Archives For physics

Here’s a conundrum for you: Using only technology available hundreds of years ago, how could you determine the speed at which light travels? We know now that light travels at 299,792,458 m/s, or, to put it simply, “very, very fast.” In fact, we are so sure of this value that we use it to define the meter, where one meter is equal to the distance that light travels in 1/299,792,458 of a second. Today, we have access to technology which allows us to calculate this value. Time-of-flight devices pulse bright flashes of light which are reflected off a mirror, and the difference in time (down to nanoseconds) combined with the distance from the source/detector and the mirror provides an accurate measurement of the speed of light. Additionally, one can take advantage of cavity resonators or interferometers to obtain the same value. However, these devices did not always exist, yet estimates for the speed of light predate their existence. How was this accomplished?

In a first account of the discussion on light propagation, Aristotle incorrectly disagreed with Empedocles, who claimed that light took a finite amount of time to reach Earth. Descartes, too, claimed that light traveled instantaneously. Galileo, in Two New Sciences, made the observation that light appears to travel instantaneously, but that the only observation is that light must travel much faster than sound:

Everyday experience shows that the propagation of light is instantaneous; for when we see a piece of artillery fired, at great distance, the flash reaches our eyes without lapse of time; but the sound reaches the ear only after a noticeable interval.

To determine the speed of light, Galileo devised a time-of-flight experiment similar to the one described above, where two individuals with lanterns would stand at a distance, uncover and recover them upon seeing a flash from the opposing partner, and calculate times between flashes. By starting very close to account for reaction times and eventually moving very far away, one could see if there is a noticeable change in latency. However, this experiment is challenging, to say the least. Is there a simpler method?

Enter Danish astronomer Ole Roemer. Known in his time for accuracy in measurement, arguments over the Gregorian calendar, and firing all the police in Copenhagen, he is best known for his measurement of the speed of light in the 17th century.

While at the Paris Observatory, Roemer carefully studied the orbit of Io, one of Jupiter’s moons. Io orbits Jupiter every 42 and a half hours, a steady rate. This discovery was made by Galileo in 1610 and well-characterized over the following years. During this time, Io is eclipsed by Jupiter, where it disappears for a time and then reemerges sometime later. However, Roemer noticed that, unlike the steady state of Io’s orbit, the times of disappearance and reemergence did change. In fact, Roemer predicted that an eclipse in November 1679 would be 10 minutes behind schedule. When he was proved right, the Royal Observatory remained flabbergasted. Why was this the case?

The figure above, from Roemer’s notes, highlights Earth’s orbit (HGFEKL) around the sun (A). Io’s orbit eclipses (DC) are shown, defined by Jupiter’s (B) shadow. For a period of time, at point H, one cannot observe all eclipses of Io, since Jupiter blocks the path of light. However, when Earth is at positions L or K, one can observe the disappearances of Io, while at positions G and F, one can observe the reemergences of Io. Even if you didn’t follow any of that, note simply that while Io’s orbit does not change, the Earth’s position relative to Jupiter/Io does change as it orbits the Sun. One observing Io’s eclipse at point L or G is closer to Jupiter than one observing an eclipse when the Earth is at point K or F. If light does not travel instantaneously, observations at points K and F will lag, because light takes a bit longer to reach Earth from Io.

In order to calculate the speed of light from this observation, Roemer needed information from his colleagues on the distances from the Earth to the Sun. Additionally, there are other complications. Nonetheless, using the measured distance from the Earth to the Sun at the time (taking advantage of parallax), Roemer announced that the speed of light was approximately 220,000 km/s. While more than 25% lower than the actual speed of light, it remains astounding that one could estimate this speed using nothing but a telescope, a moon, and a notebook.

Giovanni Cassini, a contemporary of Roemer, was not convinced at first. However, Isaac Newton noted the following in his Principia, from Roemer’s observations:

“For it is now certain from the phenomena of Jupiter’s satellites, confirmed by the observations of different astronomers, that light is propagated in succession and requires about seven or eight minutes to travel from the sun to the earth.” 

In other words, philosophers now began to accept that light travels in a finite amount of time.

