Basically the ``Bohr radius'' for the dark matter is
rB=ℏ^2 / GN M m^2
where m is the dark matter mass, M is the total amount of dark matter mass contained inside the ``gravitational atom'', and ℏ and GN are constants as usual.
Taking the dark matter to be of constant density ρ0=1.3×10−22 kg/m3, calculate the lower bound on the dark matter mass m. To do this, you will need to calculate how much mass M is inside this atom using the given density. Give your answer to two significant figures.
To calculate the lower bound on the dark matter mass (m), we need to determine the mass (M) contained inside the "gravitational atom" using the given density (ρ0). We can then substitute these values into the formula for the Bohr radius (rB) and solve for m. Let's break down the steps.
Step 1: Determine the mass (M) inside the "gravitational atom"
To calculate the mass, we can use the formula:
M = ρ0 * V
Where ρ0 is the density and V is the volume of the atom. The volume of a sphere with radius rB is given by:
V = 4/3 * π * rB^3
Step 2: Substitute the values into the Bohr radius formula
Now that we have the value for M, we can substitute it along with the other given values into the Bohr radius formula:
rB = ℏ^2 / (GN * M * m^2)
Step 3: Rearrange the formula to solve for m
Rearrange the formula to isolate m:
m^2 = ℏ^2 / (GN * M * rB)
m = √(ℏ^2 / (GN * M * rB))
Step 4: Calculate the lower bound on the dark matter mass (m)
Now, plug in the values for ℏ (Planck's constant), GN (Newton's gravitational constant), M (calculated mass), and rB (Bohr radius) into the formula and calculate m.
Remember to use appropriate units for the constants and ensure the units are consistent throughout the calculations.
The given density (ρ0) is 1.3 × 10^-22 kg/m^3, and we want the answer to two significant figures.
Follow these steps to calculate the lower bound on the dark matter mass (m).