This argument has been widely treated on Section 2 of our previous paper [1] . Here we remember:

escape speed of a particle m at a distance s from the mass M, with U the gravitational potential acting on m; if M is a real mass, s becomes the distance m-C_p with C_p the Centre of Potential of M, that is the point where |U| has the max value), while the escape velocity of m which has to be referred to C_p,

may be called as absolute escape velocity of m, absolute as referred to C_p; then

is the potential due to two masses; then the escape speed from two masses becomes

yielding

thus

is the total escape speed, with U the total potential; u(absolute escape velocity)has to be referred to the centre of potential of all the masses, C_p. Now, if S is a Source emitting a particle m, we may call

as relative escape velocity of m from S, (relative as u is referred to S). We assume now

showing that c depends on U; on Earth, U is practically constant, see the final “Conclusions”.

Equation (8) is supported by a cosmological reason, as better explained between the Equation (5) and Equation (6) on [1] : in short, c > u implies the masses dispersion, c < u implies a gravitational collapse, where as c = u, appears to be the right speed of the light to avoid the two said events (collapse or dispersion).

-This chapter is a deep revision/improvement of chapter4 of our previous paper [1] -

We assume now the light as composed of particular particles (photons), giving a Newtonian reason to the (apparent) constancy of c, defined as follows:

“Longitudinally-extended, elastic and massive particles having speed equal to the total escape speed u, and moving along rays, (continuous succession of photons)”.

(More photons emitted by a source, during an emission time T, need an equal number of rays).

Along each ray, every tail of a photon corresponds to the front of the next photon.

Now, on Figure 1, let S be a Source, (moving from an Observer O with velocity v_OS), starting to emit, at t = 0, (when S is coincident with O), a photon (with front A), along the direction S-O; therefore

represents, see Equation (7), at t = 0, the relative escape velocity of the front A from O, while the velocity of the front A, with respect to S, that is v_SA, turns out to be

Now, being S_T be the position of S at t = T, we get

where λ_O is the photon AB emitted with S in motion from O, whereas λ (ºuT) would be the photon AO if, during T, v_OS= 0. Thus, at t =T, if the source is receding from the front A, as in Figure 1, the photon length λ_O, (for the Observer O), from Equation (11), becomes

, (with) (12)

where, with Δλ (=v_OST) the path O-S_T covered by S during T. For instance, if v_OS = 0, the (12) gives with λ the photon length as seen by S.

The Equation (12) shows that the length of a photon during its emission, given u and T, depends on v_OS, meaning that Observers in reciprocal motion state different length for the same photon.

Now, referring to an Observer O, the speed of a photon, (since its length could vary), has to be defined considering its length λ_O referred to its transit time T_O, leading to

Returning to Figure 1, the transit time T_O of the photon AB, for the Observer O, is given by the time the front A needs to cover the path λ, that is T (=λ/u), plus the time the tail B needs to cover the path S_T − O = Δλ (=v_OST) which is ΔT = v_OST/u, giving

(with) (14)

Thus, see Equation (13), referring to the Observer O, the speed of the photon AB becomes

showing that c is invariant, under the Newtonian laws, in spite of any speed Source- Observer.

Anyhow, a photon, once emitted, has a constant length for every Observer, hence its speed has to vary for Observers in reciprocal motion, but we point out that the measurements of c via the method d/t implies that the light has to cross (or to be reflected by) an Observer O (taking the initial time); this means that O becomes the source S of photons emitted by a source at rest with O(v_OS = 0) who will state a length λ_O = λ, a transit timeT_O = T giving to c a constant measured value.

