Protostar - Thermal energy change

[Fleet]

Homework Statement

I have to calculate the change in a protostar's thermal energy $$\Delta E_{term}$$ on a year.
I am given the protostar's luminosity $$L=3,5\cdot 10^{28}\,W$$
Furthermore I am told that the energy that the star radiates comes from the lost potential energy under the star's contraction.

Homework Equations

The virial Theorem: $$E_{total}=E_{kin}+E_{pot}=\frac{1}{2}E_{pot}=-E_{kin}$$

The Attempt at a Solution

I can calculate the "radiated" (lost) energy: $$3,5\cdot 10^{28} W\cdot 365,25\cdot 24\cdot 60^2 \frac{s}{y}=1,105\cdot 10^{36} J$$
But I don't know how if this is right or how to get to it by arguing from the virial theorem.

I really hope you can help me.

Best regards.

 

[anathema]

Hello there,

To calculate the change in a protostar's thermal energy over a year, we first need to understand the relationship between the protostar's luminosity and its thermal energy. The luminosity of a star is the amount of energy it radiates per unit time. In the case of a protostar, this energy comes from the lost potential energy due to its contraction.

The virial theorem states that the total energy of a system is equal to twice the kinetic energy of the system. In the case of a protostar, this means that the total energy (E_total) is equal to twice the kinetic energy (E_kin) plus the potential energy (E_pot). This can be written as:

E_total = 2E_kin + E_pot

Since the protostar is contracting, its potential energy is decreasing. This means that the change in potential energy (ΔE_pot) is equal to the negative of the change in kinetic energy (ΔE_kin). In other words:

ΔE_pot = -ΔE_kin

Now, we can rewrite the virial theorem as:

E_total = 2ΔE_kin - ΔE_pot

Since we are interested in the change in thermal energy (ΔE_term) over a year, we can write this as:

ΔE_term = -ΔE_pot = -E_total + 2ΔE_kin

Substituting in the given luminosity (L) and the time period of one year (t = 365.25 days), we get:

ΔE_term = -(L*t) + 2ΔE_kin

= -(3.5*10^28 W * 365.25 days * 24 hours * 60^2 seconds) + 2ΔE_kin

= -1.105*10^36 J + 2ΔE_kin

Therefore, to calculate the change in thermal energy (ΔE_term) over a year, we need to calculate the change in kinetic energy (ΔE_kin). This can be done by using the given luminosity and the time period of one year in the following equation:

ΔE_kin = L*t = 3.5*10^28 W * 365.25 days * 24 hours * 60^2 seconds

= 1.105*10^