**Rosa M. Ros, Ederlinda Viñuales**

**"EAAE Summerschools" Working Group**

**Technical University of Catalonia - University of Zaragoza (Spain)**

# Abstract

Our main goal in this work is to deduce the mass of Jupiter by means of your Galilean satellites. To get this result we need to calculate radii of the orbits and orbital periods of the four most famous Jupiter' satellites.

Tools that we will use for getting our proposal will be a set of Jupiter' photographs in which the satellites Io, Europe, Ganymede and Callisto appear and some elemental mathematical calculations.

# Contents

At the beginning of the 17th century, when Galileo looked at Jupiter with his telescope, he saw four stars circling him. This discovery was in contradiction then with the general belief that all heavens bodies turned round the Earth. This fact represented a scientific revolution in that time and helped to the Heliocentric Copernican system was accepted.

As Jupiter and their Galilean satellites are a good model of a planetary system our intention is to study it using a set of photographs which we had taken previously.

The set of photographs should be extent enough because of the different Behaviour of the satellites.

To study the Europe and Io motions we need some photographs taken the same day into the interval of one hour; for Ganymede it is more convenient to us to take photographs every four or five days and finally for Callisto every eight dais as minimum.

In this way in the set of photographs we can have four of them in which, respectively each one of the satellites appear in their further positions to the planet. The more accurate of the maximum position is also the final results are better, because one of the aims that we want to calculate is the orbits of each one of the Galilean satellites.

For recognising each one of satellites over a photograph, we need to use a computer which draws the relative positions of Jupiter and their Galilean satellites for one day and specific time.

We present in this paper a selection of sufficient photographs for carrying out the proposal practice, because in them appear all satellites in their further position. In each photograph the visible satellites are identified (Photo 1 to 7).

Under the assumption that all Galilean satellites describe circular orbits turning round Jupiter, first at all we compute radii of this orbits for each one.

To carry out this calculation we establish a proportion between real values of the orbit radius a and the Jupiter real radius, and the measurements over the photograph of the further position of the satellite respect to de planet ro and his radius R (Fig. 1).

When the satellite rotates round Jupiter its positions change. In particular in figure 1, positions pass through P', P'', Po, P''' and so on, where the further position of the satellite is the Po position, so, the distance from this position to the centre will be the orbit radius ro.

For a specific satellite we choose among all photograph that where the position of the satellite is further from the planet. This distance, in centimetres, will be the radius of the orbit ro. The radius of Jupiter R is not possible to be measured over the photographs 1 to 7 because the image of the planet is burnt, so we have to do that over the photograph 8, where we gave an inferior exposition time when this was taken.

If we assume that the Jupiter radius is known and measures 71000 km, we deduce that:

a = 71000 r_{o}/R

where:

a = real radius of the satellite orbit (in km).

r_{o} = radius of the orbit or else the maximum distance that there are, in all photograph 1 to 7 between the satellite and the planet (in cm).

R = radius of Jupiter over the photograph 8 (in cm).

If we repeat this process for each one of satellites we will be able to complete the following Table 1.

**Table 1**

Satellite | Radius of the orbit a (in km) | Orbital period P (in days) |

Io | ||

Europa | ||

Ganymede | ||

Callisto |

Then from the set of photographs that we have we deduce periods of each one of the Jupiter's satellites. We want to relate the lapse of time passed between two photographs of the one determinate satellite and the central angle corresponding to the way gone over his orbit. The two positions of the satellite which we have to take account will be different to Po, this is, the corresponding to the maximum distance from the satellite to the planet.

For these positions P', P'' or P''' (Fig. 1), their respective central angles are not null and can be deduced in the following way

θ'= arccos (r'/r_{o})

θ'' = arccos (r''/r_{o})

θ''' = arccos (r'''/r_{o})

where:

θ', θ'', θ''' = central angle (in degrees).

R', r'', r''' = distance from the satellite to the planet for each case in all photograph 1 to 7 between the satellite and the planet (cm).

r_{o} = maximum distance to the planet from among all observations (cm).

and for each position P' , P'' or P''' we know the just moment t', t'' or t''' respectively in which photographs were taken.

It is necessary to study separately if positions of the satellite are anterior or posterior to the position Po in the photographs which we are using in our work or else, one of these is anterior and the other posterior. In the figure 1 the first case is represented by positions P' and P'' , whereas the second case correspond to positions P' and P'''.

If we consider the satellite moving from P' to P'' (Fig. 1), the interval of time passed is t''-t' and the covered angle is θ''-θ'. Therefore making use of a simple proportion, the period of revolution of satellite is obtained by:

P = 360 (t''-t')/(θ''-θ')

where:

P = the period of revolution of satellite (in days).

t', t'' = time corresponding to two photographic observations, both anterior or posterior to the maximum position Po (days).

θ', θ'' = respective central angles of the two photographic observations, both anterior or posterior to the maximum position P_{o} (deg).

If the satellite is moved from P' to P''' (Fig. 1), the interval of time passed is t'''-t' but the covered angle is now θ'''+θ'. Then we can express the period by the equation

P = 360 (t'''-t')/(θ'''+θ')

where:

P = the period of revolution of satellite (in days).

t', t'' = time corresponding to two photographic observations, both anterior or posterior to the maximum position P_{o} (days).

θ', θ'' = respective central angles of the two photographic observations, both anterior or posterior to the maximum position P_{o} (deg).

We obtain this period using two of the seven photographs (Photos 1 to 7). It is convenient to repeat the method again with other photographs and finally to calculate the medium of all values obtained. We proceed in the same way for each one of satellites and we place results in the Table 1.

Finally, making use of the third Kepler' law, we can determinate the mass of Jupiter M_{J} because we know already the radius of the orbit a (in u.a.), and the period of revolution (in days) of each one of satellites:

M_{J} = a^{3}/P^{2}

In the following, we will use values of the Table 1 and to compute the mass of the planet through the below expression where first at all, we have done a change of units taking into consideration that 1 u.a. of distance is equivalent to 150 million of km. and expressing the period in years.

M_{J} = 0,0395.10^{8}.a^{3}/P^{2}

where:

a = orbit' radius of satellite (in km).

P = period of revolution of satellite (in days).

M_{J} = mass of Jupiter (in solar masses).

For each one of satellite we will have a value of the planet mass, if we calculate the medium of these four values we will know the mass of Jupiter in an approximate form.