Along one ray, the frequency of photons, for an Observer O, is their number crossing O during a time t, that isn = n/t; thus for t = T_O, (transit time of one photon, for an Observer O), from Equation (14),

with the sign + when S and O, during the emission, are receding from each other. We point out that for β = 0, (v_OS/u = 0), the photons frequency as stated by a source S (n_s), has to be equal to the one stated by O (n_O), which is also valid if O and S belong to different potential, (e.g., the equality of the number of balls falling from the top of a tower with respect to an Observer at the tower base), and this, for v_OS = 0, always implies n_s = n_O. Figure 1, as well as Equations ((12) and (16)) represent the longitudinal Doppler effect for the light while, in general, this effect, with α the angle between the direction of S and OS, (and with S receding from O), see Figure 2, can be written as

(with) (17)

As for the Transverse Doppler effect, see Figure 3, in general, we have

As for a source circling around an Observer O, see Figure 4, it is always λ_O > λ as follows:

while their photons transit time is

where r is the orbit radius, ω the angular speed, yielding, to every photon, the speed c_O = c.

Also this argument has been widely treated on our previous paper [1] : here we point out that as the energy of light is valid for any mass, it has to be also valid for the light (massive on our bases) and therefore, writing we have to argue that each photon is provided with an internal energy equal to its kinetic energy K_c. Toward the infinity, (where c_∞ → 0) both K_c and K_i → 0 and therefore, since E = hn,

represents the energy of one ray of light, (where photons are flowing) and where

is the mass of light, having frequency ν, passing along one ray in 1s, while the constant

where

thus the Planck’s constant turns out to be the energy of one photon.

Now, since m is the mass of light passing along one ray in 1s, the term, with n_r the number of rays emitted by a source S, becomes the energy emitted by S in 1s; this unitary (unit of time) energy shall be equal to the supplied power P during 1s,yielding

therefore

is the total mass lost per second by a source of light; e.g. for a 1W lamp, we get, while the number n_r of rays is

in our case, n_r ≅ 3 ´ 10¹⁸ rays. Then, for a given power P, the higher is the frequency, the lower is the number of rays, as shown by (27) written as n_rν = P/h. The number of photons emitted in 1s is

which, for P = 1 W, gives n_γ = h⁻¹ (=1.5 ´ 10³³ photons/s), thus the inverse of Planck’s constant turns out to be the number of photons emitted in 1s by a source of unitary power, and this great number of photons allows the light to be treated as a wave.

Now, during inelastic impacts, (like on absorption or photoelectric effetcs), both kinetic and internal energy of the light are involved, so the momentum transferred to the electron is

while, during elastic impacts, the momentum transferred to the electron is

either for incident or for reflected photons, with Tʹ the total impact time for this interaction, as we show on Section 3.7, Compton effect; at this regard, via the (30), as well as via our Doppler effect equation, see (12), we get the Compton equation, which can not be obtained by the Relativity via their Doppler effect equations.

Also this argument has been widely treated on our previous paper [1] . Here the main results: referring to Figure 5(a), where h is the tower height, and M_E is the mass of Earth, writing the Equation (8) as c² = −2U, we can obtain

(valid for) (31)

that is the variation of c from the tower base to its top, where c₀ and c_h are the corresponding values of c, is the variation of potential; hence c_h < c₀, with.

Let now S be a Mossbauer source and A a related absorber, both, for instance, at the tower base, therefore in resonance. Then, see Figure 5(b), taking A to the top, and since A and S are now at rest, the frequency of the photons emitted by S, has to be equal to the one observed by A, that isn_h=n₀, therefore the Equation (31), as for the photons emitted by S, can be written as

and since c_h < c₀, it must be λ_h < λ₀, so, contrary to Relativity, A observes a blue-shift effect.

Thus, to restore the resonance via Doppler Effect (DE), S and A, see Figure 5(c), have to recede from each other, in order that the photon length should increase, see Equation (12), from λ_h to with β = v_AS/c º v/c, so, equating the photon length variation (Δλ/λ = −v/c due, see (17), to DE), to Δλ/λ, due to the altitude, as given by (32), we get, yielding

(for). (33)

This value is also predicted by GR which, implying a decrease of n for the light moving from the base to the top, predicts an opposite direction of v with respect to the one shown on Figure 5(c); at this regard, Pound-Rebka and Pound-Snider, [2] [3] [4] , gave no clear indication about the direction of the compensating velocity.

Moreover, see Figure 5(b), with S on the base, emitting upward, A goes out of resonance and since on our basesn_h = n₀, the non-resonance is physically related to a variation of λ.

