really is 1.7 times the speed of sound at the point where the parachute was deployed. Quite an aerodynamic feat.

  6. #6 Goofy
    May 27, 2008

    I agree it is uninformative, but might the speed of sound on Mars have relevance to parachuting on Mars? They might have cribbed the statistic from some engineer’s report saying that this was like a mach-1.7 parachute deployment?

  7. #7 Anthony
    May 27, 2008

    Continuing the pedantry, your post suggests that the low pressure of the Martian atmosphere would result in a reduced speed of sound, which isn’t true — it’s the low temperature and high mean molar weight of the Martian atmosphere which results in a reduced speed of sound (at the surface, should be about 30% lower. At the altitude of the probe, since temperature varies with altitude, no idea). Plus, the probe is heating the atmosphere, so its shock wave won’t be traveling at the speed of sound anyway.

    In terms of actual aerodynamics, local speed of sound is certainly important.

  8. #8 Scott Belyea
    May 27, 2008

    Well, since pedantry seems to be the flavour of the day …

    As you’ve hopefully all heard by now,

    ARRRRRGGHH!! No, no … that’s not what you mean. You actually mean something like, “I hope that you’ve all heard…”.

    You don’t know anything about my state of mind … why speculate when you have no data?

    Pedantically yours …

  9. #9 Coin
    May 27, 2008

    You can talk about speeds compared to the speed of light – because there’s a meaningful upper bound – the speed of light in a vacuum.

    Perhaps what the press release was trying to communicate was that Phoenix was moving at 1.7 times the speed of sound in a vacuum.

  10. #10 KevinS
    May 27, 2008

    Just to further the nitpicking:

    … the person with the rock was about 10 feet away. The time difference was noticeable. It was very small – but distinctly noticeable. The speed of sound in air is 1260 feet per second; so a sound takes roughly 1/10th of a second to move 100 feet.

    Surely you meant that the person holding the rock was 100 feet away? Otherwise, the time difference would be more like 1/100th of a second, and I can’t believe that humans manually timing things are likely to reliably register that small a difference.

  11. #11 Lowk
    May 27, 2008

    It didn’t seem confusing to me, I just read “1.7 times the speed of sound” to be an attempt to give the speed in units that a layperson would recognise (the speed that sound travels in day to day life).

    The speed of sound at sea-level on earth is a pretty widely recognised “high speed”. I think saying that it traveled 1.7 times that speed gets across the idea of “it was traveling really fast” more effectively than an explicit numeric value for the speed would.

  12. #12 Jérôme ^
    May 27, 2008

    #2: miles?! Aargh. This is a good way to emulate the Mars Climate Orbiter crash…

    … I guess that since that accident, NASA has learnt to use as few units as possible and is trying to use [ natural units » to avoid problems. Alas, the speed of sound is not very meaningful, and they might just crash another probe because some contractor understood that they meant « speed of sound in Mars’ upper atmosphere ».

  13. #13 Anthony
    May 27, 2008

    Re #12: If NASA specifies a Mach number to an engineer, it will be clear that they’re talking about the local speed of sound, because that’s how the Mach number is defined. The problem is, this is a press release, so we don’t know that 1.7 is a Mach number.

  14. #14 Jouni K
    May 27, 2008

    From SFGate:

    Then the spacecraft’s red-and-white striped parachute deployed about 7.8 miles above the surface, slowing the lander’s speed from 1,100 mph to a mere 120 mph in a little more than 3 minutes.

    From that I’d say the speed went from ~190 m/s to ~21 m/s, indicating that the factor 1.7 relates to local speed of sound, not the NTP version.

  15. #15 Jouni K
    May 27, 2008

    #14: Replying to myself here… way to goof up the math, man!
    1,100 mph -> 120 mph maps to ~491 m/s -> ~54 m/s, so the conclusion is not quite that clear. Even so, the starting speed is still low enough to suggest that the local speed of sound is meant here.

  16. #16 Gerald
    May 27, 2008

    Consider yourself lucky that they did not write “at an altitude equivalent to 7300 people standing on top of each other”…

  17. #17 rimpal
    May 27, 2008

    The speed of sound in air is 1260 feet per second; so a sound takes roughly 1/10th of a second to move 100 feet. The speed of sound in stone is in the range of 21,000 feet per second – which is virtually instantaneous to a human being at a range of 100 feet. So you’re looking at a roughly…

    More pedantry…Here goes…

    Mark, could you please use metric, no even more pedantry, please use S.I. units. :-)

  18. #18 Sizik
    May 27, 2008

    #9: *facepalm*
    No sound in a vacuum.

  19. #19 anonymous Martian
    May 28, 2008

    This could be in error, but the Phoenix page refers to the speed as a Mach number here (meaning the multiplier is w/respect to the local speed of sound),

    http://phoenix.lpl.arizona.edu/phases04.php

  20. #20 eddie
    May 28, 2008

    @ #18

    Yes, I too had to go back and read #9 again before I got the joke.

