A quasar with an observed luminosity of L=1012L⊙ is shining at the Eddington limit, with a net efficiency of 10%.
What is the mass accretion rate (in units of M⊙year−1)?
Oh, well, isn't that a quasar with a luminosity that's shining like a superstar? And it's hitting that Eddington limit, like someone desperately trying to keep up with societal expectations. Let me do some calculations here...
Hmm, with a net efficiency of 10%, it's like that quasar is giving it its best shot, but only hitting the target partially. Now, let's see, with a luminosity of L=10^12L⊙, that's like a shining beacon in the cosmos.
So, if we take into account the efficiency, the mass accretion rate of that quasar would be...drumroll, please...***insert clown drumroll***...
M⊙year−1! Yeah, that's right. It's mass accretion rate is in units of M⊙year−1! It's gobbling up matter like there's no tomorrow, but still winking at the Eddington limit. Keep up the good work, you luminous quasar, you!
But hey, remember, these calculations can be approximate, so take it with a pinch of stardust. And never forget to wear your cosmic clown nose when dealing with the wonders of the universe! 🌟🤡
To calculate the mass accretion rate (Ṁ) of a quasar shining at the Eddington limit, we can use the Eddington luminosity (L_Edd) and the efficiency (η) of the accretion process.
The Eddington luminosity is given by:
L_Edd = 4πGMm_p / σ_T
Where G is the gravitational constant, M is the mass of the central black hole, m_p is the mass of a proton, and σ_T is the Thompson scattering cross-section.
The net efficiency (η) is defined as the ratio of the luminosity (L) to the Eddington luminosity (L_Edd), which can be written as:
η = L / L_Edd
In this case, L = 10^12 L_⊙ and η = 0.1. We can now substitute these values into the equation to solve for the Eddington luminosity:
0.1 = (10^12 L_⊙) / L_Edd
Solving for L_Edd, we get:
L_Edd = (10^12 L_⊙) / 0.1
Now, we can substitute this value of L_Edd back into the equation for the Eddington luminosity:
L_Edd = 4πGMm_p / σ_T
And rearrange it to solve for M:
M = L_Edd * σ_T / (4πGm_p)
Substituting the known values, we get:
M = [(10^12 L_⊙) / 0.1] * σ_T / (4πGm_p)
Now, we can use this expression for M to calculate the mass accretion rate (Ṁ). The mass accretion rate is the rate at which mass is being accreted onto the black hole, and it can be calculated as:
Ṁ = L / (ηc^2)
Where c is the speed of light.
Substituting the known values:
Ṁ = (10^12 L_⊙) / (0.1 * c^2)
To convert the units to solar masses per year, we need to relate L to L_⊙ and time to years. We know that 1 L⊙ is equal to the luminosity of our Sun, which is approximately equal to 3.828 × 10^26 watts. We also need to convert the time unit to years, which means we need to multiply by the number of seconds in a year (3.154 × 10^7).
Finally, substituting the values and calculating, we get:
Ṁ = [(10^12 * 3.828 × 10^26 watts) / (0.1 * (3.154 × 10^7 m/s)^2)] * [(1 M_⊙ / 3.828 × 10^26 watts) / (1 year / 3.154 × 10^7 seconds)]
Simplifying, we get:
Ṁ = 31.5 * 10^12 M_⊙ / year
Therefore, the mass accretion rate of the quasar is 31.5 * 10^12 solar masses per year.
To calculate the mass accretion rate of a quasar, we can use the Eddington luminosity limit and the net efficiency. Here's the step-by-step process to get the answer:
Step 1: Calculate the Eddington luminosity
The Eddington luminosity (LEdd) is given by the formula: LEdd = 4πGMmₕc/κ,
where G is the gravitational constant, M is the mass of the black hole, mₕ is the mass of a hydrogen atom, c is the speed of light, and κ is the opacity of the accretion material.
Step 2: Calculate the mass of the black hole
We can rearrange the Eddington luminosity formula as: M = (LEdd * κ) / (4πGmₕc).
Step 3: Calculate the mass accretion rate
The mass accretion rate (Ṁ) is related to the Eddington luminosity and the net efficiency (η) by: LEdd = ηṀc².
Rearranging the formula, we get: Ṁ = LEdd / (ηc²).
Now, let's plug in the given values and calculate the mass accretion rate:
Given:
Observed luminosity, L = 10^12 L⊙
Net efficiency, η = 0.1
Step 1: Calculate the Eddington luminosity:
We need specific values for G, κ, and mₕ to calculate the Eddington luminosity. However, for illustrative purposes, let's assume approximate values:
G ≈ 6.67 × 10^-11 m³kg⁻¹s⁻² (gravitational constant)
κ ≈ 0.34 m²kg⁻¹ (opacity)
mₕ ≈ 1.67 × 10^-27 kg (mass of a hydrogen atom)
c ≈ 3 × 10^8 m/s (speed of light)
Using these values, plug them into the Eddington luminosity formula:
LEdd ≈ 4π × (6.67 × 10^-11) × M × (1.67 × 10^-27) × (3 × 10^8) / (0.34)
Step 2: Calculate the mass of the black hole:
Rearrange the Eddington luminosity formula to solve for M:
M ≈ (LEdd * κ) / (4πGmₕc)
Step 3: Calculate the mass accretion rate:
Ṁ ≈ LEdd / (ηc²)
Now, with the calculated values from the previous steps, plug them into the Ṁ formula to get the mass accretion rate in units of M⊙year⁻¹.