Now, see Figure 6(a), with A on the base, taking S to the top, the experience shows that the absorber goes out of resonance. Indeed, with S on the top, the (31) shows c_h < c₀, but what about the initial parameters of the photons emitted in altitude, n_h and λ_h?

Well, see Figure 6(b), since S and A are at reciprocal rest, the frequency of the photons arriving to the base, is n_h-0 = n_h, hence along the path top-base, the photon length λ_h-0 has to increase (following the increase of c from c_h to c₀); thus we have to argue that taking the source on top, see Figure 6(a), the photons initial length must be λ_h = λ₀, in order that after the path top-base, it becomes λ_h-0 > λ₀ (inducing the absorber to go out of resonance).This implies

showing that, taking S to the top, n_h < n₀. Now, along the path top-base, Δλ/λ has opposite sign with respect to (32), yielding, and since λ_h = λ₀, we get

showing an increase of λ from the top to the base. Hence the absorber, on the base, will state a red-shift so, to compensate it via Doppler shift, see Figure 6(c), S and A have now to move relative to each other; on the contrary, according to Relativity, A and S should recede from each other.

Time dilation: Atomic clocks in altitude (h-clocks) are ticking faster than identical clocks on the ground (g-clocks): indeed, on our bases, at height h, see (30), we have, while taking a clock to a GPS satellite, see also Figure 6(a), from (34), one finds

with T_h = 1/n_h the counted time of one photon emitted by a h-clock, while T₀ to a g-clock. Thus the variation of the counted time between the two clocks, for every emitted photon, becomes

Now, the frequency of photons is their number emitted in 1 s along one ray, there- fore the term

(with) (38)

is the number of photons (atomic transition of Cs 137) that constitute a one-second, so that

is the variation ofn_1s emitted in1 s by a h-clock; now, see (31), ΔU = M_EGh/r_hr₀ and sincer_h ≅ 26,600 km , r₀ ≅ 6400 km, with h ≅ 20,200 km , the increase of counted time in one day () of a h-clock, with respect to a g-clock, becomes

Now, being v = 2(2πr_h/86,400 = 3870 m/s the orbital speed of GPS satellites, (corres- ponding to two orbits/day), it turns out that the variation of the counted time, between a h-clock and an Observer E at the centre of Earth, due to their relative motion, is given by Equation (20) which, in our case, with β = v/c, becomes

(valid for) (41)

with T_E the photon counted time for the Observer E; then, with v₀ the Earth’s rotational speed, and since, we can write T₀ ≅ T_E with T₀ the counted time of one photon for a g-clock; so, the difference between the two transit times T_h and T₀ ≅ T_E given by (41), is

Then, as above, is the variation of the number of photons emitted by a h-clock in 1 s; so the variation of the counted time, in one day, becomes

showing a decrease of the counted time for a g-clock due to the clocks relative motion; hence

which is also predicted, (with different reason), by GR. To prevent the two said effects, before launching, the daily counted time (T_1d) of clocks, has to be decreased by ≅38 μs; this adjustment is sufficient to obtain synchronization between h-clocks and g-clocks: indeed, on our bases, the frequency of photons, emitted by a h-clock, does not change along the straight path satellite-ground, whereas as for the Relativity, because of its predicted increase of the frequency along the straight path satellite-ground, a g-clock should go, (for this 3^rd effect), out of synchronization.

Gravitational redshif

Referring to our previous paper [1] , we summarize, hereafter, the differences between the Relativity and our results: as for the Relativity, the only way to explain high cosmological redshifts is the Doppler effect, (which implies an incredible universe expansion at a speed v_u ≅ c), whereas, on our results, disregarding the reciprocal motion between a (far) source and an Observer on Earth, which impliesn = n₀, we get c/λ = c₀/λ₀, where n₀, c₀ and λ₀ are the values on Earth, showing that for c₀ > c it has to be λ₀ > λ, that is a red shift. In general, the shifts observed on Earth can be therefore expressed as

with U₀ the potential on Earth, U the one on the source. Thus, apart from Doppler effects, z turns out to be the variation of c (as well as λ) during the path of light toward a different potential. For s < ≅ 45 Mpc, [5] , (corresponding to −0.01 < z < ≅0.01) if U (potential on the source) is, in absolute value, higher than the potential on Earth U₀, the (45) gives, on Earth, z < 0 (blue shift), and vice versa for |U| < |U₀|; thus, for s < ≅45 Mpc, these red/blue shifts indicate that the potential, may increase or decrease. In the range ≅0.01 < z < ≅0.20, (where z follows the Hubble’s law), the (45), written as

