Does browsing /sci/ make me smarter than everyone else?
>>8182142
It makes you just smart enough to qualify to be my sandwich maker.
Can you prove the formula for volume of sphere
If yes then you're probably a fucking genius
>>8182142
nah, not just browsing... you have to fight with someone, and believe that your right
>>8182148
i did this when i was 8, what's so hard about it?
>>8182142
Volume of a sphere? Let's go one step further...
Let’s determine the gravitational energy of a galaxy of mass m at the edge of a region with radius R:
Et = ½ mv^2 - GMm/r
We can multiply both sides by a positive number without changing anything and divide by m, as m is also a positive number, so we’ll multiply by 2:
2/m Et = v^2 - 2GM/r
As the velocity of a galaxy at a given distance is proportional to the distance, and the constant of proportionality is H, then v^2 is equal to H^2 R^2.
Now, what is the total mass of a sphere with radius R?
Well the volume of a sphere with radius R is 4/3 π r^3 and as the mass of a sphere with radius R is equal to volume times the average density, then we can write the equation as:
2/m Et = H^2 R^2 - 2G [(4π/3) ρR^3] /R
In order to further simplify, as R2 is a positive quantity we can divide by it, which leaves us with:
2/mR^2 Et = H^2 - 8 πG/3 ρ
Now, let’s refer to the constants 2, m and Et as minus kappa, so that 2/m Et is represented by –k:
-k/R^2 = H^2 - 8 πG/3 ρ
This is Einstein’s equation for an expanding universe, derived with all the factors of π and 3, which will ultimately determine the evolution of the universe.
Minus kappa is a constant related to the total energy of the galaxy, which in general relativity is the curvature of the universe.
>>8182506
What book is this from?
>>8182148
Formula of a sphere := 3
Formula of a sphere = 3
voila
>>8182666
A short book that I wrote on the history of cosmology.
Would you like to know more?
8182737 yes pls
>>8182747
Ok let me just get my computer (I'm on my phone).
The nature of the expansion of the universe is dependent on its curvature.
Therefore, if the total energy of the galaxy in question is positive then kappa is negative, which indicates that it will continue expanding infinitely.
If the total energy is negative then kappa is positive, which indicates that it will collapse.
However, if the final value is zero then it will continue expanding, slowing down but never quite stopping, so that at ∞ v = 0.
The same is true of the universe as a whole.
Now, in order to determine the total energy of the universe we must determine the density of the universe and measure it against the critical density.
If the density of the universe is equal to the critical density then the universe is flat.
If it is greater than the critical density then the universe is closed.
If it is less than the critical density then the universe is open.
The critical density of the universe will be defined by ρc.
ρ > ρc = closed universe
ρ < ρc = open universe
ρ = ρc = flat universe
ρ/ ρc = Ω
Ω = >1 = closed universe
Ω = <1 = open universe
Ω = 1 = flat universe
Ω = >1 = collapse
Ω = <1 = expand forever
Ω = 1 = expand forever, slowing down but never quite stopping so that at ∞ v = 0
A flat universe for example, would therefore be defined as:
0 = H2 - 8 πG/3 ρc
>>8182761
How do we determine the density of our universe?
Well, Kepler showed that the square of the velocity of planets from the sun is inversely proportional to the distance from the sun:
v^2 = 1/R
Newton showed that the force of gravity is equal to the gravitational constant times the masses of the objects in question divided by the distance between their two centres:
F = GM1M2/r^2
Cavendish showed that it was possible to calculate G, which modern physicists have calculated to be:
6.67428 × 10^−11N-m^2/kg^2
The existence of a universal force of gravity allows us to state that:
v^2 = GM/r
Since we have a value for the strength of gravity, then the velocity of the moon around the earth can be measured and compared to its distance from the earth, subsequently allowing us to calculate the mass of the earth, which is:
5.972 × 10^24 kg
The same can be done for the sun, however only to an accuracy of 1 part in 1000, due to the difficulty in measuring such a small effect size as gravity.
In fact, gravity is tremendously weak.
