Context : Hot star winds are laboratories for 3-dimensional radiative hydrodynamics and for X-ray sources with wind accretion .
In this context analytic models presented here are helpful .

Aims : The CAK-method ( Castor , Abbot & Klein , 1975 ) has been succesful for giants and supergiants of normal OB-stars but has failed to explain the weak winds of main sequence low luminosity OB-stars ( ‘weak wind problem’ ) .
Further , CAK has never been applied seriously to WR-stars and was considered as a mission impossible due to the ‘momentum problem’ .
The aim is to reevaluate the analytic CAK-method , to recalculate proper force multipliers , numerically solve the wind equations for a sample of O- and Wolf Rayet ( WR ) -stars and to obtain their mass loss rates and wind velocities .
The secondary aim is to solve the \it { weak wind and momentum problems } of hot star winds .

Methods : The wind in the supersonic part was modelled by photoionized plasma and radiative force ( force multiplier ) using the XSTAR-code ( Kallman & Bautista , 2001 , Stevens & Kallman , 1990 ) .
The force multiplier FM was computed as a function of the absorption parameter t , ionizing parameter \xi , particle number density N , chemical composition and the ionizing source spectrum .
The force was included in the momentum equation , and together with the mass conservation solved numerically in the supersonic part of the wind for a sample of O- and WR- stars ( WN-type ) .
The input parameters were the basic stellar parameters ( mass , radius , luminosity , chemical composition ) .
The results depend also on the boundary condition of subsonic part and the velocity law .
Fitting with the \beta -law with fixed \beta =0.6 and v _ { in } = 10 km/s approximate these and gave the mass loss rate and wind velocity as outputs .
Mass clumping was introduced by the volume filling factor F _ { vol } scaling \xi by F _ { vol } ^ { -1 } .
Velocity clumping was approximated by the velocity filling factor FVEL modifying the force multiplier ( following Sundqvist et al. , 2014 ) .

Results : Force multipliers based on blackbody radiators can be used for O-stars , while cut blackbodies ( flux below 230 Ã cut to zero ) approximate well those of WR-stars .
O-stars require moderate clumping F _ { vol } = 0.13 to match the canonical Vink-prediction ( Vink et al .
2001 ) .
The low mass loss rates of main sequence late O-stars ( weak wind problem ) can be explained by velocity clumping ( FVEL = 0.1 ) .
The momentum problem of WR-stars is shown to be due to wrong treatment of the input ionizing spectrum resulting in too small force multiplier .
Due to heavy absorption in WR-winds the flux below 230 Ã ( He II ionization ) is zero enhancing greatly the number of absorbing heavy element lines , and consequently the force multiplier , by eliminating the suppression by soft X-rays .
The computed mass loss rates and terminal wind velocities for 40 OB-stars and 55 WR-stars ( WN type ) are given in Tables A1 and A2 and Figs .
10-14 .

Conclusions : A possible solution for the weak wind problem of low luminosity late O-stars was quantitatively studied and explained by a small velocity filling factor FVEL .
The momentum problem of WR-winds was solved by proper computation of the line force with correct radiator ( cut to zero below 230 Ã ) .
The problem is an opacity problem of simply identifying enough lines ( Gayley et al .
( 1995 ) ) .
The present paper is the first comprehensive and self consistent treatment and numerical solutions of hot star wind equations .
Starting from the basic stellar parameters ( mass , radius , luminosity , chemical composition ) the wind equations were solved fitting with the \beta -law ( with fixed \beta =0.6 ) giving mass loss rate and wind velocity as results .
The computed mass loss rates match well with the observed/predicted ones .
The effects of free parameters \beta , F _ { vol } and F _ { vel } = FVEL were quantitatively estimated .
The eventual X-ray suppression on the face-on side of Cyg X-3 may act like lowering of FVEL , leading to decreasing of both mass loss rate and wind velocity .