(valid for) (46)

shows that, for, U depends linearly on z, as Hubble’s law;then, for s→∞, U→0, hence z→∞, see Table 1.

For s > ≅45 Mpc, z is always positive, hence we may argue that our galaxy is close to the centre of potential C_p of all the masses, (where |U_Cp| has the max value), practically close to the middle of the masses of universe.

On our basis, (light composed of our photons), the interaction light-matter requires that to move a circling electron toward outer orbits, the impact photon-electron has to occur, see Figure 7(a), in a radial way, (giving origin to the radial velocity w), otherwise, some impacts could cause the electron fall into the nucleus, due, for instance, to an impact where photons-electron have contrary direction.

To be radial, the impact must occur in a specific point (Impact Point, fixed to the electron) which, during the electron revolution, has to face its nucleus, (up to its removal), giving to the electron one rotation every revolution. Thus we can infer that the electric charge of the electron, has to correspond to the Impact Point (I_p), and we have also to argue that each photon front is provided with a positive charge, while its tail with an equal negative one.

Moreover, in case of more impacts, as it happens, for instance, on Absorption/ Emission effect, where the impacts move a circling electron toward higher orbits, the impacts photons-electron have to occur all around the electron orbit, thus the impacting photons have to approach the nucleus, as shown on Figure 7(b), perpendicularly to the electron orbit plane, providing, to the electron, a radial velocity w.

The Figure 8(a) represents an H atom with both the electron mass m_e and the proton mass m_p circling around B (centre of mass of the system electron-proton), and where, on our bases:

r_B is the ground-state orbit of the electron charge,

r_e the electron radius,

r_p the proton ground-state orbit,

v_e = |v_e|, the electron ground-state (orbital) speed,

v_p the proton ground-state speed.

Hence, the ground-state orbit of the electron barycentre turns out to be

, (with). (47)

Now, to apply properly the equality between the electron centrifugal force and the Coulomb force, we have to consider the configuration (c) where the electron reduced mass

(with), (48)

is circling around the proton fixed as origin (O). Now, the total energy of the configurations (a) and (b) are:

Configuration (a), where;

Configuration (b), where;

hence for T_a = T_b we get = v_e and = v_p. (As for T_c = (T_a = T_b), next chapter).

Now, see Figure 8(b), can be found from the relation yielding

thus

Now, see Figure 8(c), equating along a circular orbit, the electron centrifugal force to the Coulomb one

and calling the necessary and sufficient energy to move the electron charge from r toward ¥, and assuming U_¥ = 0, we get

where is the potential due to electrostatic attraction. So the (51) becomes

Now, the orbital kinetic energy of m_r is, thus, with W the related ionization energy, (that is the electron extraction work), we may write

The term E = hn, see (21), is the energy of light passing along one ray, thus, if the ionization energy W () is supplied by one ray of light (with energy E = hn) it must be

Therefore, substituting 2hn () into (51) and solving by r we get

Now, plugging into (56) the value of the highest H-atom spectrum frequencyn₀ = c/λ₀ = cR_H where R_H (=1/λ₀) is the Rydberg constant, whose experimental value is R_H = 10,967,758 m^-1, we get, as for H atom, see Figure 8(c), the ground-state orbit (referred to the proton) of the reduced electron

with α = e²/2ε_ohc the fine structure constant. Then writing the (50) as, we get

corresponding to the Bohr radius, with, whereas, on our bases, r_B corresponds, see Figure 8(b), to the ground-state orbit of the electron around the electron-proton centre of mass B.

