Hyperphysics: Hafel-Keating experiment

[Rasalhague]

http://hyperphysics.phy-astr.gsu.edu/HBASE/relativ/airtim.html#c5

I don't understand the approximation T0 = -TS that they make in the final step of the section "Kinematic Time Shift Calculation". From this, and the other equations in this section, I get

-T_0=T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )

-1=1+\frac{R^2\omega^2}{2c^2}

c^2=\frac{R^2\omega^2}{-4}

c=\pm \frac{R\omega}{2i}

but how can this be when c is a constant positive real number, not dependent on the product of the rotation of the earth with its radius? And

T_A=T_S-T_S\left ( \frac{2R\omega v+v^2}{2c^2} \right )

T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=-T_S\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )

T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=T_0\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )

4c^2= 2R\omega v+v^2 - R^2\omega^2

c=\frac{\sqrt{(R\omega+v)^2-2R^2\omega}}{2}

which can't be right, since c doesn't depend on these arbitrary variables: radius of the earth, etc.

[starthaus]

http://hyperphysics.phy-astr.gsu.edu/HBASE/relativ/airtim.html#c5

I don't understand the approximation T0 = -TS that they make in the final step of the section "Kinematic Time Shift Calculation". From this, and the other equations in this section, I get

-T_0=T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )

-1=1+\frac{R^2\omega^2}{2c^2}

c^2=\frac{R^2\omega^2}{-4}

c=\pm \frac{R\omega}{2i}

but how can this be when c is a constant positive real number, not dependent on the product of the rotation of the earth with its radius? And

T_A=T_S-T_S\left ( \frac{2R\omega v+v^2}{2c^2} \right )

T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=-T_S\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )

T_0\left ( 1+\frac{R^2\omega^2}{2c^2} \right )=T_0\left ( -1+ \frac{2R\omega v+v^2}{2c^2} \right )

4c^2= 2R\omega v+v^2 - R^2\omega^2

c=\frac{\sqrt{(R\omega+v)^2-2R^2\omega}}{2}

which can't be right, since c doesn't depend on these arbitrary variables: radius of the earth, etc.

HK is poorly explained using SR, a correct explanation requires GR. I am quite sure that I gave a GR-based explanation for HK somewhere in this forum. It is simply calculating the proper time \tau by integrating the expression in coordinate time t. The expression can be derived straight from the general Schwarzschild metric setting:

dr=d\theta=0,
\frac{d\phi}{dt}=\omega +\frac{v_1}{R}
\frac{d\phi}{dt}=\omega -\frac{v_2}{R}

depending on the direction of plane motion