If you were to fall from the top of a building, you wouldn’t even make a dent in pavement below and this is due to the fact that gravity is so much weaker than electromagnetism.
Gravity may have accelerated you 50m towards the earth, however the electromagnetic forces resulting from electrons in your body interacting with electrons in the concrete would stop you in a fraction of an inch.
Most materials do not stop you in your tracks because they are solid, as in reality they are mostly empty space: it’s the electromagnetic interaction that brings you to a halt.
In fact, electromagnetism is almost 40 orders of magnitude stronger than gravity.
The suns mass has been calculated to be approximately:
1.989 × 10^30 kg
>>8182773
We can measure the mass of a galaxy similarly.
Our solar system is moving around the edge of our galaxy at a rate of one orbit per 200 million years and we can use this information to measure the mass of our galaxy.
The velocity of the sun around the galaxy is 220km/sec and its distance is 8kpc light years, therefore the initial calculation of the mass comes out as:
10^11 solar masses
This is equivalent to 100 billion stars, which is great as it corresponds to our observations.
However, we want to do better than that so we take a look at objects that are further out.
Now, as we know we are at the edge of our galaxy and velocity should fall off at the square root of the distance from the centre.
After looking at satellite galaxies, molecular clouds and globular clusters that are up to ten times the distance from the centre of the galaxy as we are, we find that the velocity doesn’t fall off, but instead remains constant.
What does this mean?
After all:
V^2 = GM/r
If GM/r is constant and r is ten times bigger, then M must be ten times bigger, which means that there is ten times more mass enclosing our galaxy than expected.
The same is true of other galaxies.
The rotation curve does not fall off in relation to the location of the stars, but instead remains constant.
>Pic related
>>8182777
Therefore, either gravity breaks down although we have no reason to believe it does on the scale of galaxies, or there is ten times more mass enclosing galaxies than we would expect.
This is what we refer to as dark matter.
In fact, there is so much dark matter in the universe (10x all the protons and neutrons in the universe) that it may turn out to be a new type of elementary particle, which means that it is likely to be everywhere, including here on earth.
This means that we can build experiments to try and detect it, although it interacts so weakly that it passes right through the earth, therefore the equipment used will have to be extremely sensitive.
Now, these rotation curves, do they continue to remain flat forever?
We are only able to measure rotation curves up to a certain distance, due to our limited powers of observation, therefore this calculation cannot tell us how much mass is in the universe, it can only tell us the lower limit on that value.
If we want to know the value of Ω then we have to measure mass on larger scales, which we can achieve using gravity or more specifically: gravitational lensing.
We know that gravity curves space-time and subsequently light, therefore it is theoretically possibly for gravity to bend light in such a way that it acts as a lens, thereby magnifying and potentially duplicating an image.
In this respect, an image of a galaxy cluster 5 billion light years away may include multiple images of a galaxy 10 billion light years away, due to gravitational lensing induced by the mass of the cluster.
>pic related
>>8182778
We can use general relativity to discover how much mass is in this system and where it is distributed, in order to produce such an image.
This is achieved by undertaking a mathematical inversion process, which then allows us to produce an image of the mass of the system:
>pic related
The resulting image indicates that there is forty times more mass in the system than would be expected.
We can then use this data and extrapolate to estimate the mass of the universe, due to the uniformity of the universe.
An initial calculation produces a result of:
Ω = 0.30 ± 0.1 (95%)
This value being less than 1 indicates that we are living in an open universe.
However, this estimate is only based on the mass around clusters of galaxies.
>>8182781
However, this estimate is only based on the mass around clusters of galaxies.
How do we measure the total mass of the universe?
We do so by measuring the geometry of the universe, which involves finding a triangle.
On a flat plane the sum of the angles of a triangle is 180 degrees, however on surface with positive curvature the sum is greater than 180 degrees, just as one with negative curvature produces a triangle with a sum of less than 180 degrees.
>pic related
If we can find a big enough triangle, then we can measure the curvature of the universe.
>>8182786
The largest image we have of our universe comes in the form of the cosmic microwave background, which is essentially a baby picture of our universe at the tender age of 380,000 years old.
>pic related
This is an image of the cosmic microwave background.