Now, the speed of m_r along the orbit r₀, from (55),becomes

and given the frequency of the electron m_r along r₀ that is n_e0 = v_r0/2πr₀, the ratio 2n₀/n_e0, with r₀ (=α/4πR_H) as given by (57), becomes

hence it is consistent to assume 2n₀/n_e0 = 1 (exactly). This ratio, written 2T_e0 = T₀, implies that, on H atom, the light-electron impact time T₀ lasts for two electron orbits.

Now, it is known that the admitted wavelengths, along circular orbits, have to satisfy the relation λ_n = n²λ₀, with an integer, so we can also write

withn_n the photons admitted frequency along circular orbits. Then from (56) we can write

representing the radius of each circular orbit. Then, see (59), the orbital speed of the electron m_r along any circular orbit is

while its frequency is

Then, dividing (61) by (64) and because of the ratio 2n_o/n_eo = 1, we get

Now, asn (= n/t) is the number n of photons passing along one ray during t, for t = 2T_e one gets n = n/2T_e = nn_e/2, which equals the (65), so the integer n of (65) also represents the number of photons (of the same ray) absorbed (or emitted) by the electron during 2T_e and this number, for all the n circular orbits of H atom, is an integer starting with 1 along the two orbits related to the photon n₀. [Between two circular orbits, the photons frequency are shown on Section 3.5].

The (65) written as nT_n = 2T_en shows that the impact time nT_n of n photons (with frequency n_n) equals the time needed by the electron, along the orbit r_n, for two orbits.

For n = 1, the Equation (65) corresponds to 2n₀2πr₀/v_r0 = 1 and substituting here r₀ (=e²/8πε₀hn₀), as given by (57), we get 4πn₀e²/8πε₀hn₀v_r0 = 1, that is

same value given by (59). Now, comparing (59) to (66) we find

matching the R_H experimental value.

Still referring to H atom, to apply properly the Conservation of momentum (CoM) to the impact photon-electron, we need, see Figure 9, a proper configuration where its nucleus should be fixed in the common centre of gravity B orbited by an equivalent electron mass m, as shown on Figure 9(d).

The Figure 9 shows the necessary passages, from Figures 9(a)-(d), to obtain such a configuration.

Referring to Figure 9(c), with m_r circling around, with orbital speed v₀, the (51) gives

and substituting, see (50), , we can write

then calling

electron impact mass, we get

and therefore

we can now write

where (73)

showing, see Figure 9(d), m circling along the orbit, implying an electron/proton reduced charge e_r. Now, referring to Figure 9(c), we have, and since, see (66), v_r0 = αc, the orbital speed of m_r along the orbit becomes

which, given ε_m and ε_r, leads to v₀ = c/137, as shown on next chapter.

We compare now the total energy T of the system electron-proton along the configurations of Figure 9: as for Figure 9(a) and Figure 9(b), the total energy of the system is; as for Figure 9(c), along r₀, we get; on 6(d),.

The conservation of momentum, applied before and after an inelastic impact photon-electron, since photon and w, see Figure 7, have same direction, the (29), for a generic atom, gives

representing the electron radial speed originated by an impact of one photon during the impact time T, while for n photons (with frequency ν) we have

Regarding now the H atom and referring to Figure 9(c), meaning to consider the electron reduced mass m_r circling along r₀, the (75) forn = n₀, becomes

and since, see (59), , as shown by (66), we get

while considering the configuration 9(d), where the electron m_i is circling along, we get

(valid for) (79)

close to c/137² = 15,972.743 m s^-1; on next chapter, via r_e, we get w₀ = c/137² and v₀ = c/137.

Now, referring to Figure 10, let us consider an electron (m_e) circling, with velocity v around its nucleus with mass, which can therefore be considered as fixed in the atom centre of gravity B; its removal may happen when its radial speed w equals v (=|v|) that is

In particular, as for H atom, and referring to Figure 9(c), let us consider the electron m_r circling along r₀; comparing (78), that is w₀ = α²c, with (66) that is v_r0 = αc, we get.

