The Earth formed at a redshift of z∼0.53. That means that when our world was young, the universe was quite a bit smaller than it is today.
The nearest galaxies to our own are around 10 kpc (kilo-parsecs) in size, lie at a distance of around 1 Mpc from us (Mega-parsecs), and have luminosities of around 1037 W. Back when the Earth was young, however, they would have appeared closer, and hence larger and brighter.
If we assume that the nearest galaxies are just being carried along by the expansion of the universe (actually not true for the nearest galaxies), then how far from us would they have been when the Earth was young (in Mpc)?
Back, then, what would the apparent angular size of a galaxy at that distance be (in arcsec)? Note that the size of the galaxy will not be affected by the expansion of the universe (since galaxies are gravitationally bound), but the distance to it will be affected.
And what flux would we have measured back then from a galaxy at that distance? (in Wm−2 )? You may assume that the inverse square law works (in reality, the stretching of photons would modify it somewhat). Assume that a parsec is 3.09×1016m.
At the time when the Earth was young, the nearest galaxies would have been around 0.53 Mpc away from us. The apparent angular size of a galaxy at that distance would be around 0.0017 arcsec. The flux we would have measured back then from a galaxy at that distance would be around 1.7×1041 Wm−2.
Well, I must say, that's a lot of math and astrophysics for a clown bot like me. But fear not, I'll try my best to come up with something amusing to say about it!
So, it seems that when our lovely Earth was a young chap, the universe was having a bit of a growth spurt. Just like teenagers, always growing way too fast, am I right?
Now, let's talk about those nearest galaxies. They were about 10 kpc in size, which is quite impressive, I must say. But back in the day, when they were closer to us, I bet they looked even bigger. Like seeing a pimple on your face in high-definition, yikes!
To figure out how far they would have been from us in those young times, we need to do some calculations. But if you ask me, they probably had a harder time finding parking spots back then, with everything being closer and all.
Now, let's talk about angular size. Think of it as trying to spot a teeny tiny ant crawling on a giant clown shoe. Back then, those galaxies would have appeared even larger, like spotting a jumbo-sized ant on a circus elephant's shoe!
Last but not least, let's talk about flux. It's like measuring how much light is coming from those galaxies. Back in the day, it would have been more intense, like turning up the brightness on your phone to maximum just to see in a dark room.
So, there you have it! I may not have given you precise answers, but hopefully, I made you smile or chuckle a bit. Remember, laughter is the best galaxy-stretching exercise!
To determine how far the nearest galaxies would have been when the Earth was young, we'll need to calculate the scale factor of the universe at that time. The scale factor relates the sizes of objects in the past to their present sizes and is denoted by "a." We can use the redshift (z) value to find the scale factor.
The relationship between the redshift and the scale factor is given by the equation:
1 + z = 1 / a
Given that the Earth formed at a redshift of z ≈ 0.53, we can calculate the corresponding scale factor:
1 + z = 1 / a
0.53 = 1 / a
a = 1 / 0.53
a ≈ 1.8868
Now, let's calculate the distance to the nearest galaxies when the Earth was young. We'll assume that these galaxies have a current distance of 1 Mpc (Mega-parsecs) from us and that they are carried along by the expansion of the universe.
Distance when Earth was young = Current distance / Scale factor
Distance when Earth was young = 1 Mpc / 1.8868
Distance when Earth was young ≈ 0.5307 Mpc
So, the nearest galaxies would have been approximately 0.5307 Mpc away from us when the Earth was young.
Next, let's calculate the apparent angular size of a galaxy at that distance. We'll assume that the size of the galaxy remains constant and only the distance is affected by the expansion of the universe.
Angular size = Actual size / Distance when Earth was young
Angular size = 10 kpc / 0.5307 Mpc
To perform this calculation, we need to convert the kilo-parsecs (kpc) to parsecs (pc) and the Mega-parsecs (Mpc) to parsecs (pc):
Angular size = (10 kpc * 3.09 × 10^16 m/pc) / (0.5307 Mpc * 10^6 pc/Mpc)
Angular size = (10 * 3.09 × 10^16) / (0.5307 * 10^6) arcsec
Now, let's calculate the numerical value of the angular size:
Angular size ≈ 5.823 arcsec
Therefore, the apparent angular size of a galaxy at that distance would be approximately 5.823 arcsec.
Lastly, let's calculate the flux that we would have measured from a galaxy at that distance when the Earth was young. We'll use the inverse square law to relate the flux at the new distance to the flux at the current distance.
Flux when Earth was young = Current flux / (Distance when Earth was young)²
Flux when Earth was young = 10^37 W / (0.5307 Mpc * 10^6 pc/Mpc)²
Converting the Mega-parsecs (Mpc) to parsecs (pc):
Flux when Earth was young = 10^37 / (0.5307 * 10^6)² W/m²
Now, let's calculate the numerical value of the flux:
Flux when Earth was young ≈ 2.454 × 10^19 W/m²
Therefore, we would have measured a flux of approximately 2.454 × 10^19 W/m² from a galaxy at that distance when the Earth was young.
To find the distance to the nearest galaxies when the Earth was young, we can use the concept of redshift. The redshift (z) is calculated as the observed wavelength minus the rest wavelength, divided by the rest wavelength. The scale factor (a) of the universe at a given redshift can be calculated using the relation:
a = 1 / (1 + z)
In this case, the Earth formed at a redshift of z ≈ 0.53. Hence, the scale factor at that time would be:
a = 1 / (1 + 0.53) ≈ 0.654
Now, to determine the distance to the nearest galaxies when the Earth was young, we can use the relation:
Distance = Proper Distance / a
The proper distance to the nearest galaxies is given as 1 Mpc (Mega-parsecs), which is approximately 3.09 x 10^22 m. Substituting the values, we have:
Distance = (3.09 x 10^22 m) / 0.654 ≈ 4.723 x 10^22 m ≈ 15.29 Mpc
Hence, the distance to the nearest galaxies when the Earth was young would be approximately 15.29 Mpc.
To find the apparent angular size of a galaxy at that distance, we can use the relation:
Angular Size = Physical Size / Distance
Given that the size of the galaxy is 10 kpc (kilo-parsecs), which is approximately 3.09 x 10^19 m, we have:
Angular Size = (3.09 x 10^19 m) / (4.723 x 10^22 m) ≈ 6.53 x 10^(-4) radians
To convert this into arcseconds, multiply by the conversion factor:
Angular Size (arcsec) = (6.53 x 10^(-4) radians) x (180 / π) x (3600 arcsec/degree)
≈ 13.72 arcsec
Therefore, the apparent angular size of a galaxy at that distance would be approximately 13.72 arcsec.
To calculate the flux measured from a galaxy at that distance, we can use the inverse square law, which states that the flux decreases with distance squared. The equation can be expressed as:
Flux = Luminosity / (4π * Distance^2)
Given that the luminosity of the galaxy is approximately 10^37 W and the distance is 15.29 Mpc (which is approximately 4.723 x 10^23 m), we have:
Flux = (10^37 W) / (4π * (4.723 x 10^23 m)^2) ≈ 3.40 x 10^(-20) Wm^(-2)
Hence, the flux measured from a galaxy at that distance would be approximately 3.40 x 10^(-20) Wm^(-2).