    @ #9;

    “Perhaps what the press release was trying to communicate was…”

    Not the greatest press release, then ;¬)

  21. #21 Ian
    May 28, 2008

    Maybe they’re just trying to say it made a sound landing on Mars…!

  22. #22 DouglasG
    May 28, 2008

    But what is it in football fields per hour? A football field is a standard press release metric isn’t it?

  23. #23 Devyl
    May 28, 2008

    If Nasa (or the people posting comments) mentions velocity at all they should know that velocity is a vector consisting of speed AND direction. By mentioning only speed (even if they think they are right on that), they are missing half of the equation!
    Simple example: The velocity of my car at 9 am this moring was 35 mph northeast

  24. #24 Jonathan Vos Post
    May 28, 2008

    I’m happy with parameterizing the speed of sound compared to the speed of light. That is routinely done in models of the universe shortly after the big bang. Back then, what was to be Mars and what was to be Earth were the same place, anyway.

  25. #25 Sili
    May 28, 2008

    Scott (#8),

    There’s nothing at all wrong with sentence adverbials.

    Don’t let the crazies win!

    To those saying the press release is clear: why then are we having this discussion in the first place? The sensible thing woulda been to give the absolute value of the speed (in imperical units for the sake of congress …).

  26. #26 Goofy
    May 28, 2008

    If the Mach number is meaningless in the case of a parachute deployment on Mars, aren’t all dimensionless numbers similarly meaningless?

  27. #27 Mark C. Chu-Carroll
    May 28, 2008

    Goofy:

    A mach number isn’t a dimensionless unit. It’s a value with an underspecified unit. “3” has no unit. “3 thingamabobs per hour” is underspecified, because you don’t know what a thingamabob is.

  28. #28 Mark Dow
    May 28, 2008

    NASA and JPL in general do a good job of communicating through graphics and images.

    This event would have been good timing for teaching moment, by including in the press release a simple graphic showing the parachute unfurling, a diagrammatic bow shock, and a velocity vector showing it’s relationship (angle) with the surface of Mars, and perhaps the ultimate landing site. And maybe a few numbers with appropriate units. I would be interested in the estimated atmospheric density and altitude.

  29. #29 Jonathan Vos Post
    May 28, 2008

    See: Exponential Atmosphere.
    The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude….

    Terminal velocity
    From Wikipedia, the free encyclopedia
    [see that for the full formula, and caveats]

    An object reaches its terminal velocity when the downward force of gravity (F_g) equals the upward force of drag (F_d). The net force on the body is then zero, and the result is that the velocity of the object remains constant.

    As the object accelerates (usually downwards due to gravity), the drag force acting on the object increases. At a particular speed, the drag force produced will equal the object’s weight (mg). Eventually, it plummets at a constant speed called terminal velocity (also called settling velocity). Terminal velocity varies directly with the ratio of drag to weight. More drag means a lower terminal velocity, while increased weight means a higher terminal velocity. An object moving downward at greater than terminal velocity (for example because it was affected by a downward force or it fell from a thinner part of the atmosphere or it changed shape) will slow until it reaches terminal velocity.

    Examples
    For example, the terminal velocity of a skydiver in a free-fall position with a semi-closed parachute is about 195 km/h (120 mph or 55m/s)[1]. This velocity is the asymptotic limiting value of the acceleration process, since the effective forces on the body more and more closely balance each other as the terminal velocity is approached. In this example, a speed of 50% of terminal velocity is reached after only about 3 seconds, while it takes 8 seconds to reach 90%, 15 seconds to reach 99% and so on….

    Not to be confused with:

    Terminal Velocity (1994)
    Directed by Deran Sarafian. With Charlie Sheen, Nastassja Kinski, James Gandolfini. A maverick skydiver and a former KGB agent team up to stop the Russian…

  30. #30 rimpal
    May 28, 2008

    Not to be confused with:

    Terminal Velocity (1994)…

    Darn, it was just getting interesting!

  31. #31 Flane
    May 28, 2008

    How about altitude above MSL, Earth! That’s a big number.

  32. #32 Torbjörn Larsson, OM
    May 28, 2008

    No, absolutely not meaningless. The fully inflated parachute had only 10 meter radius, so obviously they were referring to deployment at supersonic speed – and so the absolute speed would have been meaningless instead.

    For my 0.02 $ John Miller (#5) nailed it – it was presenting an aerodynamic feat [in thin atmosphere, mind], albeit poorly.

  33. #33 Anonymous
    May 28, 2008

    http://en.wikipedia.org/wiki/Dimensionless_number

    I note that Mark C. Chu-Carroll is wrong about mach not being a dimensionless number.

  34. #34 TruePath
    May 28, 2008

    Alright if we are going to be pedantic let’s be pedantic.

    The sentence in question is *vague* not meaningless. Not only did it parse in a perfectly meaningful way it told the reader the speed was one of a small number of alternatives and as a result allowed the reader to get an order of magnitude approximation if they put in enough effort. I agree the report should have been more clear but compared to some ‘science’ the BBC gets away with reporting this doesnt rate a raised eyebrow for me (tho I don’t begrudge you your pet peeves).
    ———

    To the commenter complaining that velocity is a vector the truth is more complicated.

