2 nd scenario

PURPOSE OF EXERCISE  To find out the existence of Jupiter’s moons (satellites)  To calculate the radius and period of their orbit around Jupiter  To calculate the mass of Jupiter.

 We need the Gravitation low  Formula of centripetal force  Equating the two  formulas

 Knowing the relation between the linear and angular velocity is:  And that:

 We finally have:  So, if we determine the radius of the orbit and the period of one of Jupiter’s moons, we can estimate the mass of Jupiter.

 We will use 6 images of Jupiter and its moons: jup5.fits jup6.fits jup7.fits jup8.fits jup9.fits jup10.fits and we will analyze the images using SalsaJ.  Open the 6 images with SalsaJ.  Use the Image > Adjust Brightness/Contrast to see the moons in each image

 Select auto

 To determine the radius and angular speed of one moon, we need:  To measure the distance of the moons to Jupiter in the different images and know the time interval between images.  It is easier if we first gather all the information of the different positions of the moons in one single image.  As you shall see, this is done by subtracting and adding the images:

 Start with: and get  The Image Calculator creates a new image called Result of Jup5.fits.  The central part shows that Jupiter subtracts well, which confirms that the images are aligned.  You can identify “white moons” from jup5 and “black moons” from jup6.  Notice that the 2 moons on the left side and moving away from Jupiter; the closest moon on the right side is also moving away from Jupiter, but the other one is moving towards Jupiter.  For each image, write down the time of day of the exposure: Image > Show Info… and look for UT (Universal Time) Image > Show Info… and look for UT (Universal Time)

Identifying the Galilean moons  Analyze the movement of the moons.  Try to order the moons starting with the closest one to Jupiter. with the closest one to Jupiter.  Argue with considerations about their velocities - notice that the their velocities - notice that the images were all taken with images were all taken with 1 hour interval so the difference 1 hour interval so the difference in position tells us something in position tells us something about the velocities about the velocities

 Moon D seems to be reaching the turn-around point; this is clearly the smallest orbit, so moon D should be Io. the smallest orbit, so moon D should be Io.  Ordering the velocities of the moons: velocity moon B > velocity moon Γ > velocity moon A velocity moon B > velocity moon Γ > velocity moon A  Considering that we expect that moons that orbit closer to Jupiter have greater velocities, we can say that have greater velocities, we can say that  Moon A is Callisto  Moon B is Europa  Moon C is Ganymede  Moon D is Io  Note: Moon D is the fastest, although it doesn’t show. That’s because it is near the turning point and most of its velocity is because it is near the turning point and most of its velocity is perpendicular to us. The other moons are not in that situation. perpendicular to us. The other moons are not in that situation.

Analising Io  We will use the fact that moon D is near the turning around point to establish the radius of the orbit for this moon in pixels.  The maximum distance of moon D from Jupiter is a good approximation to its radius. approximation to its radius.  We will use the different distances to Jupiter to get its velocity  Open a spreadsheet with Excel and make the following table:  Open a spreadsheet with Excel and make the following table: Time intervalX coordinate Y coordinate Distance to Jupiter ΘΔθ Jupiter Io jup5 Io jup6 Io jup7 Io jup8 Io jup9 Io jup10

Jupiter coordinates  To determine the position of Jupiter use the linear selection X coordinate: draw a horizontal line that passes through Jupiter (press SHIFT while drawing a line and it will make (press SHIFT while drawing a line and it will make it horizontal); it horizontal);  point to the beginning of the line and write down the x coordinate write down the x coordinate

 Go to Analyze > Plot Profile and determine the width of Jupiter. Divide by 2 and add the width of Jupiter. Divide by 2 and add to the x coordinate of the line you draw. to the x coordinate of the line you draw /2= /2= =80

Jupiter and moon D coordinates  You might want to do the same procedure with a few lines, to make sure you have the right x coordinate.  Do the same procedure (with the necessary adaptation) to determine the y coordinate for Jupiter.  Then repeat the procedure to get x and y coordinates for moon D on the different positions. Remember, you can zoom the image.  Fill in the data in the spreadsheet…

 Calculate the distance, in pixels, between the moon and Jupiter in each image. each image.  Use the formula:

To get the angular velocity, consider the picture.  In the time interval Δt between jup5 and jup6, Io moved Δθ=θ2-θ1 radians, so its angular velocity was ω1 =Δθ/ Δt  With basic trigonometry we find that θ1=arcsin(d1/d5), θ2=arcsin(d2/d5)

Angular velocity and period  Calculate all angular displacements. Since the time interval between all images Since the time interval between all images is the same, 1 hour, the angular velocity is the same, 1 hour, the angular velocity is ω=Δθ rad/h. is ω=Δθ rad/h.  Calculate the average angular displacement and you have the average angular velocity in rad/h; express the average angular velocity in rad/days;  use T=2π/ω to get the period in days (T).

Then you fill the spreadsheet

Radius  We have considered the position of moon D in the image jup10 as the turning point, so we have the radius of its orbit in pixels.  We need to transform pixels into real distance.  We use Image > Show Information and we get Plate scale 0,67''/pixel and we get Plate scale 0,67''/pixel  Transform the radius in pixels into degrees and then into radians.  Use the known distance from Earth to Jupiter, D Earth-Jupiter =7,8x10 11 m, and convert the D Earth-Jupiter =7,8x10 11 m, and convert the radius into meters. radius into meters.

Results

Comparison with real data