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|strength2=1,300,000 infantry<br/>3,600 tanks<br/> 2,400 aircraft
|casualties1=500,000 dead, wounded, and captured<br/>500 tanks<br/>200 aircraft
|casualties1=200,000 dead, wounded, and captured<br/>500 tanks<br/>200 aircraft
|casualties2=607,737 dead, wounded, and captured<br/>1,500 tanks<br/>1,000 aircraft
|casualties2=607,737 dead, wounded, and captured<br/>1,500 tanks<br/>1,000 aircraft
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Revision as of 19:40, 6 April 2006

Battle of Kursk
Part of World War II
File:T34 kursk.jpg
Disabled Soviet T-34 being towed by a turretless armoured recovery tank, under enemy fire
DateJuly 4, 1943August 23, 1943
Location
Result Strategic Soviet victory
Belligerents
Germany Soviet Union
Commanders and leaders
Erich von Manstein
Walther Model
Georgy Zhukov
Konstantin Rokossovsky
Nikolai Vatutin
Strength
800,000 infantry
2,700 tanks
2,000 aircraft
1,300,000 infantry
3,600 tanks
2,400 aircraft
Casualties and losses
200,000 dead, wounded, and captured
500 tanks
200 aircraft
607,737 dead, wounded, and captured
1,500 tanks
1,000 aircraft

The Battle of Kursk, also known as Operation Zitadelle (Operation Citadel in English) by the German Army, was a significant battle on the Eastern Front of World War II. It remains the largest armored engagement of all time, and included the most costly single day of aerial warfare in history. Though the Germans planned and initiated an offensive strike, the Soviet defense managed to frustrate their ambitions and launch a successful counter-offensive and retook Oryol (August 5), Belgorod (August 5) and Kharkov (August 23), the first such success during summer since 1941.

Background

In the winter of 19421943 the Soviets conclusively won the Battle of Stalingrad. One complete German army had been lost, along with about 500,000 Germans and Axis allies, seriously depleting the Axis strength in the east. With an Allied invasion of Europe clearly looming, Hitler realized that an outright defeat of the Soviets before the western Allies arrived had become unlikely, and he decided to force the Soviets to a draw.

In 1917 the Germans had built the famous Hindenburg line on the Western Front, shortening their lines and thereby increasing their defensive strength. They planned on repeating this strategy in Russia and started construction of a massive series of defensive works known as the Panther-Wotan line. They intended to retreat to the line late in 1943 and proceed to bleed the Soviets against it while their own forces recuperated.

In February and March 1943 German Field Marshal Erich von Manstein had completed an offensive during the Third Battle of Kharkov, leaving the front line running roughly from Leningrad in the north to Rostov in the south. In the middle lay a large 200 km wide and 150 km deep Soviet-held salient (bulge) in the lines between German forward positions near Orel in the north, and Manstein's recently captured Kharkov in the south.

German plans

Von Manstein pressed for a new offensive based on the same successful lines he had just pursued at Kharkov, when he cut off an overextended Soviet offensive. He suggested tricking the Soviets into attacking in the south against the desperately re-forming 6.Armee, leading them into the Donets Basin in the eastern Ukraine. He would then turn south from Kharkov on the eastern side of the Donets River towards Rostov and trap the entire southern wing of the Red Army against the Sea of Azov.

The OKH did not approve von Manstein's plan, and instead turned their attention to the obvious bulge in the lines between Orel and Kharkov. Three whole Soviet armies occupied the ground in and around the salient, and pinching it off would trap almost a fifth of the Red Army's manpower. It would also result in a much straighter and shorter line, and capture the strategically useful railway town of Kursk located on the main north-south railway line running from Rostov to Moscow.

The eastern front at the time of the Battle of Kursk.

In March the plans crystallized. Walther Model's 9.Armee would attack southwards from Orel while Hoth's 4.Panzer-Armee and Armee-Abteilung "Kempf" under the overall command of Manstein would attack northwards from Kharkov. They planned to meet near Kursk, but if the offensive went well they would have permission to continue forward on their own initiative, with a general plan to create a new line at the Don River far to the east.

Contrary to his recent behavior, Hitler gave the General Staff considerable control over the planning of the battle. Over the next few weeks they continued to increase the scope of the forces attached to the front, stripping the entire German line of practically anything remotely useful for deployment in the upcoming battle. They first set the attack for May 4, but then delayed it until June 12, and finally until July 4 in order to allow more time for new weapons to arrive from Germany, especially the new Panther tanks.