    There is a *technical term* velocity that is used in physics and engineering that indeed is a vector quantity. However, there is also a common language term that is a synonym for speed and no amount of pedantry by your HS physics teacher changes the fact that this is a widespread deeply ingrained usage and thus valid English.

  35. #35 Goofy
    May 29, 2008

    “A mach number isn’t a dimensionless unit. It’s a value with an underspecified unit. “3” has no unit. “3 thingamabobs per hour” is underspecified, because you don’t know what a thingamabob is.”

    Mach is a dimensionless unit. In the OP, the text was “1.7 times the speed of sound”. People who deal with fluid dynamics read ‘The speed of sound’ as the dimensionless number or unit “Mach 1″, and the science PR guy who wrote it probably thought “Mach” sounded too thingamabobby so they substituted “speed of sound”.

    If you are saying ‘Mach 3′ is underspecified because you can’t know what Mach is in SI units, fine. But using the Mach number as a speed measurement unit in fluid science is a good thing, since it accounts for the disparities in temperature, density, and composition that one would expect between planets, or decelerating through an atmosphere.

    Your first interpretation of what it means “That would make sense as a measurement,” makes sense, but the next phrase is almost unreasonable: “but be totally uninformative to us back on earth, since we don’t know the speed of sound in the upper atmosphere of Mars.” How could we know the SI speed of the lander at parachute deployment? Interpolation between periodic on-board GPS fixes? High power radar from the Orbiter? Some calibration of the measured Mach number based on a model of the unmeasureable temperature, density, pressure, and composition of the local upper atmosphere of Mars?

  36. #36 SteveM
    May 29, 2008

    Mach is a dimensionless unit. In the OP, the text was “1.7 times the speed of sound”. People who deal with fluid dynamics read ‘The speed of sound’ as the dimensionless number or unit “Mach 1″, and the science PR guy who wrote it probably thought “Mach” sounded too thingamabobby so they substituted “speed of sound”.

    No, “speed of sound” is not a unitless number, it is a normalized unit. “Mach” is a unit of speed (not velocity) the same as c is a unit of speed. It can be converted to other units just as c can be converted to meters per second or furlongs per fortnight. The exact conversion depends on the medium (just as for c) Now, the fine structure constant, that is a unitless number.

  37. #37 Prometheus
    May 30, 2008

    @ Sili (#25)

    Of course there’s nothing wrong with sentence adverbials… except in the case of “hopefully”, where its meaning has been eroded to the point of ambiguity.

  38. #38 Reinier
    May 30, 2008

    The Mars lander lands 3.048 meters away, and somebody isn’t using the metric system. Does it make a sound?

  39. #39 Torbjörn Larsson, OM
    May 30, 2008

    Now, the fine structure constant, that is a unitless number.

    Only because it is defined as a normalized ratio. :-\

    @ Reinir:

    The Mars Polar Lander did. :-P

    The follow up question is, what did it sound like? Like 3.048 car crashes?

  40. #40 Sili
    June 3, 2008

    Prometheus (#37),

    What’s ambiguous about it? It’s simply speaker oriented. A way to avoid saying “I”.

  41. #41 Mike
    June 4, 2008

    Haha, awesome. I notice the weird things these people say too.
    I heard another NASA representative on a local news show mention that it was currently, “morning here in the Northern Hemeshpere.”
    Pointing this stuff out is not pedantry. Lol, maybe we should have these NASA jobs?

  42. #42 Paul G
    June 6, 2008

    On the whole, journalists are much better at writing than they are at doing science. However, I would agree that Mark is being a little bit pedantic here – clearly they meant “the speed of sound on Earth in air at standard temperature and pressure”, and the lay reader would understand it to mean that.

    However, as has already been said, the use of “velocity” is egregious. There is no such thing as a velocity of 1.7 times the speed of sound. Unfortunately, this is just the typical sort of mistake that hacks make when talking about science. Compare the newspaper story that talks of a temperature of 80 degrees F one day being twice the temperature of 40 degrees F of the previous day, when, if the temperatures had been measured in Celsius, today’s temperature would have been six times yesterday’s; on a related subject, consider an article that reports a rise in temperature of 10 degrees C and gives the equivalent rise in degrees F as 50. (They never seem to realise that a rise of 14 degrees F could not possibly be a “rise” of -10 degrees C.) These are not invented cases – I have seen both numerous times before in newspapers.

    But these are NASA’s press people, and they should certainly know better. They are deserving of Mark’s wrath. Bring it on!

  43. #43 Stephen
    June 10, 2008

    I hope it was the local mach number. Aerodynamics guys have to compute it anyway, and for some reason, it matters. A problem on Mars is that, until Phoenix lands, it’s not clear that they knew the density, and therefore the speed of sound. So, it’s a big guess. And they’ve guessed wrong about Mars before.

    As a kid, the Mars atmosphere was 6 millibars. But Phoenix said it was 8. Well, it’s summer.

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