The Doppler shift of the Human Torch

In this post we are going to revisit our flaming superhero the Human Torch. Previously we have looked at the emission spectrum of the Human Torch and derived his surface temperature (link to previous post). Now we are going to inspect what we observe, when the Human Torch is flying. Because the Human Torch is flying we will see a shift in the color of his flames. This is due to the Doppler effect.

The Doppler effect is a phenomenon that occurs when a source that emits waves is moving relative to the observer. Due to the relative movement there will be a shift in the observed frequency. Depending on the direction of the relative movement the observer will notice an increase in frequency or a decrease in frequency. An easy and intuitive explanation for this effect is the folowing. When a source of waves moves towards an observer, each succesive wavefront will need to travel a shorter distance, but each wavefront still moves with the same speed and starts with the same time interval between succesive wavesfronts. It appears that the wavefronts get packed together in front of the source and stretched out behind the source. To visualize this you can look at this link where you can play with some animations.

Now that we know what is going on we can look at the effect on the emitted wavelength by our moving Human Torch. In a previous post we established that the peak-wavelength of the emission spectrum of the Human Torch is around 720 nm. The speed of light is 299 792 458 m/s. From this we get that our emission frequency is equal to:

$f_0 = \frac{c}{\lambda} = \frac{299792458m/s}{720\cdot 10^{-9}m}\approx 4.1638 \cdot 10^{14} Hz = 416.38 THz$

Now it is time to look at the formula for the observed frequency:

$f_{obs} = f_0\left(\frac{c\pm v_{obs}}{c \pm v_{s}}\right)$

Where v_{obs} is the speed of the observer. We will assume the observer is standing still so his speed is 0. The speed of the source v_s will be around 140mph or 62.59 m/s. This is the speed of the Human Torch. For reference this lies between a fast moving car and the speed of a train. The different signs in the equation are distinguished by the direction of movement. We are only interested in the magnitude of change in frequency so both signs will give us the same final answer. For now we will assume that the Human Torch is flying away from us, so we will take the positive sign. Our observed frequency becomes:

$f_{obs} = f_0\frac{c}{c+v_s} \approx 4.1638 \cdot 10^{14} Hz$

$\lambda_{obs} = \frac{c}{f_{obs}} \approx 720.0001 nm$

The shift in frequency is so small it gives no perceptible difference in color. The Human Torch is simply not fast enough. But now we can reverse the question: How fast does he need to go to appear blue or even green? First of all we need the frequencies of blue and green light. The wavelength of green light is around 520 nm and for blue light around 470 nm. So converting to frequencies we get:

$f_g = \frac{c}{\lambda_g} = 5.7652\cdot 10^{14} Hz$

$f_b =\frac{c}{\lambda_b} = 6.3786\cdot 10^{14} Hz$

These frequencies are higher than our emission frequency so to get these the Human Torch will need to move towards us, so we need to pick the minus sign this time. To find the speed of the Human Torch we can rearrange our Doppler formula.

$f_{obs} = f_0\left(\frac{c}{c - v_{s}}\right)$

$v_s = c\left(1-\frac{f_0}{f_{obs}}\right)$

Now we can find the speed needed for the Human Torch to appear green or blue by using our values for the emission frequency and observing frequency.

$v_g = c\left(1-\frac{4.1638 \cdot 10^{14} Hz}{5.7652\cdot 10^{14} Hz}\right) \approx 0.278c \approx 8.327\cdot 10^7 m/s$

$v_b = c\left(1-\frac{4.1638 \cdot 10^{14} Hz}{6.3786\cdot 10^{14} Hz}\right) \approx 0.347c \approx 1.041\cdot 10^8 m/s$

So the Human Torch needs to move at 28% of the light speed towards us to appear green and at 35% of the speed of light to appear blue. This is incredibly fast. These speeds are in the same order of magnitude as the escape velocity of a neutron star and the speed of light in diamond. So in conclusion the Human Torch is fast but not fast enough to change his color drastically while flying.

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