The basic concept behind the German offensive was the traditional (and usually successful) double-envelopment, a Cannae-style attack the German Army had long favored. The tools of blitzkrieg made these types of tactics even more effective. Blitzkrieg depended on mass, shock and speed to surprise an enemy and defeat them through disruption of his command and supply rather than by destroying all his forces in a major pitched battle. Blitzkrieg-type breakthroughs were easier to achieve if they hit an unexpected location -- hence the Germans had attacked through the Ardennes in 1940, and towards Stalingrad and the Caucasus in 1942.

The OKH's conception of the attack on the Kursk salient, Operation Citadel violated the principle of surprise. Anyone with a map could confidently predict the obvious point of attack: the German plan reflected traditional German thought. A number of German commanders questioned the idea, notably Heinz Guderian who asked Hitler:

Was it really necessary to attack Kursk, and indeed in the east that year at all? Do you think anyone even knows where Kursk is? The entire world doesn't care if we capture Kursk or not. What is the reason that is forcing us to attack this year on Kursk, or even more, on the Eastern Front? Perhaps more surprisingly Hitler replied: I know. The thought of it turns my stomach. The interview ended categorically, Operation Citadel was postponed till mid-June.

Simply put, Operation Citadel embodied a traditional German plan. However, there was no reason to doubt whether it would be successful. Until Kursk, no major WW2 German offensive had ever been defeated. It is only in retrospect that the outcome seems preordained; at the time, most leaders on both sides expected that the German attack would probably succeed, at least in the breakthrough stage. Guderian's objections were at the strategic level, i.e., whether Germany should attack at all, not whether the plan itself was sound.

Soviet plans

The Red Army had also begun planning for their own upcoming summer offensives, and had settled on a plan that mirrored that of the Germans. Attacks in front of Orel and Kharkov would flatten out the line, and potentially lead to a breakout near the Pripyat Marshes. However, Soviet commanders had considerable concerns over the German plans.

All previous German attacks had left the Soviets guessing where it would come from, but in this case Kursk seemed obvious for the Germans to attack. Moscow received warning of the German plans through the Lucy spy ring in Switzerland. This was almost unnecessary since Zhukov had already correctly predicted the site of the German attack as early as April 8th, when he wrote his initial report to Stavka. In this report he also recommended the strategy eventually followed by the Red Army.

Stalin and a handful of the Red Army Stavka (General Staff) wanted to strike first. The pattern of the war up until this point had been one of German offensive success. The blitzkrieg tactic worked against all opposing Armies, including the Soviet Army. No Army had succeeded in stopping a German blitzkrieg breakthrough. On the other hand, Soviet offensive actions during both winters showed their own offensives now worked well. However the overwhelming majority of the Stavka, and notably Georgi Zhukov, advised waiting for the Germans to exhaust themselves in their attack first. Zhukov's opinion swayed the argument.

The German delay in launching their offensive gave the Soviets four months in which to prepare, and with every passing day they turned the salient into one of the most heavily defended points on earth. The Red Army laid about one million landmines and dug about 5,000 kilometers of trenches, with positions as far back as 175km from the front line. In addition they massed a huge army of their own, including some 1,300,000 men, 3,600 tanks, 20,000 artillery pieces and 2,400 aircraft. The Red Army could build up forces faster than the Germans; each month they pulled further ahead in men and weapons systems.

Many of the troops assigned to the defense of the salient were recent veterans of the Battle of Stalingrad, but the Red Army also added over one million new men to its ranks in the first half of 1943. Thus the Army was larger than in 1942, even after the losses at Stalingrad. The long delay between the identification of the likely site of the German attack and the beginning of the offensive gave the new units an unusually long time to train.

The density of artillery in the salient was unusual; there were more artillery regiments in the salient than Infantry regiments. The Soviets were determined to attrit attacking German units with a combination of mines and artillery fire. Indirect fire from howitzers would stop the German infantry while direct fire from 45mm, 57mm, and 85mm towed Anti-Tank guns and 76.2mm divisional field guns would destroy the tanks. In the 13th Army sector (facing the German Ninth Army on the northern face of the salient) the density of anti-tank guns was 23.7 guns per kilometer of defended front. In the 6th and 7th Guards Armies sectors in the south, the density was lower at about 10 guns per kilometer.

