Notes

Center of Mass


If we want to measure a planet's mass, we need to measure it via motions of its star. A star and a planet will orbit their mutual center of mass, and the position of that center will depend on the masses of the two objects involved in the interaction.

Set up a system that maximizes the size and speed of the "wobble" of the star as the planet orbits it.

Describe this system.

Star Mass
LOW

Planet Mass HIGH
Good job! The more massive the planet is, the closer the center of mass will be to the planet. Reducing the mass of the star also helps shift the center of mass towards the planet. The farther the center of mass is from the center of the star, the more the star will move, visibly, as the planet orbits it.
Doppler effect
Good work! Light that is approaching you will be compressed, resulting in shorter wavelengths, while light that is receding from you will be extended, resulting in longer wavelengths. If you'll recall, different wavelength sizes correspond to different colors. So let's apply colors to these generic photons.
(shorter = bluer wavelengths)

(no color change)

(longer = redder wavelengths)
A lack of a color change, however, doesn't always mean that the object is not moving!


It just means that it is not moving towards you or away from you.
Great job! The higher the Doppler shift, the faster the star is moving. If the star mass and orbital distance of the planet are unchanged, this must mean that the planet has more mass, therefore pulling on the star harder and making it wobble faster.
e've now discovered the following:


- Stars and planets orbit a center of mass


- This causes the star to appear to wobble


- Stars in motion can have their light blueshifted when moving towards us, redshifted when moving away


- This phenomenon is called the Doppler effect


- The strength of the Doppler effect is directly related to the star's velocity, which is determined by the planet's mass


- These color changes are often too small to see


- However, we can observe the Doppler effect on spectral lines, which will not appear at the positions they are supposed to if the object is moving
To determine a planet's mass, we must first convert the Doppler shift to a star velocity, which is easy enough to do:


Star Velocity = (max line shift / rest wavelength) x 300,000,000 m/s

Basically, this means that we take how far the line has shifted from its expected location, divide by the expected location, and multiply by the speed of light.


So this method will work on any spectral line, not just the H-alpha line. You just have to substitute in the correct rest wavelength.
The star's velocity is going to be dependent on the mass of the planet that's tugging on it according to the following relationship:


Earth masses:

mass = (max star velocity2 x orbital radius x star mass)1/2 x 11.177


Jupiter masses:

mass = (max star velocity2 x orbital radius x star mass)1/2 x 0.0352


These are simplified down from laws of conservation of momentum, Kepler's 3rd law, and various conversions to account for different units.


For these equations, you can enter the star velocity in meters per second (m/s), orbital radius in astronomical units (AU), and star mass in solar masses (MS). The constants at the end correct for the units so that you end up with either Jupiter masses (MJ) or Earth masses (ME), depending on which equation you use.
To determine a planet's mass, we must first convert the Doppler shift to a star velocity, which is easy enough to do:


Star Velocity = (max line shift / rest wavelength) x 300,000,000 m/s


The star's velocity is going to be dependent on the mass of the planet that's tugging on it according to the following relationship:


Earth masses:

mass = (max star velocity2 x orbital radius x star mass)1/2 x 11.177


Jupiter masses:

mass = (max star velocity2 x orbital radius x star mass)1/2 x 0.0352


These are simplified down from laws of conservation of momentum, Kepler's 3rd law, and various conversions to account for different units.


For these equations, you can enter the star velocity in meters per second (m/s), orbital radius in astronomical units (AU), and star mass in solar masses (MS). The constants at the end correct for the units so that you end up with either Jupiter masses (MJ) or Earth masses (ME), depending on which equation you use.

rest wavelength = 656.300000 nm


reddest wavelength = 656.300030 nm

bluest wavelength = 656.299970 nm


Use either the reddest OR the bluest wavelength as the maximum reading.


Shift = |rest wavelength - reddest (or bluest) wavelength|

Shift = abs(656.300000 nm - 656.300030 nm)

Shift = 0.000030 nm


Velocity = (0.000030 nm / 656.300000 nm) x 300,000,000 m/s

Velocity = 13.7 m/s
PAAAARAT@2

Star Mass = 1 MS

Star Velocity = 13.7 m/s

Planet Orbital Distance = 1 AU


(in Earth masses)

Mass = ((13.7 m/s)^2 x 1 AU x 1 MS)1/2 x 11.177

Mass = 153.1 ME


(in Jupiter masses)

Mass = ((13.7 m/s)^2 x 1 AU x 1 MS)1/2 x 0.0352

Mass = 0.482 ME
The Doppler shift of spectral lines is strongest at a 90° inclination. That's because during the course of an orbit that we are viewing edge-on, all of the star's velocity is directed at us when the star is moving toward/away from us.

An animation showing the host star swinging towards you as the planet swings behind the star, and then retreating into the distance as the planet swings towards you.25
Conversely, if the system has an inclination of 0° or 180° (face-on), then even if the star is wobbling, none of that velocity is directed at us, and so we would see no Doppler shift. The star could be moving a significant amount, but it is neither moving towards us nor away from us.

An animation showing a top-down view of a planet orbiting its star. Although the star is moving a great amount around the star-planet system's center of mass, the size of the star isn't changing because it is not moving towards or away from you.
And of course, you can have angles in-between. The Doppler effect will be noticeable because at least some of the velocity will be directed towards us. But the effect won't be as strong as when the system is at 90° inclination, where all of the velocity is directed towards us.
When the system has a different inclination, the equations gave us a smaller mass than was true.


This is because at inclinations that are not 90°, some of the star's velocity is not directed towards us. The star looks like it is moving slower than it actually is, and so the planet looks like it has less mass than it actually does.


In this example, we got tricked into thinking that a really large planet was an Earth-like planet, when in reality it wasn't!
We often do not know what the inclination of the system is because we don't have an easy way of measuring it, since we can't see the planets.


As a result, masses for the radial velocity method are reported as "minimum mass".


This means that the planet is at least this massive, but could be more massive if the system has an inclination other than 90°.
To keep things simple for the class, though, when running our calculations, we'll assume that all our planetary systems are inclined at 90°.


Just realize that this is a simplification of reality.