**Stellar Binary system.**

**The basis knowledge**

A binary star system is a system of two stars that orbit around a common
center of mass, bound by gravitational force: the study of this kind of
system is a good opportunity to present the gravitation in a concrete
application.

As well known the gravitation law is:

$$\overrightarrow{F} = - G\frac{m_{S_{1}}m_{S_{2}}}{r^{2}}\hat{r}$$

where $m_{S_{1}}$ is the mass of Star 1, $m_{S_{2}}$ is the mass of Star
2, *r* is the distance between them and *G* is gravitational constant
($G = 6\text{.}\text{67} \cdot \text{10}^{- \text{11}}$ *Nm^2^/Kg^2^*).
This relationship inform us that:

-   *The force is radial.*

-   The two stars fell the same force with *opposite towards*.

-   ![](media/image1.png){width="4.427083333333333in"
    height="3.21875in"}The force is always *attractive*.

The gravitation force between two masses can be thought as a direct
interaction between them: we can say that a star is a kind of
'mass-probe' which 'feels a force' in the point (*x~Star~;y~Star~*);. If
the star S~2~ disappear, what happens at that point? We can imagine that
the star S~1~ modifies the space around it trough a invisible *field*:

$$\overrightarrow{g} = - G\frac{m_{\text{Star}}}{r^{2}}\hat{r}$$

that, when we put in that point a mass, causes a *force*. This is *the
gravitational field* made by the stellar mass *m~Star~*; it's a
*vectoral quantity*.

Now we imagine to have two stars *S~1~* and *S~2~* with the same mass
but different distance from the Sun (*r~1~* and *r~2~*, with
*r~1~\<r~2~* (Fig. 2)): if we want remove them, we must spend more
energy to remove the star closer to the Sun. This objects have two
different *potential energies*, depending by the distance *r* from the
Sun:

$$U = - G\frac{m_{\text{Sun}}m_{\text{Star}}}{r}$$

![](media/image2.png){width="4.5in" height="3.251388888888889in"}The
minus sign means that the gravitational force is attractive; the
potential energy is a *scalar quantity*.

When two masses come near, the system gains negative potential energy:
if we want to separate these masses we need to do work against the
gravitational force. As we have defined the gravitational field, we can
introduce the concept of *gravitational potential* made by a mass
*m~Star~*. The *gravitational potential* is the potential energy of a
unitary mass, when it is at the distance *r* from *m~Star~*:

$$V = - G\frac{m_{\text{Star}}}{r}$$

If we plot this function, we don't forget that the distance *r* is
always positive, and so we use the \[2.2\] with $|r|$ (Fig. 3).

![](media/image3.png){width="5.074305555555555in"
height="2.9402777777777778in"}

Knowing that the gravitational force is a radial one, we can project the
two dimensional gravitational potential (Fig. 4) on the plan (x;y) (Fig.
5); we can see that the lines with the same potential have a circular
shape; these lines are called *equipotential lines*. The reality is more
complex, because we have a three dimensional gravitational potential; so
we can have spherical surface on which the gravitational potential is
constant: they take the name of *equipotential surface*.

> ![](media/image4.png){width="3.335433070866142in"
> height="3.525196850393701in"}![](media/image5.png){width="3.7125984251968505in"
> height="3.5070866141732284in"}

The binary system that we will study is called KPD1930 + 2732 and
according many research it is a candidate Type Ia Supernova
progenitor[^1]: in fig. 6-7 you can see the plot of the gravitational
potential function:

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

where *d* is the distance between two stars.

![](media/image6.png){width="3.5208333333333335in"
height="3.125in"}*L~1~* point is called *internal Lagrangian point* and
it represents the point in which the *S~1~* potential is the same of the
*S~2~* potential: the two surface that cross through the *L~1~* point
are called *Roche lobes*. The gravitational potential is a good
instrument to understand the dynamic and the evolution of a star's
binary system. We create a model around it that induce a lot of
considerations in the student's mind: the stimulation of them is
![](media/image7.png){width="3.25in" height="3.125in"}our target.

**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 the
student are incite:

-   to understand the theoretical structure

-   to use this mathematical instruments

-   to imagine the "physics consequences" of this model.

We would like to put in evidence that some scientific statements 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 presented with growing difficulty. The challenge for the
students is to resolve them using only the data and the formula
collected in Tables. 1 & 2. With this data in the next lesson we will
build a concrete model of a star binary system.

![](media/image8.png){width="4.9125in" height="2.4451388888888888in"}

![](media/image9.png){width="4.9in" height="2.588888888888889in"}

1.  The students calculate the gravitational force between two stars
    S~1~ and S~2~.

2.  Jupiter's mass is about 1/1000 of solar mass; if its mass were 1/10
    of solar mass, the temperature in its core would be enough to begin
    the thermonuclear reactions (for this reason we can say that Jupiter
    is a would-be star); in that case our planetary system would have
    two stars. Have the students calculate the gravitational force
    between the Earth and the Sun; between the Jupiter and the Sun;
    between the Jupiter and the Earth.

> The students repeat the same calculations imaging that Jupiter is a
> star with a mass equal to 1/10 of solar mass. What kind of
> observations can they make?

3.  The students plot, before on paper and after using some software as
    Excel or Derive, the function of the gravitational field made by the
    Sun, by the Earth and by the Jupiter. If they superpose all graphs,
    what kind of observations can they do?

4.  A NASA probe with a mass of 100 Kg take place in the point of
    gravitational equilibrium between the Sun and the Earth, how many
    kilometers is it from the Earth?

5.  If we imagine to drag away the Earth from the Sun gravitational
    influence, how much energy do we need?

6.  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. If they superpose all
    graphs, what kind of observations can they do? What do they observe
    if they compare the graph of the function of the gravitational field
    with the graph of the function of the gravitational potential?

7.  Where is the point of gravitational equilibrium between the stars
    *S~1~* and *S~2~*? The student plot on the plane (*x;y*) the
    equipotential lines made by *S~1~* star at the distance of
    $\text{100}´\text{10}^{6}\text{km}$,
    $\text{200}´\text{10}^{6}\text{km}$ and
    $\text{400}´\text{10}^{6}\text{km}$ from the center of the star
    *S~1~*. The student plot on the plane (*x;y*) the equipotential
    lines made by *S~2~* star at the distance of
    $\text{100}´\text{10}^{6}\text{km}$,
    $\text{200}´\text{10}^{6}\text{km}$ and
    $\text{400}´\text{10}^{6}\text{km}$ from the center of the star
    *S~2~*.

8.  Have 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 the gravitational potential made by the star *S~1~*
    and *S~2~* is the same?

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

> $$V_{\text{tot}}\text{=-}G\frac{m_{S_{1}}}{|r|} - G\frac{m_{S_{2}}}{|r - d|}$$
>
> where *d* is the distance between two stars. 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 create 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]: 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.