The engineer preparation of the battlefield was thorough. Soviet reports indicate that 503,993 Anti-tank mines and 439,348 anti-personnel mines were laid in the defended area. On average, 1,500 antitank mines and 1,700 antipersonnel mines were laid per kilometer of front. In the sectors that were eventually attacked, densities were never lower than 1,400 mines per kilometer and sometimes reached as high as 2,000 per kilometer. Soviet engineers also constructed miles of trenches, laid barbed wire, built antitank obstacles, and constructed thousands of gun and mortar positions. Dummy positions were built to attract German artillery fire. Camouflage of these positions and minefields was excellent; the first warning most German units would have of the presence of Soviet minefields or dug-in guns was their own vehicles exploding.

Set in the larger vista of the war on the eastern front, Kursk is significant because it demonstrated that the Soviet high command and staff worked more effectively than the OKH - largely due to Stalin finally being prepared to act on the advice of his professional staff officers and his intelligence staff; the defeat of the blitzkrieg in summer campaigning weather - albeit at a very high price; and the ability of the Soviet forces to move from defensive to offensive operations due to better staff work, larger reserves and better planning. In these senses Kursk, and not Stalingrad, can be viewed as the turning point in the war: certainly the initiative passed decisively from the OKH to the Stavka.

Operation Citadel

Preliminary Actions

File:Totenkopf-Kursk-01.jpg
Waffen-SS Panzergrenadiers and Tiger tanks of the SS Panzergrenadier Division Totenkopf during the start of Operation Zitadelle

It took four months before the Germans felt ready, by which time they had collected 200 of the new Panther tanks (only 40 available at the beginning of the battle due to technical problems with the new type), 90 Elefant Panzerjägers and every flyable Henschel Hs 129 ground attack aircraft, as well as 270 Tiger Is, late model Panzer IVs and even a number of captured T-34s. In total they assembled some 2,700 tanks and assault guns, 1,800 aircraft and 800,000 men. It formed the greatest concentration of German fighting power ever put together. Even so, Hitler expressed doubts about its adequacy.

By this time Allied action in Western Europe was beginning to have a significant impact on German military strength. Although actions in North Africa hardly constituted the Soviets' longed for second front, the battle there did begin to tell and in the last quarter of 1942 and the first half of 1943 the Luftwaffe lost over 40 per cent of its total strength in the battles over Malta and Tunisia. Luftwaffe air superiority was no longer guaranteed.

The start date for the offensive had been moved repeatedly as delays in preparation had forced the Germans to postpone the attack. Finally, On July 1 the orders to attack on July 5 were issued. The following day, Marshal Vasilevsky warned the Front commanders (Vatutin, Rokossovsky and Konev) that the long-awaited German offensive would begin sometime betwen July 3 and July 6. For months, the Soviets had been receiving detailed information on the planning of the offensive, i.e. from their espionage organisation Red Orchestra [not chapel] with sources including officers in the Nazi administration, among others in Goering's aviation ministry.

Preliminary fighting started on 4 July 1943 in the south. The Fourth Panzer Army had elected to try to take the Soviet outposts prior to the main assault on July 5th. Thus they deliberately sacrificed tactical surprise. However, the Soviet forward positions were on small hills overlooking German assembly areas, so it is likely surprise would have been lost in any case.

In the afternoon Junkers Ju 87 Stukas bombed a two-mile-wide gap in the front lines on the north in a short period of 10 minutes, and then turned for home while the German artillery opened up to continue the pounding. Hoth's armored spearhead, the III.Panzerkorps, then advanced on the Soviet positions around Zavidovka. At the same time the Panzergrenadier-Division Großdeutschland attacked Butovo in torrential rain, and the 11.Panzer-Division took the high ground around Butovo. To the west of Butovo the going proved tougher for Großdeutschland and 3.Panzer-Division, which met stiff Soviet resistance and did not secure their objectives until midnight. The II.SS-Panzerkorps launched preliminary attacks to secure observation posts, and again met with stiff resistance until assault troops equipped with flame-throwers cleared the bunkers and outposts.