It is incredibly uniform, however not completely as it features hot and cold spots.
The hot spots are approximately 1/1000th of a degree hotter than the average and the cold spots are 1/1000th of a degree colder.
This slight variation is what allowed for the creation of matter; these are the primordial lumps, created at the beginning of time, which went on to become galaxies, stars, planets and ultimately us.
>>8182791
This diagram features the surface of the cosmic microwave background at a distance of one degree in angular size, which corresponds to around 380,000 light years in size:
>pic related
The fact that the CMB is so uniform is very puzzling.
An angular distance of one degree is roughly equal to a lump that is 380,000 lightyears across, which means that, as the universe itself was only 380,000 years old, light from one particular lump would not have had enough time to reach another and, subsequently, information would not have been able to be communicated between distant regions of the universe.
Therefore, there is no reason to expect the CMB to be so uniform.
This implies that some form of order preceded creation of the CMB.
In relation to our search for a method for measuring the curvature of the universe, we only need to take a look at the size of the primordial lumps present in the CMB.
This is because the largest lumps that could possibly form would have been 380,000 light years across, as, due to the age of the universe, if they were any larger gravity would not have been able to act on them and therefore they would not have been able to collapse.
Now, the apparent angular distance of a lump 380,000 light years across (1 degree), is dependent on the curvature of the universe.
In an open universe, light rays would bend outward as time reverses, therefore the lumps would appear smaller than it actually is, such as 0.5 degrees.
Likewise, in a closed universe light rays would bend inwards as time reverses, therefore the lumps would look larger, say 2 degrees.
However, in a flat universe light rays travel in a straight line and therefore the lumps would appear to be approximately 1 degree.
Therefore, all one has to do is simulate universes featuring lumps of 0.5, 1 and 2 degrees respectively, and then compare them with the image of the CMB.
>>8182799
This is exactly what physicists have done and the results are extraordinary.
>pic related
This image of the CMB compared with simulated universes featuring lumps of varying sizes, allows us to conclude that we live in a flat universe; the lumps appear to be 1 degree in angular distance.
However, all the mass in the universe amounts to 0.3 as shown earlier, therefore both the visible matter and dark matter combined only amount to 30% of the mass required for a flat universe.
Therefore, we have to ask: where’s the other 70% coming from?
The odd thing is that it seems to be situated in empty space.
That is to say, there’s energy where there is nothing; most of the energy in the universe resides where there is nothing.
If you take a region of space and remove all the particles, the radiation and just everything, then it will still weigh something.
It turns out that when you combine quantum mechanics and relativity, empty space is a boiling – bubbling – brew of virtual particles and fields popping in and out of existence on a time scale far too small for us to observe them, and this is happening everywhere.
For example, less than 5% of the mass of a proton can be accounted for by its 3 quarks.
However, although we may not be able to observe them directly, we can measure their effects indirectly.
This is known as dark energy.
>>8182142
My IQ since browsing sci
>>8182804
A line segment without a basis? Cool.
>>8182142
>>8182803
Should I continue?
>>8182808
That just further proves his iq degradation
>>8182808
The chart makes even more sense now, doesn't it?
>>8182142
I wouldn't be surprised if the average iq on /sci/ was below 100...
>>8182148
>Freshman calculus question
>Genius
le kek
>>8183194
ya that's my bad
>>8182148
nigga i can fucking use gauss and stokes, get real you little shit
>>8182148
mfw this is a triple integrals question
>>8182506
good thing that the universe isn't an sphere
>>8183250
How? Just use a volume of revolution and you're done.
>>8183309
It's much easier using spherical coordinates. You don't even have to think.
>>8182803
I laughed at the pictures name
i enjoyed this anon, where i can read more
>>8183322
I'll make a new thread tomorrow and post this again and more.
>>8183307
We use a sphere as it is essentially the set of points that are all at the same distance r from a given point, in three-dimensional space.
We can then measure the curvature of space and discern what shape the universe actually is.
Protip: it's flat.
>>8182812
Do it
my mind is blown :)
>>8182812
Pls
>>8182737
I would
