Calculate the escape velocity from a red giant's atmosphere (use the formula for escape velocity from chapter 2). Assume that the star's mass is 1 M and its radius is 100 R How does this compare with the speed at which a planetary nebula shell is ejected?
To calculate the escape velocity from a red giant's atmosphere, we can use the formula for escape velocity:
v_escape = sqrt((2 * G * M) / R)
Where:
- v_escape is the escape velocity
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3⋅kg^-1⋅s^-2)
- M is the mass of the red giant
- R is the radius of the red giant
Given that the red giant's mass is 1 M (where M is the solar mass) and its radius is 100 R (where R is the solar radius), we need to convert these values into appropriate units.
1 M = 1 * 1.989 × 10^30 kg (mass of the Sun)
100 R = 100 * 6.957 × 10^8 m (radius of the Sun)
Now, we can substitute these values into the escape velocity formula:
v_escape = sqrt((2 * G * M) / R)
= sqrt((2 * 6.67430 × 10^-11 m^3⋅kg^-1⋅s^-2 * 1 * 1.989 × 10^30 kg) / (100 * 6.957 × 10^8 m))
Calculating this expression will give us the escape velocity from the red giant's atmosphere.
On the other hand, the speed at which a planetary nebula shell is ejected is typically much higher than the escape velocity. The specific speed depends on various factors such as the mass and energy released during the red giant's transition to a white dwarf, as well as the nebula's structure. Unfortunately, without more specific information, it is challenging to provide an exact comparison between the escape velocity and the speed at which a planetary nebula shell is ejected.