At 22:30 the Soviets hit back with an artillery bombardment in the north and south. This barrage, by over 3,000 guns and mortars, expended up to one-half of the artillery supply for the entire operation. The goal was to delay and disorganize the German attack. In the northern face, the Central Front artillery fired mostly against German artillery positions and managed to suppress 50 of the 100 German batteries they targeted. The result was much weaker German artillery fire on the opening day of the attack. Also, German units attacked at staggered times on July 5th due to the disruption caused by this bombardment. In the south, the Soviets chose to fire largely against the German Infantry and tanks in their assembly areas. This was partially successful in delaying the German attack, but caused few casualties.

Main Battle

The Northern Face

The real battle opened on 5 July 1943. The Soviets, now aware even of the exact time of the planned German offensive, commenced a massive attack by the Soviet Air Force on the Luftwaffe airbases in the area, in an attempt to turn the tables on the old German "trick" of wiping out local air support within the first hour of battle. The next few hours turned into probably the largest air battle ever fought. Neither side was able to establish air superiority over the battlefield.

The 9.Armee in the north fell far short of its objectives on July 5th. The attack sector had been correctly anticipated by the Soviet Central Front. Attacking on a 45-kilometer-wide front, the Germans found themselves trapped in the huge defensive minefields, and needed engineering units to come up and clear them under artillery fire. Although a few Goliath and Borgward remote-control engineering vehicles were available to clear lanes in the minefields, they were not generally successful. Even when the vehicles cleared mines, they had no on-board marking system to show following tanks where the cleared lanes were. Soviet units covered the minefields with small arms and artillery fire, delaying and inflicting losses on German engineers clearing mines manually. Thus German losses in the Soviet minefields were high. For example, the German 653rd Heavy Panzerjager Battalion began the attack with 49 Ferdinand self-propelled guns and lost 37 of them in minefields before 5pm on July 5th. Although most of the lost vehicles were mobility kills rather than permanent losses, they were out of action until they could be repaired. While idle they added nothing to German combat power and were easier for Soviet artillery to knock out permanently. Since the Germans were advancing, any repairable vehicles could be fixed and put back into action.

There are a number of factors that explain the German Ninth Army's lack of progress. Soviet defensive planning and preparation was the major factor. The Soviet Central Front under Marshal Rokossovski had correctly anticipated the likely areas of German attack and had fortified those areas very heavily, holding other areas more thinly. The 13th Army, which bore by far the heaviest weight of the German attack, was far stronger in men and antitank guns than the other Central Front units, and indeed held the strongest defensive positions in the entire salient. Ironically, a major Soviet planning error was their expectation that the main weight of the German attack would come in the north on the Central Front. Thus they concentrated more strength there. Also, the Central front chose to defend the tactical zone (to a depth of 20km) very heavily, leaving far fewer units in the depths of the defense. Model's army had fewer tanks than Manstein had in the south. The Ninth Army also committed major units piecemeal due to the disruption caused by the Soviet preparatory fire. Finally, the Ninth Army led with reinforced Infantry Divisions that were already in the line facing the Soviets, rather than attacking with uncommitted units.

Review of attack frontages and depth of German penetration shows clearly that the Soviet defensive tactics were succeeding. Beginning with a 45-kilometer-wide attack frontage on July 5th, on the 6th the Ninth Army attacked on a 40-kilometer front. This dropped to 15 kilometers wide by July 7th, and only 2 Kilometers each on July 8-9. Each day, the depth of the German advance slowed: 5 kilometers on the first day, 4 on the second, never more than 2 Km each succeeding day. By the 10th the Ninth Army was stopped in its tracks.

After a week the Wehrmacht had moved only 12 km forward, and on the 12th the Soviets launched their offensive against the 2.Armee at Orel. The 9th Army had to withdraw, their part in the offensive over. Their casualty rate versus the Red Army stood at about 5:3 in their favour. German tank losses were about 300 Pzkw III & IV, half a dozen Tiger I's and 50 tank destroyers. However, this fell short of the usual figures, and failed to keep up with the steady influx of new soldiers and matériel for the Red Army. Few Soviet guns were captured, and those Soviet units that did retreat did so on orders; they were not overrun. The German attack failed to penetrate beyond the Soviet tactical zone.

