Distribution of Mass
An interesting feature of the Orbital Method is that we employed the same equation used to calculate the mass of the Sun
to calculate the mass of a galaxy within a certain radius. This leads to the following question: By using this equation for a galaxy, aren't we implicitly assuming that the mass of a galaxy is concentrated at its centre, like the mass of the Solar System?
The answer to this question is that we do not make this assumption. The equation
applies to any spherically symmetric distribution of mass, including both spread out and localized distributions. Thus, it applies to a galaxy in which mass is spread over a large volume.
The reason the equation applies to any spherically symmetric distribution of mass is that such a distribution produces a gravitational field outside of its radius that is identical to one that would have been produced if all of the mass had been located at the centre of the distribution. Thus, the outside gravitational field for such a mass is independent of the distribution of mass and only depends on the total mass M contained within the distribution.
We implicitly use this fact when calculating the gravitational attraction between any two extended bodies (e.g., the Sun and Earth) using the centre-to-centre distance as the distance r in Newton's law of universal gravitation (e.g., between the Sun and Earth).
A second issue related to the Orbital Method and mass distribution is why the Orbital Method only measures the mass of a galaxy within a certain radius? That is, why doesn't it also measure the mass of the galaxy farther out? An answer to this question can be found in Appendix C.
Doppler Effect
The basic principle underlying how physicists measure the speed of a star using the Doppler effect is the same as that for measuring the speed of an ambulance from the Doppler shift of its siren, as in Figure 10. In the latter case, we measure the siren's apparent frequency f and calculate its speed v (towards or away from us) from knowing its actual frequency f0 and using the formula
where vm is the speed of sound in air and the ± sign is minus when the ambulance is moving towards us and plus when it is moving away.
Physicists use a similar equation to find the speeds of stars within galaxies, but there are two differences. First, physicists do not directly measure the frequency shift of light waves emitted by a star. Instead, they measure the frequency shift of radio waves emitted by hydrogen gas orbiting at the same speed as the star. This allows them to calculate the speed of the gas and thus the speed of the star.
The second difference is that physicists use a slightly different equation from Equation 3.1 because electromagnetic radiation travels at the speed of light and must be handled using Einstein's theory of relativity, as detailed in Appendix D.
Note that in the animation of Doppler shifted stars in the video we have greatly exaggerated the colour changes associated with the Doppler effect in order to highlight them.
Doppler Effect and Orientations of Galaxies
It is important to note that the Doppler effect only measures the speed of a star towards or away from us. It does not measure any sideways or transverse motion with respect to Earth.
If a galaxy is oriented so that we can only see the edge of its orbital plane (an edge-on galaxy) some of its stars are moving directly towards us and some are moving directly away from us, as in Figure 11. We can measure the orbital speeds of these stars using the Doppler effect.
However, if a galaxy's orbital plane faces Earth directly (a face-on galaxy) then all of its stars move with purely transverse velocities relative to Earth, as in Figure 12. In this case, both the stars and hydrogen gas do not exhibit any frequency shift and we cannot measure their speeds using the Doppler effect. Only a small fraction of galaxies are face-on.
Most galaxies lie somewhere in between the two extremes of being face-on or edge-on. They have orbital planes tilted at some angle towards us. So, physicists can use the Doppler effect to measure a component of the velocities of their stars. Physicists then find the total velocities by determining the angle of the tilt and adjusting the measured Doppler speeds accordingly.
Where did the data for Triangulum come from?
The speed (v = 123 km/s) and radius (r = 4.0 x 1020 m) values for Triangulum used in the video came from recent measurements of this galaxy. The complete set of data is in Appendix F.
Measuring the Orbital Radius of a Star
Another question that arises from this chapter of the video is how physicists measure the radius of the orbit of a star in a distant galaxy. Appendix E answers this question.
