Calculating Earth's Accretion Time: Solving the Formulas for Astronomy Homework

[Poop-Loops]

My Astronomy homework has a problem asking to find the accretion time of Earth.

The forumula's the professor gave (since the book doens't have ANY) are:

$$\frac{dm}{dt}=\pi s^2\frac{\sigma \omega}{2}$$

And

$$\frac{ds}{dt}=\frac{\sigma \omega}{8 \rho} (1+ \frac{V_{esc}^2}{V_{relative}^2})$$

Where sigma = surface density of the accretion area (I'm guessing here, by the way), omega is the angular acceleration (which means radians/second around the sun, right?) and s is the radius of Earth.

I have no idea what rho is, probably the density of Earth.

Relative velocity = 1/3rd escape velocity, so those cancel. I know the accretion area is from halfway to Mars to half way to Venus from Earth and the total mass of that volume is 2 Earth masses.

My problem is: how do I calculate that? Since s is constantly changing, does that end up being a differential equation? Looks pretty ugly. Or do I assume that it is constant (say, the size of Earth), since it is so much smaller than the accretion area?

I have no idea what to do from here. If I can figure out s, I can then find dm/dt and then just divide Earth's current mass by that, right? So that shouldn't be a problem.

 

[nate808]

Dear student,

Thank you for reaching out with your question about finding the accretion time of Earth. I can understand that the formula provided by your professor may seem daunting, but let me break it down for you.

Firstly, let's define the terms used in the formula:

- dm/dt: This represents the rate of change of mass with respect to time. In other words, it tells us how much mass is being added to Earth over a given period of time.
- pi: This is a mathematical constant that is used to calculate the surface area of a circle.
- s: This represents the radius of Earth, which is the distance from the center of Earth to its surface.
- sigma: As you correctly guessed, this represents the surface density of the accretion area. This is the amount of mass per unit area that is being added to Earth.
- omega: This is the angular acceleration, which is a measure of how quickly Earth is rotating around the Sun.
- rho: This represents the density of Earth, which is the amount of mass per unit volume.
- Vesc: This is the escape velocity, which is the minimum velocity needed for an object to escape Earth's gravitational pull.
- Vrelative: This is the relative velocity, which is the velocity at which the accreting material is approaching Earth.

Now, let's look at the first formula: dm/dt = pi * s^2 * (sigma * omega)/2. This formula tells us that the rate of change of mass is equal to the surface area of Earth (pi * s^2) multiplied by the product of the surface density (sigma) and the angular acceleration (omega), divided by 2.

The second formula, ds/dt = (sigma * omega)/(8 * rho) * (1 + (Vesc^2)/(Vrelative^2)), tells us that the rate of change of Earth's radius (ds/dt) is equal to the product of the surface density (sigma) and the angular acceleration (omega), divided by 8 times the density of Earth (rho), multiplied by the quantity (1 + (Vesc^2)/(Vrelative^2)). This quantity takes into account the relative velocity of the accreting material and the escape velocity.

To calculate the accretion time, you will need to use these formulas in conjunction with the given information about the accretion area and the total mass of that volume. You are

 

[Blueshift5]

I would suggest approaching this problem by breaking it down into smaller, more manageable steps. First, let's clarify some of the terms in the equations given by your professor.

- Sigma (σ) is indeed the surface density of the accretion area. This refers to the amount of mass per unit area that is being accreted onto Earth.
- Omega (ω) is the angular velocity, which is measured in radians per second. This is the speed at which Earth is rotating around the sun.
- Rho (ρ) is the density of Earth, as you correctly guessed.
- Vesc is the escape velocity, which is the minimum speed needed for an object to escape the gravitational pull of Earth.
- Vrelative is the relative velocity between Earth and the accretion area. In this case, it is given as 1/3 of the escape velocity.

Now, let's address the issue of s (the radius of Earth) changing over time. This is indeed a differential equation and will make the problem more complex. However, we can make some simplifying assumptions to make the calculations easier. For example, we can assume that s remains constant throughout the accretion process, as the change in radius over the accretion time period is minimal compared to the size of Earth.

With these assumptions in mind, we can now calculate the accretion time for Earth. We can start by using the first equation given to find the rate of change of Earth's mass (dm/dt). This will give us an equation in terms of the variables we know, such as sigma, omega, and s.

Next, we can use the second equation to find the rate of change of Earth's radius (ds/dt). Again, we will have an equation in terms of the known variables.

Now, we can use these two equations to solve for the accretion time (t). This will involve integrating both equations and setting them equal to each other, since both dm/dt and ds/dt are changing over time.

Once we have the value for t, we can use it to calculate the total accreted mass of Earth by multiplying it by dm/dt. This will give us the total mass that was accreted onto Earth during the time period.

I understand that this may seem like a daunting task, but with some careful calculations and assumptions, you should be able to find a reasonable estimate for the accretion time of