4
$\begingroup$

Suppose those people engage in similar activity, walking on surface etc.

$\endgroup$
  • 8
    $\begingroup$ What then about people killed by not having air to breath? $\endgroup$ – SE - stop firing the good guys Mar 25 '17 at 13:47
  • $\begingroup$ @Hohmannfan your new name is "Mr. Buzzkill." :-) . Just try to pretend for a moment that we are all gigantic tardigrades. $\endgroup$ – Carl Witthoft Mar 27 '17 at 13:26
  • $\begingroup$ "If you can dodge a ball you can dodge a meteor" $\endgroup$ – Carl Witthoft Mar 27 '17 at 13:34
10
$\begingroup$

Three millions.

For this answer to be any fun, let's first assume that before removing the atmosphere, we have equipped all humans with oxygen supplies. In return, they are not allowed to go indoors or otherwise protect themselves.

oxygen

The size of meteors — as well as most other things in the Universe — has a distribution that is well described by a power-law. The larger the meteor, the rarer they are. Small meteors can be lethal only by a direct hit, while large meteors in the real world can be lethal due to the effect they have on the environment; dust can cover the sky for years and prevent the Sun from keeping us warm. However, since we have removed the atmosphere, the debris will fall down immediately, and the only way to kill us is by the explosion. But, as will be shown below, the dangerous ones are the small ones because the rate at which large ones falls off so quickly.

The number $N$ of meteors entering Earth's atmosphere every year with a diameter $D$ (measured in meters) or larger is given by (Brown et al. 2002) $$ \frac{N(>D)}{\mathrm{year}} \simeq 37 \left( \frac{D}{\mathrm{m}} \right)^{-2.7}. $$

Minimum size needed to be lethal

What makes a meteor lethal is really outside astronomy, and someone may be able to come up with a better answer, but for the sake of the argument, let's assume that the energy needed to kill a human by a direct hit is equal to the energy of a bullet fired from the world's smallest firearm, the 2mm Kolibri, which has a muzzle energy of $E_\mathrm{lethal} \simeq 4\,\mathrm{J}$.

Earth orbits the Sun at 30 km/s. Depending on the direction from where the meteor arrives, it will have different speeds, but it will at least enter with the escape velocity of Earth, i.e. 11 km/s, and a typical speed is $v_\mathrm{impact} = 17\,\mathrm{km/s}$ (Marcus et al. 2010). Thus, the minimum mass to kill by a direct hit is as small as $$ m_\mathrm{lethal} = 2 \, E_\mathrm{lethal} \, / \, v_\mathrm{impact}^2 \simeq 30 \,\mu\mathrm{g}. $$ At an average density of 4 g/cm$^3$, this means that a small grain of a diameter of 0.2 mm will kill you. According to the formula above, the number of such grains (or larger) that hit Earth every year is $2\times10^{11}$. Since the surface area of Earth is 510 million $\mathrm{km}^2$, that's $n_\mathrm{hits}\sim5\times10^{-4}\,\mathrm{hits}\,\mathrm{m}^{-2}\,\mathrm{year}^{-1}$.

Some 7 billion people live on Earth. They each occupy roughly $1\,\mathrm{m}^2$ (and weren't allowed to protect themselves, remember), so in total they occupy an area of $A_\mathrm{folks,tot} = 7 \times 10^9\, \mathrm{m}^2$.

Thus, the number of people being killed by direct hit meteors every year is $$ N_\mathrm{kill} = n_\mathrm{hits} \, \times \, A_\mathrm{folks,tot} = \mathbf{3\,million\,year^{-1}}, $$ or one person every 10 seconds.

Death by explosion

How about non-direct hits? Let's assume that the minimum size to kill you would have an energy equal to that of a hand grenade holding 100 g explosives, and that you're killed if it hits within a few meters of you, so that you "occupy" 10 m$^2$. Similar to above, you can work out that this corresponds to a meteor of ~1 cm being lethal, and that the rate at which such meteors is only $2\times10^4$ per year, so only 2-3 people would be killed in this way. Even larger explosions are even more rare. Thus, the probability of killed by a explosion is negligible compared to direct hits, so the final answer would be the above, i.e. 3 million per year.

Conclusion: Let's not remove the atmosphere.

$\endgroup$
  • 3
    $\begingroup$ That's awesome, also relevant for colonizing the moon and maybe Mars, but meteors hit those 2 bodies on average a little slower. I think many 4J. impacts would simply leave a thin hole in a person and not be fatal, the heat, perhaps cauterizing the wound (perhaps not) but small punctures would close more quickly, less bleeding out. Impacts wouldn't be fun but I think fewer than 3 million would be deadly. And, I love your answer!!! XKCD worthy. $\endgroup$ – userLTK Mar 26 '17 at 2:32
  • 1
    $\begingroup$ Thanks @userLTK. And yes, I think you're right, I probably overestimated the death rate. It's also dubious whether the N-D relation can be extrapolated to such small sizes. $\endgroup$ – pela Mar 26 '17 at 2:36
  • $\begingroup$ Great answer to this Fermi problem. Just a small note, you might want to specifically inlude in your units for $n_{\mathrm{hits\:per\:area}}$ a $\mathrm{year}^{-1}$. $\endgroup$ – zephyr Mar 27 '17 at 12:52
  • $\begingroup$ I think your $1 m^2$ coverage only applies to those who are lying down (and in certain cases, that applies to a pair of people lying down heh heh). People standing up have a cross-sectional area closer to roughly $ 0.3 m^2$ I would guess. And if a meteor only hits our shoulder or toes, that would hurt but not kill us. $\endgroup$ – Carl Witthoft Mar 27 '17 at 13:29
  • 1
    $\begingroup$ @CarlWitthoft: Regarding the "hits that don't kill", I assumed that since they're not allowed to go indoors, including the hospital, treating exsanguination is so difficult that we can assume that most hits result in death. $\endgroup$ – pela Mar 27 '17 at 20:31

Not the answer you're looking for? Browse other questions tagged or ask your own question.