Now, along r₀, (ground-state orbit), we have n = 1 (meaning one photon along the double orbit r₀), hence the ionization, requiring w₀/v_r0 = 1, cannot happen along r₀. Referring now to Figure 9(d), along the ionization double orbit (in short d-orbit) # n_i, where the photons incident frequency, see (61), is, the impact due to n_iphotons would produce, considering the electron impact mass m_i, an electron radial speed, see (76), equal to

and since along this orbit (n_i), see (63), is v_i = v₀/n_i, where v₀ is given by Equation (74), we find

hence the ionization, which requires w_ni = v_i, with only n_i photons along the n_i^thd-orbit, would never happen; thus we must infer that there are 137 progressive d-orbits, where the electron is circling n times along every d-orbit; thus the number of photons admitted along n d-orbits turns out to be n², yielding to the radial speed, along n ionization d-orbits n_i, the value which becomes

giving the same radial speed w_o for any circular d-orbit, see Table 2.

Now we can obtain the radius (r_e) of the electron: indeed, along the ionization d-orbit n_i, the ionization condition becomes w₀ = v₀/n_i, and plugging v₀ as giving by (74) we get

where w₀ is given by (77) and since, see (67), we have

but n_i has to be an integer so we can infer n_i = 137, giving

yielding

Now, the correct values of m_i, w_o and v_o from (70), (79), (74), with ε_r given by (86), become

Now, the (84), that is n_i = v₀/w₀, through (89) and (90) yields

where n_i, on Absorption effect, is a specific constant representing the number of circular orbits as well as the numbers of admitted photons along theionization orbit.

Referring to Figure 11, where we represent the Absorption of photons from acircling electron, let us assume the nucleus mass, so to consider the nucleus fixed in the atom centre of gravity B. Now, the expression of the total energy of the system photon-electron is given by

where E(=mc²) is the energy of the incident light, U_r(=−e²/4πε₀r) is the potential due to electrostatic attraction acting on the electron, K_e(=m_ev²) is the electron orbital kinetic energy, and K_r(=m_ew²) its radial kinetic energy (related to its radial speed w).

Regarding the Absorption/Emission effect (elements on gaseous form), see Figure 11, along circular orbits it is K_r = 0 and since, at the end of absorption, along the orbit r₂, (where the photons have been absorbed), it is E₂ = 0, between two circular orbits r₁ and r₂, the (92) gives

Now, from (53), U_r = −m_ev², and since K_e = ½m_ev², we get so from (93) we have; thus, as E₁ = hν, and since m_ev² = e²/4πε₀r we find

Then, according to (62) we have r₁ = r₀n² and r₂ = r₀k² (with k > n as r₂ > r₁), thus

and plugging the (57) written as n₀ = e²/8πε₀hr₀, we find

which is the photons frequency between two circular orbits, where n, as showed on Section 3.2, is an integer representing the (progressive) number of each circular orbit, and where k turns out to have the values which is, one by one, the number of the remaining external circular orbits.

Claimed fall of a circling electron into its nucleus: an electrical current emits an electro-magnetic radiation and therefore it is claimed that the circulating electrical charge of an electron should also emit an e.m. radiation yielding the electron, in a short time, to fall into the nucleus; but on our results, a free electron, moving, for instance, along a copper wire under an electrical potential difference, when entering into an atom influence, (at that moment the electron charge will return to face the atom nucleus), will release the necessary photons to reach the atom energy level corresponding to the energy previously received (during the absorption effect). Indeed, along circular orbits, it is w = 0, therefore the absorption/emission of photons may only start/finish along these orbits, thus the circling electrons are absorbing/emitting photons only between circular orbits, so the e.m. radiation related to an electrical current is due to the emitted photons during their re-entry to an atom; by the way, the photons emission is necessary for the electron not to fall into the nucleus.

Between the electron ground-state orbit r₀ and its extraction orbit r ® ¥ (intending on microscopic scale), the (92), valid for every interaction light-matter, gives

with E' the energy of re-emitted light, w_ae the electron radial speed after its extraction, (its kinetic energy), while the other terms have been defined referring to (92).

