# Hubble's law and the age of the universe

Just did a test today (so no results back yet) (year 12 physics) and it annoyed the heck out of me, in part because to answer these questions you had to make preposterous assumptions, assumptions which in hindsight I probably should have just stated as opposed to writing out explanations as to why the answer I expected wasn't the real answer. Note: Part of the tradition of tests (though not exams) at my school (and state, and probably country as well) is that we do get to contest if we believe the marker has marked us unfairly, be it whether the test contradicts itself, or the marker simply counted up the marks wrong.

One question was on relative qualities such as time, length and mass. However to answer to the question only required length contraction, but it turns out that was all you needed despite the question giving you a time dilation formula in the question. So yay.

Another question involved Hubble's law. It gave us a rough graph of redshift (in km/s) versus distance (in Mpc) with various points above and below a line with gradient Hubble's constant (68 km/s Mpc, which I presume is meant to say 68 km/s/Mpc) and a patch in the middle which contained several outliers, was circled and labelled "Virgo's Cluster". The first question asked:

Question: What is the implication of Hubble's constant?

Expected Answer (the answer I expected the question wanted): Hubble's constant is the inverse of the age of the universe! 68 km/s/Mpc = 68 / (3 × 1019) km/s/km = 2.3 × 10-18 and henceforth 1/H = 4.4 × 1017 seconds = 1.4 × 1010 years. So Hubble's law implies that universe is 14 billion years old.

Not the age of the universe! Who the heck told you that? Consider Velocity = Distance × Hubble's Constant, this can be rewritten as:
ds/dt = s × H0 where s is the displacement of the object from us and t is time
∫1/s ds = ∫ H0 dt
ln|s| + c = H0t, c being some unknown constant
t = (ln|s| + c)/H0
Which is NOT H0
Note that this doesn't actually give a satisfactory answer to what is the age of the universe, but it goes to show that you can't just say H0 is. It is preposterous to assume that an object travelling at a speed proportional to it's distance is travelling at a constant speed, to do so goes against the entire point of Hubble's law.

What I want to know: Will I, and more importantly should I get marks for this? The calculation I've done here is in the sort of calculus I barely know, in fact that's the extent of my knowledge (thus far). From the moment I saw a question like this in class I thought it had something to do with the area under the curve, but which curve? What the heck has this little algebraic manipulations done to my poor graph? The reason I ask this is to find out, what is c? My first instinct would just be to set it to 0, but looking at it in context... I don't have a clue what it is.

The last question of the test, still referring to the same graph asked:

Question: Estimate the diameter of the Virgo Cluster

At the time of the test I expected the answer to be along these lines: Calculate the redshift of the highest and lowest redshift star in the virgo cluster and their difference from their expected redshift value using Hubble's constant. This difference is in fact their speed relative to the galaxy their revolving. This gives 2∏r/t = v, however, as I didn't actually go through this calculation, I failed to realise that you couldn't actually do it this way either due to not knowing the revolution time of the galaxy.

At the end of school I expected the answer to be along these lines:
Find the difference between the furthest and closest stars in the cluster. That's its diameter.

What I wrote: Due to the nature of gravity and the fact that an object subtends the same area of an ellipse as it revolves the centre of the galaxy, the fastest stars would be at the centre of the galaxy (such as in the globular cluster right next to the super-massive black hole at the centre of the galaxy) and the slowest would be further out. The slowest star in Virgo was just above the regression line so by measuring it's redshift and distance on the graph and comparing it with the expected value using hubble's constant I found its speed relative to the centre of the galaxy to be about 64 km/s [though 64000 would make more sense... I can't remember exactly if it wasn't either].
Fc = mv2/r
Fg = GM1M2/r2
mv2/r = GM1M2/r2
mv2r2 = GM1M2r
v2r = GM2
r = GM2v2
r = 6.67 × 10-11×642 × M2
= 2.7 × 10-7 × M2
Of course all this was for nought because we don't know the mass of the Virgo cluster, but if we did know, you'd just multiply it by 5.4 × 10-7 and you'd have the diameter of the galaxy.

What I want to know: Do I deserve any marks for the above?

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Oops:
v2r = GM2
r = GM2/v2
r = 6.67 × 10-11 × M2 / 642
= 1.6 × 10-14 × M2
Hence multiply by 3.3 × 10-14
This mistake is restricted to the calculation on this forum.

Dratting edit expiration, sorry about the triple post.
I'm moving this to differential equations.