Southern Face

In the south the Voronezh Front fared less well against the Germans. The German II SS Panzer Corps attacked on a narrower frontage against two Soviet rifle regiments. The armored spearhead of Hoth's 4.Panzer-Armee forced its way forward, and by the 6th had reached some 15km past the lines. Again, Soviet planning played a big role. In the south the Soviets had not been able to pinpoint the German attack sectors; this forced them to spread out their defenses more evenly. For example, three of the four Armies of the Voronezh Front had about 10 antitank guns per kilometer of front; this contrasts sharply with the Central Front's distribution of guns, which was twice as heavy in the active sectors. Also, the Voronezh Front made the decision to hold the tactical zone much more thinly, leaving a much higher proportion of units in deeper positions compared to the Central Front. Finally, the Voronezh Front was weaker than the Central Front, yet it faced much stronger German forces.

The German forces made steady progress against the Soviet defenses, but, as in the north, attack frontages (width) and penetration depth tended to drop as the attack proceeded. The trend was not as marked as in the north, however. Beginning with a 30-kilometer-wide attack frontage on July 5th, this dropped to 20-kilometers wide by July 7th and 15 km by July 9th. Likewise, the depth of the penetration dropped from 9 km on July 5th to 5km on July 8 and 2-3 km each day thereafter until the attack was cancelled.

Soviet minefields and artillery were again successful in delaying the German attack and inflicting losses. The ability of dug-in Red Army units to delay the Germans was vital to allow their own reserves to be brought up into threatened sectors. Over 90,000 additional mines were laid during the battle by small mobile groups of engineers, generally working at night immediately in front of the expected German attack areas. There were no large-scale prisoner caches nor any great loss of artillery, again indicating that Soviet units were giving ground in good order.

German losses can be seen in the example of the Grossdeutchland Division, which began the battle with 118 tanks. On July 10th, after five days of fighting, the Division reported it had 3 Tigers, 6 Panthers, and 11 Pzkw-III and Pzkw-IV tanks operational. The XLVIII Panzer Corps reported, overall, 38 Panthers operational with 131 awaiting repair, out of the 200 it started with on July 5.

Nevertheless it was obvious that the threat of a German breakthrough in the south had to be reckoned with. The Steppe Front had been formed in the months prior to the battle as a central reserve for such an eventuality. Units of the Steppe Front began movement to the south as early as July 9th. This included the 5th Guards Tank Army and other combined-arms armies.

The German flank, however, stood unprotected as the Soviet 7th Guards Army stalled Kempf's divisions, aided by heavy rain, after the Germans had crossed the Donets River. The Fifth Guards Tank Army, reinforced with two additional Tank Corps, moved into positions to the east of Prokhorovka and had started to prepare a counterattack of their own when II.SS-Panzerkorps arrived and an intense struggle ensued. The Soviets managed to halt the SS - but only just. Little now stood in the way of the 4th Panzer Army, and a German breakthrough looked like a very real possibility. The Soviets decided to deploy the rest of the 5th Guards Tank Army.

Prokhorovka

Main article: Battle of Prokhorovka

On the morning of July 12, SS-Panzerkorps advanced on Prokhorovka at the same time that 5th Guards Tank Army launched a series of attacks in an attempt to catch the Germans off balance. The SS and Guards units collided west of Prokhorovka in open country punctuated by farms, rolling hills and gullies. What happened next is open to debate with the release of new information from archives.

The battle can best be described as a very costly tactical draw but an operational success for the Soviets. Neither the Fifth Guards Tank Army nor the II SS Panzer Corps accomplished their missions that day. Tank losses have been a contentious subject ever since. Soviet losses have been claimed as low as 200 or as high as 822 tanks. Likewise, German loss claims have reached as low as 80 or into the hundreds, including "dozens" of Tigers. While the exact losses on each side cannot be established beyond reasonable doubt, the outcome is clearer. The 5th Guards Tank Army accomplished part of its mission of stopping the German attack, but did not take its terrain objectives or destroy the II SS Panzer Corps. Both units were weakened although both were committed to combat the following day.

The End in the South

Significantly, earlier in the battle the attacking German units had been squeezed into ever-narrowing frontages by the defenders. Elite Soviet Guards Airborne units were holding firm on the flanks of the very narrow German penetration. The Germans could not squeeze many units into this narrow front, nor did they have the combat power to widen the penetration. Thus as the attacking Corps moved forward, they continually lost strength due to the need to hold their own flanks.

