In an earlier post, I briefly described the redshift components of an extragalactic object. It turns out that generally an extragalactic object has two redshift components; kinematical and cosmological. In this post I’m going to describe how the different components are handled, when calculating things relating to redshifts of galaxies and their associated objects. When combining different redshift components, they are not simply added together. Redshift is a curious quantity in that sense. Let us consider a simple example of one galaxy having a cosmological and kinematical redshift components.

The galaxy we observe has a peculiar velocity with respect to a reference frame that is non-moving frame local to the galaxy. For example, the Earth moves around the Sun, Sun is in motion around the Milky Way center, and the Milky Way is in motion with respect to Local Group, and so on. All this makes the Earth to have a peculiar velocity with respect to non-moving space at location of the Earth. So, when the galaxy we are observing emits light, the first thing the light experiences is the kinematical redshift that occurs between the galaxy’s rest frame and the galaxy’s local non-moving reference frame. After that, the light starts moving through space towards us. In the space it encounters something that causes redshift also. That is the cosmological redshift component. In Big Bang cosmology, cosmological redshift component is caused by expanding space. Many alternative cosmologies have different explanations.

In the receiving end, there is the Earth’s peculiar velocity that also causes redshifting, but that is usually not kept in these analyses. Instead, the measured redshift is corrected for the Earth’s motion. Usually redshifts have been given in heliocentric reference frame, but galactocentric and Local Group -centric redshifts have also been used. Lately, the redshifts have been increasingly given in CMBR reference frame. However, it is quite common to just use the heliocentric reference frame. That contains some error in the redshift, but when dealing with objects close to each other, such as a galaxy and an object near it, then the correction due to Earth’s complex motion is approximately the same for both objects, so in practice there’s no effect for the Earth’s motion to the results in that kind of situations. Therefore it is OK for us to use heliocentric redshifts here, which is still most common frame where redshifts are given.

Now, let us derive the equation that can be used to calculate the redshift components. This derivation is given in Davis *et al.* (2003). We start with the basic redshift equation:

1 + z = Lobs/Lem [eq. 1]

where z is the measured redshift (the heliocentric redshift), Lobs is the wavelength of the light we observe, and Lem is the wavelength of the light when it was emitted. The peculiar velocity of the galaxy causes the wavelength to change. We will call this wavelength Lnm, the wavelength of the light in non-moving reference frame local to the galaxy. We will do a mathematical trick next. We will multiply the basic redshift equation by one (which doesn’t change the result so it’s a valid thing to do), but we just write 1 = Lnm/Lnm. So we have:

1 + z = Lobs/Lem * Lnm/Lnm

Then we move Lnm’s a bit:

1 + z = Lobs/Lnm * Lnm/Lem [eq. 2]

This equation now contains both redshift components. I’ll demonstrate that next. If we would be in the non-moving reference frame local to the galaxy, we would observe the wavelength of the galaxy to be Lobs = Lnm, and putting that to equation 2 we would get kinematical redshift zK:

1 + zK = Lnm/Lnm * Lnm/Lem = Lnm/Lem

=> 1 + zK = Lnm/Lem [eq. 3]

If the galaxy wouldn’t have any peculiar velocity, there wouldn’t be any kinematical redshift component, so wavelength in the non-moving reference frame would be the same as emitted wavelength, i.e. Lnm = Lem. Again from equation 2 we would then get cosmological redshift component zC:

1 + zC = Lobs/Lnm * Lnm/Lnm

1 + zC = Lobs/Lnm [eq. 4]

Now we have seen that equation 2 contains both redshift components, but it still has wavelengths in it. We would want redshifts there explicitly. We can do it by replacing wavelengths in equation 2 with redshift components from equations 3 and 4, and it becomes:

1 + zM = (1 + zC) * (1 + zK) [eq. 5]

Where zM is the measured redshift of an object. So, as you can see, when adding up redshift components, it is not done simply by zM = zC + zK. Let’s try the equation 5 with an example galaxy. Let us assume that we have measured a redshift of zM = 0.01 to a galaxy, but let’s also assume that the galaxy has a peculiar velocity of 500 km/s away from us. First thing we need to do is to convert the peculiar velocity to redshift. There is a simple equation for that: z = v/c. Speed of light, c, is 299792.458 km/s, so our kinematical redshift component is:

zK = 500 km/s / (299792.458 km/s) = 0.001668

Now we know the kinematical redshift component and the measured redshift, so only unknown in equation 5 is the cosmological redshift component. Let’s calculate it:

1 + zM = (1 + zC) * (1 + zK)

=> zC = (1 + zM)/(1 + zK) – 1 = (1 + 0.01)/(1 + 0.001668) – 1 = 0.008318

In practice it is very difficult to determine the redshift components. The kinematical and cosmological redshift components look the same in the spectra of objects. If we are only observing a single galaxy, it is practically impossible to determine the components. Situation gets better if we have two objects physically associated with each other, pair of interacting galaxies for example. Easiest situation is when there is a large galaxy interacting with a clearly smaller galaxy, and we wish to calculate the redshift components of the smaller galaxy. In that situation, we can assume that the smaller galaxy follows the motion of the large galaxy similarily as the Earth follows the motion of the Sun around the Milky Way nucleus, even if the Earth also has its own motion around the Sun. So, the smaller galaxy has its own motion around the large galaxy, but it still follows the path of the large galaxy. This means we can calculate the system so that we ignore any peculiar velocity the large galaxy has because the smaller galaxy also has the same peculiar velocity in addition to its own motion around the large galaxy. Or, to put in other words, we can calculate the redshift component that the smaller galaxy has in addition to large galaxy redshift components. So, the thorough equation for the redshift components of the smaller galaxy would be:

1 + zS = (1 + zC) * (1 + zKlarge) * (1 + zKsmall)

Where zS is the measured redshift of the smaller galaxy, zKlarge is the kinematical redshift component, or the peculiar velocity of the large galaxy, and zKsmall is the additional peculiar velocity of the smaller galaxy. zKlarge is included here because we assume that the smaller galaxy follows the motion of the large galaxy, as discussed above. Also following the discussion above, especially the equation 5, we can now just put all large galaxy components under one heading to derive an equation that we can use to calculate the kinematical redshift component of the smaller galaxy:

1 + zS = (1 + zL) * (1 + zKsmall) [eq. 6]

Where zL is the combined redshift components of the large galaxy, i.e. the measured redshift of the large galaxy. Note that equation 6 is exactly the same as equation 5, only the names of the components have changed. How can we use this equation then? In practice, we should have redshifts measured both for the large and the smaller galaxy. The measured redshift of the large galaxy is the zL, and the measured redshift of the smaller galaxy is the zM. So, let’s assume that we have a large galaxy with measured redshift of zL = 0.010, and we have a smaller galaxy right beside the large one with measured redshift of zS = 0.011. We assume that these galaxies are interacting (sometimes there are some visible signs suggesting that, but not always), and we therefore decide we can use equation 6. First, we’ll solve the equation 6 for zKsmall, because that is what we want to calculate:

zKsmall = (1 + zS)/(1 + zL) – 1 [eq. 7]

And then we will just put the numbers in:

zKsmall = (1 + 0.011)/(1 + 0.010) – 1 = 0.00099

Peculiar velocity of the smaller galaxy with reference to the large galaxy is then:

v = cz = 299792.458 * 0.00099 = 297 km/s

The kinematical redshift component of the smaller galaxy is the only redshift component that we can calculate from the system when we only know redshifts of both galaxies. In order to determine other components, we need additionally a redshift independent distance measurement for the galaxy pair. If we know the distance between us and the galaxy pair, we can calculate how big kinematical redshift component the large galaxy has, and the size of the cosmological redshift component. However, problem here is that distance measurements to galaxies are still quite inaccurate, so this calculation will have lot of uncertainty.

Let’s take NGC 289 as an example. NGC 289 has a redshift of c * zL = 1629 km/s (zL = 0.005434) in NED, and a distance of 23.4 Mpc. NGC 289 has a companion, LSBG F411-024, which has a redshift of cz = 1510 km/s (zS = 0.005037). If you look at the distance measurement section for NGC 289 in NED, you will notice that there’s two measurements, from which the NED value is derived; 19.4 Mpc and 27.4 Mpc, quite a difference, which reflects the above mentioned inaccuracy in distance measurements.

To convert the measured distance to redshift, we will use Hubble law (there are more distinguished methods to do that, but it will serve the purpose for us in this example). From Hubble law we get (using Hubble constant of 72 (km/s)/Mpc):

vDIST = H0 * d = 72 (km/s)/Mpc * 23.4 Mpc = 1684.8 km/s

If we assume that this is accurate value, it would mean that NGC 289 has a peculiar velocity of:

vKlarge = c * zL – vDIST = 1629 km/s – 1684.8 km/s = -55.8 km/s

Minus sign indicates that the velocity is towards us (redshift is positive when velocity is directed away from us). From the peculiar velocity we then get the kinematical redshift component of the large galaxy:

zKlarge = vKlarge / c = -55.8 km/s / (299792.458 km/s) = -0.000186

Cosmological redshift component is (from equation 5 applied to the large galaxy):

1 + zL = (1 + zC) * (1 + zKlarge)

zC = (1 + zL) / (1 + zKlarge) – 1 = (1 + 0.005434) / (1 + -0.000186) – 1 = 0.00562

We could also have derived the cosmological redshift component directly from the redshift independent distance measurement (if we multiply the above calculated zC by c, we get 1685.2 km/s which is quite close to vDIST calculated above). Finally we will also have to calculate the kinematical redshift component of the companion galaxy using the equation 7:

zKsmall = (1 + zS)/(1 + zL) – 1 = (1 + 0.005037)/(1 + 0.005434) – 1 = -0.000395

vKsmall = c * zKsmall = 299792.458 km/s * -0.000395 = -118 km/s

Kinematical redshift component of the companion galaxy turns out to be negative too, but that was expected because the redshift velocity cz was smaller in the companion galaxy suggesting that it has a peculiar velocity towards us. Notice that the calculated value here is almost exactly the value from the measured cz’s = 1510 – 1629 = 119 km/s, so we could have calculated the kinematical redshift component also from measured cz’s directly by:

zKsmall = c * zS – c * zL / c = (zS – zL) * c / c = zS – zL

Now you might notice that this goes exactly against what I said in the beginning, that redshifts are not simply added together. It is true, it goes against what I said, and the reason is this: the equation v = cz is only an approximation that works quite well for velocities far lower than the speed of light. If we would use more precise calculation methods, we would use the relativistic doppler line of sight equation instead of v = cz:

1 + z = sqrt[(1 + v/c)/(1 – v/c)]

Using that equation, the companion redshift component calculation wouldn’t reduce back to zS – zL anymore, and our world makes sense again. 🙂

### References

Davis *et al.*, 2003, AmJPh, 71, 358, “Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects”

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