Some time ago, I wrote about anomalous redshifts ending that to a mention of *intrinsic redshift component*. I followed that post by explaining how redshift components are generally calculated. There I gave the basic equation used for that:

1 + z_{M} = (1 + z_{C}) * (1 + z_{K}) [eq. 1]

where z_{M} is the measured redshift of the object in question, z_{C} is the cosmological redshift component, and z_{K} is kinematical redshift component, which is my own term, it is generally called as redshift caused by peculiar velocity of the object.

In discordant redshift systems, the kinematical redshift component tends to be too large to be explained by peculiar velocity in traditional thinking. Too large is rather subjective concept but there is no generally accepted limit for that. 1000 km/s is largely used but in tight galaxy cluster conditions even few thousand km/s might be accepted. When the kinematical redshift component is too large, we need another redshift component, the intrinsic redshift component. The “intrinsic” comes from the assumption that the additional redshift comes from within the object in question. I’m not particularly keen on this term because it assumes a rather specific cause for the additional redshift, but as it is commonly used, I’ll do so too. Here I will deal only with the mechanics of the addition of intrinsic redshift component to the equations described in above mentioned posts, I will not discuss the possible causes of the component.

This additional redshift component is added to the equation 1 like this:

1 + z_{M} = (1 + z_{C}) * (1 + z_{K})* (1 + z_{i}) [eq. 2]

where z_{i} is the intrinsic redshift component. Problem we usually are facing is that we don’t know how large the kinematical redshift component is, so we cannot use this equation as such. As a solution, common method is to assume that there is no kinematical redshift component, so the equation 2 reduces to:

1 + z_{M} = (1 + z_{C}) * (1 + z_{i}) [eq. 3]

Basically it is the same equation as equation 1 but we just call the second redshift component by different name. It is important to remember that the assumption of no kinematical redshift is not necessarily a good assumption. When we are dealing with large redshift differences between objects, then the assumption does not distort our results much, but when the redshift differences are small, the unknown kinematical redshift component introduces a large uncertainty to our calculations, as we shall see later.

Let us now calculate the redshift components of an example system. We’ll use the famous NGC 7603 system. There are four relevant objects in this system; NGC 7603, NGC 7603B, and the two high redshift galaxies ([LG2002] 3 and [LG2002] 2) within the bridge area. Redshift of NGC 7603 (according to NED) is z = 0.029524, redshift of NGC 7603B is z = 0.055742, redshift of [LG2002] 3 is z = 0.391000, and redshift of [LG2002] 2 is z = 0.243000.

Let us now use equation 3 to calculate the intrinsic redshift of NGC 7603B. To calculate that we need to know the cosmological redshift of NGC 7603B. For that, we assume that it is at same physical distance from us than NGC 7603, and we also assume that NGC 7603 itself doesn’t have intrinsic redshift or peculiar velocity (lot of assumptions here). With these assumptions, we can use the redshift of NGC 7603 as cosmological redshift. Before we can calculate the values, we need to solve the equation 3 for intrinsic redshift like this:

1 + z_{M} = (1 + z_{C}) * (1 + z_{i}) | /(1 + z_{C})

=> 1 + z_{i} = (1 + z_{M})/(1 + z_{C}) | -1

=> z_{i} = (1 + z_{M})/(1 + z_{C}) – 1

Now we can insert the values and calculate:

z_{i} = (1 + 0.055742)/(1 + 0.029524) – 1 = 0.025466138

Using relativistic Doppler redshift equation to convert that to velocity yields 7537 km/s. We can use exactly the same equation for the other two objects, and we find out that intrinsic redshift of [LG2002] 3 is z_{i} = 0.35111 (87600 km/s) and intrinsic redshift of [LG2002] 2 is z_{i} = 0.207354 (55800 km/s).

What if NGC 7603B would have peculiar velocity? Let us assume that it does have a peculiar velocity of 1000 km/s. That corresponds to z_{K} = 0.00334. Now we have to use equation 2. Solving it for intrinsic redshift gives:

z_{i} = (1 + z_{M})/{(1 + z_{C}) * (1 + z_{K})} – 1

Then we’ll put the numbers in:

z_{i} = (1 + 0.055742)/{(1 + 0.029524) * (1 + 0.00334)} – 1 = 0.022052

When large peculiar velocity was involved, the intrinsic redshift changed from 0.025466138 to 0.022052, a change of 13 %. From this we can see that high precision studies relating to this are very difficult, remembering also the many assumptions that were needed to come this far. However, because of the large redshift difference between NGC 7603 and [LG2002] 3, a similar peculiar velocity in [LG2002] 3 would change the intrinsic redshift only by 1 %, so this approach can be used to cases that have high redshift difference between the object in question and the main galaxy.

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