On ground-state, as also shown between Equations ((93) and (94)), it is

with v₀ the electron speed along r₀ and with W_f the Work function (electron extraction

work); now, at the start of impact, w = 0 giving, while for r ® ¥, the

electron orbital speed v_¥ ® 0, so, and (97) gives

where is the total kinetic energy transferred from light to elec- tron. On Photoelectric Effect (PhE), the light scatters off an electron (K_ae ≥ 0), but it is not re-emitted, (E' = 0), so the (99), with n_f (=W_f/h) the specific threshold frequency, becomes

showing that forn = n_f there is ionization with w_ae = 0.

At frequency n_f the electron radial speed w_f, due to the impact of one photon, see Equation (75), is, and writing the (55) as v₀ = (2W_f/m_e)^1/2, we get

and since the values of W_f are in the range 2 - 6 eV, the (101) gives w_f/v₀ @ 0.0028 - 0.0048 meaning that the ionization, requiring w = v₀, at frequencyn_f needs n_f photons, as follows: the electron radial speed due to n_f photons with frequency n_f, see (76), is, so the ionization condition (w = v) becomes w_nf = v_o leading to giving

that is, on PhE, the number of photons at frequency n_f necessary for ionization(w_ae = 0). Now, if n_f photons, at frequencyn_f, are sufficient for ionization, then the frequency

is sufficient for ionization (w_ae = 0) with one photon only, meaning that ν₁ is the threshold between PhE and Compton effect which requires one photon only, as shown on next chapter.

Now let us find the number n₁ (giving w_{ae max}) of the impacting photons at frequency n₁: writing the (100) as hn = 1/2m_ew², (energy transferred from a ray of light to an electron), one gets w = (2hn/m_e)^1/2 which has to be equal to the electron radial speed due to n photons, w_n = 2nhn/cm_e, see (76); so, forn = n₁ and n = n₁, we get yielding

with n₁ the number of impacting photons at frequency n₁ and plugging n_f given by (102), we find

meaning that on PhE, the number of impacts photons-electron varies from n_f related to the frequency n_f to related to the max admitted frequencyn₁ (=n_fn_f). For instance as for caesium (Cs), having W_f @ 2 eV, since, we may infer n_f = 361 leading to n₁ =19, while as for Pt, having W_f @ 6 eV, we may infer n_f = 196 leading to n₁ = 14.

Here, see Figure 12, the incident photon (length λ, frequency n), while ejecting a circling electron is also reflected (λ', n') so the recoiling electron, emitting a photon λ' toward the Observer A, represents a source in motion from A along the direction w, implying an undoubted Doppler effect.

Now, on the basis that the scattered photon starts to be reflected at the same time when the incident photon starts to hit the electron, and since is the emission time of the reflected photon, it turns out that is also the whole interaction time, meaning that there is not a complete absorption of the incident photon followed by an emission. Now, with the whole impact time photon-electron, the momentum transferred from the incident light to the electron, as per (30), is and the same value is then transferred from the reflected photon to the electron, so the Conservation of Momentum (CoM) along the direction normal to w, becomes

giving. Then, the length of the reflected photon, for the Observer A, see Equation (12) is

where Δλ = w_AT' and where w_A = wcosθ is the component of the electron speed along the direction of the Observer A and is, for A, the photon transit time, so we get

Now the CoM along w is

giving

Then, plugging this value into (108) we get

Now, , , hence and therefore

and since, we get the Compton equation:

which cannot be obtained via the Doppler effect relativistic equations regarding the light.

Then, the (110), for cosθ = 1, equals the (75), implying the impact of one photon only.

Now, the (111), for cosθ = 1 gives, or which plugged into (110) gives

yielding, see Figure 13, w → c for n → ∞, whereas, for inelastic impacts, w is proportional ton.

Then, as for the electron radial speed after its extraction, here indicated v, from (100) we have W_f + K_ae = 1/2m_ew² where K_ae = 1/2m_en² and since as shown by (98), we get

which for w = v₀, (ionization condition) gives n = 0, as represented on the Figure.