**Stellar Binary system.**

**The goals**

-   Starting from everyday experience, introduce the concept of
    potential energy.

-   Next provide a model explaining how gravity modifies geometry.

-   Describe the motion in the gravitational field of two stars in terms
    of the model previously developed.

**The main items**

A binary star system is a formed with two stars orbiting around the
center of mass, bound by gravitational force. This system is a good
opportunity to present gravitation in a concrete application.

We start with a study of this function:

$$y = - \frac{A}{\sqrt{x^{2}}}$$

![](media/image1.png){width="6.699212598425197in"
height="4.099212598425197in"}![](media/image2.png){width="1.7251968503937007in"
height="2.3736220472440945in"}Substituting some value in the x, with A
constant, it's easy to plot a graph (Fig. 1).

This permits the student to understand that the space and the objects
can be modeled follow a precise mathematical function. We can build a
\"trumpet\" like well with polystyrene, plotting a 3D formula using a
software like Derive (Fig. 2 -- 3):

$$z = - \frac{A}{\sqrt{x^{2} + y^{2}}}$$

![](media/image3.png){width="3.2in"
height="2.9161417322834646in"}![](media/image4.png){width="3.3in"
height="2.6in"}

An object falls under the action of gravity inside this kind of
trumpet[^1]: the every day experience guide us to remark that:

1.  the closer it is to the center, the harder it is to avoid the fall.

2.  bodies fall toward the center.

3.  the closer it is the center, the higher the effort to extract them.

4.  the effort for extraction is the same, if the distance from the
    center is the same.

This observation can be 'translate' saying:

1.  the forces varies with distance from the center of the 'trumpet'.

2.  the forces are central.

3.  the object on the surface has a potential energy.

4.  there are points in which the potential is the same: these form
    equipotential curves.

The object on the surface of the trumpet function is bound in its motion
by the presence of geometrical shape of polystyrene: the polystyrene
modifies the space in which the object can move. A mass in 3D reality
can modify the space (and then the geometry) around it in the same way.
Changing the value of A modifies the steepness of the 'trumpet'
function. Likewise the mass M modify the space around it.

$$z \propto - \frac{M}{\sqrt{x^{2} + y^{2}}}$$

How does a particle fall in the presence of a star? The gravity of the
star modifies the space much in the same way as the trumpet does. How do
particles fall when two stars are far from each other? And how when the
stars are close? The actions of the two stars simply sums up:

$$z = - \frac{A}{\sqrt{x^{2} + y^{2}}} - \frac{A}{\sqrt{(x - d)^{2} + y^{2}}}$$

where d is the distance between the two stars (Fig. 4 -- 5). Connecting
the mass M of the star with the value of A, one can ask for the
equilibrium position in between the two stars (*Lagrangian point L1*)
and show how it is changed when one of the masses is changed. *The Roche
lobes* can be introduced as the equipotential curve/surface containing
*L1*.

With this geometrical approach we pay attention on the space deformation
made by gravitational force; if one wants approach the problem of the
object falling in a 'trumpet' well in a 'classical' way, he can note
that the acceleration on this surface is always direct towards the
center and closer the object to the center, the acceleration become
bigger.

![](media/image5.png){width="4.158333333333333in"
height="3.707638888888889in"}

![](media/image6.png)

**The development in the classroom**

In classroom we suggest to develop this items with a problematic
approach: after a quick theory presentation, it's important that student
are oriented towards

-   understanding the theoretical structure

-   using the mathematical tools

-   imagining the "physics consequences" of this model.

We would like to put in evidence that some scientific predictions are
not dogmatic truth, but they are the result of a logic argument based on
mathematical concepts. In order to reach this objective, we propose some
problems. The challenge for the students is to resolve them using only
the data and the formula collected in tab. 1. The binary system that we
will study is called KPD1930 + 2732 and according many research it is a
candidate Type Ia Supernova progenitor[^2]: in fig. 6 you can see the
plot of the gravitational potential function:

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

![](media/image7.png){width="3.95in" height="3.798031496062992in"}where
*d* is the distance between two stars and $r = \sqrt{x^{2} + y^{2}}$.
With this data in the next lesson we will build a concrete model of a
star binary system.

![](media/image8.png){width="6.691666666666666in"
height="1.863888888888889in"}

1.  The students plot, before on paper and after using some software as
    Excel or Derive, the 'trumpet' functions \[2.1\] and \[2.2\] varying
    the value of parameter $\begin{matrix}
    A & (M)
    \end{matrix}$. What kind of observations can they do about the
    steepness of the 'trumpet' function?

2.  Have the students plot, first on paper and then using some software
    as Excel or Derive, the function of the gravitational potential made
    by the Sun, by the Earth and by the Jupiter ($A = G´M$ where
    (gravitational constant)
    $G = 6\text{.}\text{67}´\text{10}^{- \text{11}}$. If they superpose
    all graphs, what kind of observations can they do?

3.  If a NASA probe with a mass of 100 Kg is placed at the point of
    gravitational equilibrium between the Sun and the Earth, how many
    kilometers is it from the Earth?

4.  If we imagine dragging away the Earth from the Sun's gravitational
    influence, how much energy do we need?

5.  The students plot, using some software as Excel or Derive, the
    function of the gravitational potential made by the star *S~1~* and
    *S~2~*. Where is the gravitational potential made by the star *S~1~*
    and *S~2~* the same?

6.  The sum of function of the gravitational potential made by the star
    *S~1~* and *S~2~* is:

> $$V_{\text{tot}} = - G\frac{m_{S_{1}}}{|r|} - G\frac{m_{S_{2}}}{|r - d|}$$
>
> where *d* is the distance between two stars. Have the students plot
> this function: they should note the Roche point. If a NASA probe is
> placed at this point, what is its behavior?

At the end this second lesson the student should know that:

-   Every mass creates a gravitational field and a gravitational
    potential.

-   These are functions with characteristic shape.

-   If I want to separate two masses I need to do work equal to their
    potential energy.

-   The point, in which the gravitational potentials of two stars are
    the same value, is called internal Lagrangian point L1; in this
    point we have also gravitational equilibrium.

[^1]: Within this surface shape, a body falls with velocity v such that
    $\frac{1}{2}v^{2} - \frac{\text{gA}}{\rho} = \text{const}$, where
    $\rho = \sqrt{x^{2} + y}$. This is similar to the energy
    conservation of Newtonian motion in the field created by a mass
    $M = \frac{\text{gA}}{G}$.

[^2]: P.F. Maxted, T.R. Marsh & R.C. North, \"KPD1930 + 2732 - a
    candidate Type Ia Supernova Progenitor\", Monthly Notices of the
    Royal Astronomical Society **317**, September 2000.