Astrophysics: Magnitude of sun when Jupiter crosses over it.

[Xyius]

Homework Statement

An observer 5pc away observes the sun in the plane of Jupiter orbit. He cannot resolve either object but he notices a slight dimming of the star when Jupiter passes across the sun in his vision. Find the magnitude of the sun with and without Jupiter in front of it.

Homework Equations

Magnitude formula
$$m_1-m_2=2.5log\left( \frac{\Phi_1}{\Phi_2} \right)$$ (Where Phi is the flux)
I do not know if this is the only equation or not.
Absolute Magnitude of Sun = 4.77

The Attempt at a Solution

So the case when Jupiter is NOT crossing over seems simple enough.

$$m_1-m_2=2.5log\left( \frac{\Phi_1}{\Phi_2} \right)$$
$$=m-M=2.5log\left( \frac{d^2}{10^2} \right)$$
$$=m-4.77=2.5log\left( \frac{5^2}{10^2} \right), m=3.26$$

So now I just need to find the magnitude when Jupiter is crossing the sun. My professor said to find the area of the solar disk that is visible when Jupiter is crossing by simply doing..
$$\pi (R_{sun}^2-R_{Jupiter}^2)$$
But I do not know where this fits into the above equation. How can I relate this to the flux ratio? Any help would be appreciated :]

 

[Andrew Mason]

Start with the definition of flux: $\Phi$. How is flux related to area of the source?

How does Jupiter passing in front of the sun affect the effective area of the light source (ie the sun).

AM

 

[Xyius]

Start with the definition of flux: $\Phi$. How is flux related to area of the source?

How does Jupiter passing in front of the sun affect the effective area of the light source (ie the sun).

AM

Okay so Flux is..

$$\Phi=\frac{L}{4\pi d^2}$$ Where d is the distance from the star to the observer.

I guess the only thing that changes here would be the Luminosity. Luminosity is defined to be..

$$L=4\pi R^2 \sigma T^4$$ Where R is the radius of the star.

So would this be on the right track? My concern is this isn't a "disk" as my professor hinted towards. :\

 

[Andrew Mason]

Okay so Flux is..

$$\Phi=\frac{L}{4\pi d^2}$$ Where d is the distance from the star to the observer.

I guess the only thing that changes here would be the Luminosity. Luminosity is defined to be..

$$L=4\pi R^2 \sigma T^4$$ Where R is the radius of the star.

So would this be on the right track? My concern is this isn't a "disk" as my professor hinted towards. :\

It may not be a disk. But a sphere's cross-sectional area is what one sees from a distance. How is the viewable cross sectional area of the sun affected when Jupiter passes in front of it? When does the effect reach a maximum?

AM

 

[asteriatic]

As an astrophysicist, I can provide some insight into this problem. First, we need to understand the concept of magnitude in astronomy. Magnitude is a measure of the brightness of an object, with lower numbers indicating brighter objects. The magnitude formula you have provided is correct and can be used to calculate the magnitude of the sun in both cases.

When Jupiter is not crossing over the sun, the observer is seeing the combined light of both the sun and Jupiter. Using the magnitude formula, we can calculate the magnitude of the sun as 3.26. However, when Jupiter is crossing over the sun, it blocks some of the sun's light, causing a decrease in brightness. This decrease in brightness can be quantified by calculating the ratio of the flux (or amount of light) from the sun with and without Jupiter in front of it.

To calculate this flux ratio, we can use the area of the solar disk visible when Jupiter is crossing over the sun. As your professor mentioned, this can be calculated using the formula \pi (R_{sun}^2-R_{Jupiter}^2). This represents the area of the sun's surface that is being blocked by Jupiter. We can then use this area to calculate the flux ratio using the formula \frac{\Phi_{sun}-\Phi_{Jupiter}}{\Phi_{sun}}, where \Phi_{sun} is the flux from the sun and \Phi_{Jupiter} is the flux from Jupiter.

Once we have the flux ratio, we can plug it into the magnitude formula to calculate the magnitude of the sun when Jupiter is crossing over it. This will give us the magnitude of the sun with Jupiter in front of it. Subtracting this value from the magnitude of the sun without Jupiter will give us the dimming or change in magnitude caused by Jupiter crossing over the sun.

I hope this helps clarify the problem and how to approach it. Keep in mind that in astrophysics, we often use complex equations and calculations to understand and study the universe. Don't get discouraged if it takes some time to understand and apply them. Keep practicing and asking questions, and you will get the hang of it.