**Discussing about the model**

**The goals**

After the realization of the model, the student faces the moment of the
analysis: some phenomena are expected, but other are not realistic.
After a deep criticism, they discover that the model is too simple: the
model has some defects because it doesn't take into account the relative
motion of the stars (static model) and of the third dimension. Is it
possible to realize a more complete model? With a pc software? The unit
end with a challenge!

The development in the classroom

This lesson does not require a background of astrophysical concepts. The
goal is *the stimulation of the critical sense of the students*. The
list of the state teachers is to urge sharp and smart observations about
*the model* developed in the previous lessons, the fundamental question
being : *"Is it a good model?"*. When we talk about the model, we need
not think only to the model made in polystyrene, also the mathematical
model must be discussed!

Karl R. Popper has written: "Our general analysis of the notion of
*explanation* is not obviously burdened by the failures of a particular
representation or of a model. The general analysis focalizes on all
kinds of abstract theory so as mechanic model. \[\...\] The progress of
science consists in *attempts*, in the *elimination of errors* and in
*further attempts*, led by the experience acquired during the previous
attempts and errors."[^1]

![](media/image1.png)

Guided by this attitude, we can start with the analysis of the first
lesson. In table 1 we have collected the results obtained by students
(black) together with the data appearing in the literature(red).

Around these data one can develop a wealthy discussion in the classroom,
stimulating the student to a deeper observation and to an analysis of
the relations between his prediction and data. Many questions can be
raised:

1.  What's the relationship between the predicted the luminosity and the
    measured value? Why is there a correlation?

2.  What's the relationship between the predicted masses and the values
    found in the literature ? Why is there a correlation?

3.  Why do we have a big uncertainty about the spectral class of
    Regulus, Antares, Deneb and Rigel?

4.  Why do we have a big error about the spectral class of Polaris?

The answers can lead a better understanding about the concept of
"absolute magnitude", about the relationship between mass and
luminosity, and about a critical interpretation of HR diagram. Faced
with these results and these questions, it's important that the teacher
doesn't give immediately the right solution, but he incites the student
to formulate hypothesis and interpretations. This moment is fundamental
also for a clarification of possible misunderstanding.

1.  The observed magnitude depends on the wavelength. The formula used
    for calculation of luminosity is based on an mean of Sun's
    luminosity in different bands. The absolute magnitude of the Sun is
    +4.79 around the yellow, +5.41 around the blue and +5.51 on the
    ultraviolet spectrum). This causes great errors especially when
    considering stars having spectral class different than Sun.

2.  The relationship between the mass and luminosity of a star depends
    on other important parameters, in particular the metal content,
    which determines opacity and thus the heat flow mechanism. The
    formula used is an empirical relationship that works best with main
    sequence stars.

3.  The big uncertainty about the spectral class of Regulus, Antares,
    Deneb and Rigel puts in evidence the fact that a giant star can be a
    red-giant (spectral class M/K) and a blue giant (spectral class B).
    Because it's very easy to see in the night sky these stars, is
    enough to observe them to resolve the dilemma! If it's not possible,
    a picture of this star can be a solution.

4.  About the big uncertainty of Polaris's spectral class, the argument
    is different. Students (probably) don't choose the spectral class F
    because in that region (with luminosity 3200) there are few star.
    It's an instinctive reasoning but it's important since it puts in
    evidence that the HR diagram is a statistical collection of star.
    The HR diagram doesn't say that is *impossible* to find stars
    outside the zone of main sequence, giants, supergiants or white
    dwarfs! It says us that stars outside such regions are *infrequen*t.
    Polaris is a star which fills a zone, above the main sequence and to
    the left of giants, called "variable stars": the Polaris is indeed a
    variable star.

![](media/image2.png){width="5.0in"
height="3.102755905511811in"}![](media/image3.png){width="0.4263779527559055in"
height="0.4507874015748031in"}About the arguments of the second lesson,
a simple question which can open new considerations is: "*If the correct
pattern of a gravitational potential made by a star is that described in
fig 4 (Lesson two), why don't we observe a fall of all solar system's
planet on the Sun?* (fig. 2)"

The student would discover that the mathematical model described by the
potential:

$$V_{\text{tot}} = - G\frac{m_{S_{1}}}{|r|} - G\frac{m_{S_{2}}}{|r - d|}$$

is about a *[static situation]{.underline}*, and then it's not a
realistic description. In a binary system the stars are not at rest, but
they rotate around the center of mass. The analogy with the solar
system, in which the planets make a revolution around the Sun, can
induce in a misunderstanding: in this case the Sun can be considered at
rest because the Sun is much heavier than the planets and the center of
mass is inside the Sun. If we consider a binary system of stars, their
masses are often comparable and the motion develops around the center of
mass becomes evident. For a visualization of this kind of motion, we
suggest to show this movie (Fig. 3).

![](media/image4.png)

If the potential \[4.1\] describes a static situation, it means that in
a better mathematical model we must take into account of the
'centrifugal potential'. The 'new potential' looks like:

$$V_{\text{tot}} = - G\frac{m_{S_{1}}}{|r|} - G\frac{m_{S_{2}}}{|r - d|} + P_{\text{centr}\text{.}}$$

On the left we have the potential of a 'static' system, on the right a
potential of a 'dynamic' system. (Fig. 4-5)

![](media/image5.png){width="3.5374015748031495in"
height="3.4015748031496065in"}![](media/image6.png){width="4.060236220472441in"
height="2.769291338582677in"}

In the light of the latest observations, the question about the model
made in polystyrene is: *"Is it a good model?"*. As all models, it has
some limits.

-   As we have seen, it is designed basing on the gravitational
    potential \[4.1\], and so it isn't a good model for a dynamic
    situation. In our simulation the fluid 'falls' on the secondary star
    following a direct way (Fig. 6), while in reality the material tend
    to roll up around the star. In Fig. 7 you can see a computer
    simulation about the 'falling' of material from a red giant to a
    dense star (as white dwarf).

![](media/image8.png){width="3.2125in" height="3.0in"}

-   Our model in polystyrene is a 2D representation of the gravitational
    potential. In reality, i.e. in three dimensions, we don't have
    equipotential *lines*, but equipotential *surfaces*; so we can
    imagine the Roche lobes as a 'quasi-spherical' surface on which the
    binary potential is the same. The potential function \[4.2\]
    describes a 3D potential that we can visualize with a computer
    simulation. In the next picture you can compare the 2D situation
    with the 3D one, just in the moment in which the material falls down
    (Fig. 8 - 9 - 10).

![](media/image10.png){width="3.2125in" height="2.582638888888889in"}

![](media/image11.png)

[^1]: Conference held (in German) during the European Forum of Austrian
    College at Alpbach (Tirol) on August 1948. It was published for the
    first time with the title Naturgesetze und theoretische Systeme. The
    German version was edited by Simon Moser (1949) and it was never
    translated into English.