Over the course of many years, others continued to estimate the speed of light using creative methods. James Bradley, in 1728, noticed that the positions of stars changed during rainfall, using these observations to estimate the speed of light with great accuracy (Bradley: 185,000 miles/second; Speed of Light: 186,282 miles/second). In 1850 in France, Fizeau and Foccault designed a time-of-flight apparatus like the one described in the opening paragraph. As opposed to using modern technology, the apparatus uses a rotating wheel to simulate blips of light. With a wheel of one hundred teeth moving at one hundred rotations per second, the speed of light could be calculated to within the accuracy of Bradley’s observations. Albert Michelson, in the 1870s, repeated the measurements on a larger scale, again with a series of mirrors.

What can be gleaned from this story is a powerful lesson. At times, the simplest observations can result in the most compelling findings. What it required in this case was careful note-taking and a bit of intellect. Even without those, simple observation cannot be understated.

A couple weeks ago, Felix Baumgartner set the record for the longest free fall, previously held by Captain Joseph Kittinger. To be more specific, Baumgartner dove from 39 kilometers in 2012, and Kittinger dove from 31 kilometers in 1960. However, Baumgartner traveled faster, with his total dive taking 17 seconds less than Kittinger’s. This was the true feat, as he reached 1,342 kilometers per hour, thus breaking the sound barrier.

At 39 kilometers, how high up was Baumgartner? This is 8% short of a full marathon, the distance of 4.4 Mount Everests stacked atop one another, and 3.6 times the greatest depth of the ocean. At this height, the temperature is only -25.6 C, the pressure is only 1/3 of that on the ground, and the effect of gravity was still 98-99% of what it would be at sea level. His maximum speed of 1.1 Mach (the speed of sound in dry air at 15 C and 1 atm) was just 13% short of the maximum speed of the X-1 rocket plane. In other words, he was moving very fast from an extreme height.

If you want to consider how this might affect a human, you must consider not speed, but acceleration. With the Stratos jump, it took Baumgartner 42 seconds to reach his terminal velocity. This is pretty quick. How did he do it? When in free-fall, we can consider two forces, drag and gravity. I noted above that the effects of gravity were reduced, but not by much, at this elevation (to 9.7 meters per second-squared, to be specific). The force of drag acting on a body is dependent upon its velocity. So, as you fall and gain speed, the effects of drag become greater. Eventually, drag force becomes great enough such that you cannot accelerate any further from gravity, and you reach terminal velocity. This is usually 25 m/s for most objects. However, Baumgartner was in the stratosphere. The air pressure, as I mentioned above, was only 0.33 that of what is at sea level. With such low air density, the effects of drag are reduced. The force of acceleration, relatively unchanged, provides a strong downward force (toward the Earth). This leads to a very high terminal velocity, one that can break the sound barrier. (I should note that the speed of sound is reduced at high elevation because sound propagation is dependent upon the density of the medium through which the wave travels.)

What does this have to do with acceleration, then? We know his speed was significant, and we know he did it quickly. According to the argument above, the only real force leading to acceleration is gravity, and we have an upper limit of 9.7-9.8 m/s^2. The numbers reported agree with this, with an average acceleration of about 8.8 m/s^2. Is this fast enough to hurt a human? Again, what matters is accelerationnot speed. This discussion started in WWI, where pilots reported vision problems with high-acceleration maneuvers. Today, we see it ranging from the design of rockets by NASA to safety reports in four-door sedans. If we look back at other human endeavors, we see a story of high acceleration. The now-retired shuttle missions accelerated astronauts to three times the force of gravity. The Apollo missions entered the atmosphere at six times the force of gravity upon their return home. However, the highest reported cases were with the Daisy Decelerator in the 1970s. Major Beeding was placed in this capsule, and he was decelerated at 83-times the force of gravity for approximately 0.04 seconds. He survived, emerging with a short period of shock and a bruised back. While I wouldn’t claim that most humans could survive such an extreme scenario, this demonstrates the importance of acceleration over total speed. With Baumgartner traveling at <1g (less than one times the force of gravity), this provided little danger.

Whether or not this provided a danger to the diver, it was an exciting watch. Whether or not it advanced our knowledge of the stratosphere, it gave me a fun topic for this blog post.