While the German offensive had been stopped in the north by July 10th, in the south the overall battle (of Kursk) still hung in the balance, even after July 12th. German forces on the southern wing, exhausted and heavily attrited, nevertheless had breached the first two defensive belts and believed (wrongly) that they were about to break through the last belt. In fact at least five more defensive zones awaited them, although they were not as strong as the initial belts. Soviet defenders had been weakened, and major parts of their reserve forces had been committed. Still, the available uncommitted Red Army reserves were far larger than the few available German reserves.

Allied landing on Sicily

There is a historical point of view holding that the western Allied invasion of Sicily caused Hitler to call off Citadel. On 10 July, in the midst of Citadel, US, Canadian and British forces landed on Sicily during Operation Husky. Hitler called Günther von Kluge and von Manstein to his Wolfsschanze headquarters in East Prussia and declared his intention of calling off Operation Citadel. Von Manstein argued that one final effort could win Kursk, but Hitler disregarded it, particularly as the Soviets had launched their counteroffensive in the north.

Opposing this point of view, the fact is the offensive had already failed and further attempts to break through were likely to incur very high casualties. The northern pincer had been stopped cold, and Red Army units were already breaking through in their own counteroffensive. In the south, the maximum efforts of the most elite German units had failed to achieve a breakthrough into the Soviet rear despite some tactical success.

In the end, only one German division, Leibstandarte SS Adolf Hitler, departed for Italy, leaving all their equipment behind for their sister units. However, initially the Germans had decided that to send the entire II SS Panzer Corps, the formation that had spearheaded their push in the south.

Limited attacks continued in the south, to get rid of a Soviet force squeezed between two German armies.

Soviet counteroffensive

The Soviet counteroffensive at Orel decisively changed the situation. German Ninth Army units had to be redeployed to resist this attack instead of continuing their own offensive; units from the southern pincer were given warning orders on July 15th to withdraw back to the start lines held on July 4th. The purpose of the withdrawal was to shorten the front, enabling the Germans to re-form a reserve.

To the south the Red Army needed more time to re-group after July, and could not open their counterattack until 3 August. Aided by diversionary attacks further south they took von Manstein's hard-won Belgorod. Fireworks in Moscow marked the capture of Belgorod and Orel, a celebration that henceforward became an institution with the recapture of each Soviet city. On 11 August the Red Army reached Kharkov, a city Hitler had sworn to defend at all costs. The German units had reduced manpower and shortages of equipment.

Battle ends

By 22 August utter exhaustion had affected both sides and fighting (officially) drew to a close.

The campaign was a decisive Soviet success. For the first time, a major German offensive had been stopped prior to achieving a breakthrough. This was an outcome that few confidently predicted. The Red Army had suffered substantially higher casualties than the Germans. The Germans however had failed to achieve their goals. From this point on, a new pattern emerged. The initiative had firmly passed to the Soviets, while the Germans spent the rest of the war reacting to their moves. A new front had opened in Italy, diverting German resources and attention. Both sides had their losses, but only the Soviets had the manpower and the industrial production to recover fully, as well as the appreciated aid from the American lend-lease program. The Germans never regained the initiative after Kursk.

Moreover the loss further convinced Hitler of the incompetence of his General Staff. He continued his interference in military matters progressively, so that by war's end he was involved in tactical decisions. The opposite applied to Stalin, however. After seeing Stavka's planning justified on the battlefield, he trusted his advisors more, and stepped back from operational planning, only rarely overruling military decisions.

Predictable results ensued for both sides: the German army went from loss to loss as Hitler attempted to personally micromanage the day-to-day operations of what soon became a three-front war, while the Soviet army gained more freedom and became more and more fluid as the war continued.

References

  • The Battle of Kursk by David M. Glantz and Jonathan M. House (1999) ISBN 0700609784
  • The Battle for Kursk, 1943 : the Soviet General Staff Study translated and edited by David M. Glantz and Harold S. Orenstein (1999) ISBN 0714649333
  • Kursk 1943 : a statistical analysis by Niklas Zetterling and Anders Frankson (2000) ISBN 0714650528
  • Kursk: The German View by Steven H. Newton (2003) ISBN 0306811502
  • Rossija i SSSR v vojnakh XX veka: poteri vooruzhennykh sil by G. F. Krivosheev (2001)